\section{Introduction}

Periodic comet 12P/Pons-Brooks was discovered by J. L. Pons on 1812 July 21 from Marseille, France; the comet was followed during that apparition until September 28. It was re-discovered by W. R. Brooks on 1883 September 2 from Phelps, NY, USA, and followed until 1884 June 2; during that apparition, the comet experienced several outbursts.  At its next apparition, in 1954, comet 12P also exhibited several outbursts.  Condensed observational details of all three apparitions can be found in Kronk (2003, 2009).

Despite the fairly high absolute magnitude of around 4-5 and the comet's apparent tendency to have occasional outbursts, it seems that no searches for earlier appearances have been made using historical data.  In February 2020, the first author integrated the orbit of 12P backward until about the year 1000.  The calculations used data from the apparitions of 1883-1884 and 1953-1954 (taken from the Minor Planet Center's online database).  From the integration backwards, it was apparent that the orbit for this comet is very stable and does not experience strong planetary gravitational perturbations in the covered period.  From a check of different cometographies, it was apparent that the first comet of 1457 and the comet of 1385 were almost perfect matches concerning the perihelion time.  As a next step, these backward-integrated orbits were compared with catalogued orbits for the 1385 and 1457 comets, using a planetarium software program (``GUIDE", by B. Gray; cf. website URL {\tt https://www.projectpluto.com}) that showed that the integrated orbits for 12P were fully compatible with the observed paths and the observational circumstances of the 1457 and 1385 comets. Not only did 12P appear positioned within the area indicated by the ancient observations but also the sense of movement did fit perfectly. By adjusting the perihelion time by a few days for each of these apparitions, the match could be brought even closer.

\section{The Comet of 1457}

In 1864 a manuscript by Italian cartographer and astronomer P. Toscanelli was found in the National Library in Florence, Italy.  The manuscripts contained observations of six comets seen by Toscanelli in the 15th century, with their positions drawn into celestial maps drawn by Toscanelli.  After the discovery of the manuscripts, G. Celoria performed in-depth investigations of Toscanelli's manuscripts and derived orbits from the positions drawn by Toscanelli.  The first comet of 1457 was observed daily on five nights between 1457 January 23 and 27.  Celoria published his analyses, including the derived orbits, several times (Celoria 1884, 1894, 1921).  His orbit is based on three of the five observations and via comparison with the other two observations. He correctly states that the orbit is not of much accuracy due to the very short arc.

The Toscanelli drawings show a short tail extending to about $0.5{\circ}$. There are indications of a coordinate grid in Toscanelli's drawings that at times seems incomplete, but it helped Celoria to identify the area of the sky where the comet was seen. A more contemprorary analysis of Toscanelli's maps can be found in Jervis (1985). Figures 1 and 2 show the original drawing by Toscanelli and the representation by Celoria, respectively.  It can be seen from the images that the accuracy can only be good to maybe a few degrees, and any orbit derived from them is prone to some considerable uncertainty, something that Celoria (1894) acknowledged quite clearly:  ``The observations, despite their nature, are quite well represented from my orbital elements, but they, even if of a remarkable precision for that time, are too close to each other and too few to judge with certainty how much the orbital elements themselves are close to the real ones."\\

\begin{center}
	\includegraphics[width=0.8\textwidth]{"figure1.jpg"}\\
\end{center}

Figure 1: Drawing of the positions of the comet of January 1457 by P. Toscanelli. Image courtesy F. Stoppa, Milan, Italy.\\

But Toscanelli was apparently not the only observer of this comet.  There exists a Chinese observation of the 1457 comet that, however, bears a problem.  The Chinese reports are as follows (Pankenier, Xu, and Jiang 2008) and are given for 1457 January 14:  ``7th year of the Jingtai reign period of Emperor Yingzong of the Ming Dynasty, 12th month, day jiayin [51], at night, a broom star with bright rays 5 cun long reappeared in lunar mansion BI [LM 19], slowly traveling southeastward.  Its bright rays gradually lengthened from this day through day guihai [60].  [\textit{Ming Yingzong shilu}] ch. 273".  Another Chinese source given there (\textit{Ming shi: tianwen zhi}, ch. 27) supplies basically the same information.\\

\begin{center}
	\includegraphics[width=0.80\textwidth]{"figure2.jpg"}\\
\end{center}

Figure 2: Representation of Toscanelli's drawing by G. Celoria (1894). Image courtesy F. Stoppa, Milan, Italy.\\

Stryuck (1740, p.\ 247) wrote:  ``Besides the story about the Comet one finds in the last mentioned Chronicle [{\it Anton.\ de Ripalta Annal.\ Placent., col.\ 905}] that in the Year 1456, in the Month of December, and in the Year 1457, in January, four strange Stars appeared, moving from the East to the West, almost in the shape of a Cross.  This could easily be some Fixed Stars or one or more Planets.  {\it Ludov.\ Cavitel.\ Cremonen.\ Annal.}\ (col.\ 1456, tom.\ 3, par.\ 2) tells about a Comet that was seen in the Year 1456, the 5th of December, and another one in January ..." Interestingly, both Struyck and Lubienietz (1667) mention the comets of June 1456 and June 1457, but nothing more on the comet of Jan. 1457 (other than that above).  And neither of these two cometographers mentioned the 1385 comet at all; as they both had access to many European historical materials on comets, it says a lot that these comets were not apparently widely known in Europe, so they must not have been very bright and thus not easily seen from Europe.

What can be deduced from the Chinese texts?  First, the object was seen as early as 1457 January 14 -- but possibly earlier, since the word `re-appeared' is used. Second, the tail was about $0.5^{\circ}$ long; 1 cun is about $0.1^{\circ}$.  This nicely agrees with the length derived by Celoria from Toscanelli's drawings. Third, the object was seen for about 10 days, until January 23.

The main problem with the Chinese observation (which was also discussed at length by Celoria) is the position in the lunar mansion Bi (or Pi), which corresponds to lunar mansion 19 and refers to the area around the Hyades and $\varepsilon$ Tau, also called the ``Hunting Net".  However, using the orbital elements of 12P, the comet would have been near $\omega$ Psc and $\gamma$ Peg on that date.  It can be concluded that the Chinese position is not at all compatible with the Toscanelli observations, except there is an error in the Chinese sources (see also the similar remark by Celoria).  For the Chinese comet, 12P would have been in the 14th lunar mansion, called Dongbi or Tung-bi but sometimes also given as Pi (Ho 1962).  If the 14th and 19th lunar mansions were mixed up, then 12P would be correctly placed on 1457 January 14; but, of course, this cannot be proven anymore.

Finally, another argument for the identity with 12P:  Based on the derived magnitude parameters of the apparition of 1953, the magnitude of 12P in 1457 was perhaps at mag 3-4 (assuming no outburst), with the comet being close to perihelion and about 0.95 AU from the earth.  This would explain why it was not such a conspicuous object.  If Celoria's orbit were correct, the comet would have brightened further with increasing elongation in the following days which raises the question why it was not observed further by Toscanelli. The orbit of comet 12P shows that it in fact became fainter with a slowly increasing elongation, which explains the short observation period.

As a sidenote, it should be mentioned that the 1457 comet was long suspected to be identical with what later became known as comet 27P/Crommelin (Schulhof 1885; Galle 1894, p.\ 157; Procter and Crommelin 1937).  Modern calculations were not able to confirm this and, moreover, that identity can be ruled out (Marsden 1975; Festou, Morando, and Rocher 1985).
	\\

The orbit by Celoria is given here for reference
	\\

$ 	q = 0.703\ AU, peri = 195^\circ , node = 258^\circ , i = 13^\circ   $
	\\

\section{The Comet of 1385}

For the 1385 apparition, we only have the description of the comet's apparent movement from Asian sources:  the \textit{Ming Taizu shilu}, ch. 175, and \textit{Ming shi: tianwen zhi}, ch. 27 (Pankenier et. al. 2008).  On 1385 October 23, the comet appeared near Coma Berenices, Leo, and Virgo, and	after that, moved towards $\beta$ Vir and left the area of $\beta$ and $\eta$ Vir.  On 1385 October 30, the comet entered Crater; on November 4 it 'trespassed against' an asterism in Hydra.  The comet had a 10-degree tail according to Biot (1843a; see also Carl 1864, p.\ 42).  The widely cited orbit by Hasegawa (1979) of course resembles this general movement.

The orbit of 12P is perfectly consistent with the above description and moves similar to comet C/1385 U1; the apparent path in the sky fits the description from the Chinese records even better.  Using the magnitude parameters from the	1953 apparition the brightness was perhaps around magnitude 2 (assuming no outburst), since the	apparition was very favorable due to a close approach to the earth.  This agrees well with the Chinese observations, too.
	\\

Several orbits have been calculated from the Chinese descriptions in the past. The orbits (2000.0) derived by Peirce (1846), Hind (1846) and Hasegawa  are given here for reference.\\

	Peirce: 	$q = 0.755\ AU, \omega = 155^\circ, \Omega = 270^\circ, i = 105^\circ
	$

	Hind:		$q = 0.738\ AU, \omega = 130^\circ, \Omega = 296^\circ, i =  52^\circ
	$

	Hasegawa:	$q = 0.79\  AU, \omega = 289^\circ, \Omega = 103^\circ, i = 103^\circ
	$
	\\

As noted above for the 1457 comet, neither Struyck nor Lubienietz mentioned the 1385 comet at all; since they both had access to many European historical materials on comets, it says a lot that these comets were not apparently widely known in Europe, so they must not have been very bright and thus not easily seen from Europe.

Nevertheless, the comet was not completely missed in Europe. J. Meyerus	Baliolanus (1561) gives an account of a comet seen on the feast day of Saints Cosmas and Damian, which corresponds to September 27, 1385, when the comet would have been at magnitude 5, but it may have been in outburst then. However,	his text also says that the comet appeared in October. He goes on to say that the comet did shine in many colours. A similar account can be found in a later annal by E. Sueyro (1624) with the only difference that he put it in the year 1386. From the description it could also relate to an aurora. Another mention	can be found in the annals the German city of Trier (1838). The editors of this edition remark on a manuscript which states that in 1385 a terrible comet 	appeared. The comet can also be found in annals of Iceland (1847) which simply says for 1385 that a comet appeared.

\section{Linkage}

For the linkage of the apparitions of 1385 and 1457, the following positions were derived from the descriptions in the historical sources.\\

\begin{verbatim}
	1385 UT             R.A. (2000) Decl.       Mag.
       Oct. 22.9       12 00         +12 00         2
            29.9       11 55         - 8 00
       Nov.  3.9       11 55         -35 00

   1457 UT             R.A. (2000) Decl.       Mag.      Observer
      Jan. 23.7        0 40         - 3 40         3     Toscanelli
           25.7        0 47         - 5 20                 "
           27.7        0 56         - 7 15                 "
\end{verbatim}

The observations of the apparitions of 1812 and 1883-1884 were re-reduced recently by co-author T. Kobayashi, from which a linked orbit could be derived and which was published in Green (2020a) and Nakano (2020a).

Prompted by these announcements, the comet was recovered on June 10 and 17, 2020, with the Lowell Observatory 4.3-m Discovery Telescope and the Large Monolithic Imager by Ye et. al. (2020a) with the comet at a distance of 11.9 AU. On stacked images a broad tail of 3' length was visible implying that it was already active. The correction to the orbit by Kobayashi based on the data from 1385 - 1954 was only +0.16 day (see Green (2020b).

Including these recovery observations the following linked orbital elements were derived.  These observations are listed in Table 1. His elements are based on a total of 1052 astrometric observations and include perturbations by Mercury-Neptune and Ceres, Pallas, and Vesta. Non-gravitational effects were included in the orbit computation. The weighted mean residual is 1$''$\llap.41.  The comet passed 3.71 AU from Uranus on 1819 Apr. 26 and 1.62 AU from Saturn on 1957 July 29 UT.  The comet has made numerous	close approaches to the earth (0.41 AU on 1385 Oct. 29, 0.90 AU on 1457 Jan. 10, and 0.63 AU on 1884 Jan. 9 UT).

It should be noted that a correction in ET-UT for the 1385 and 1457 observations was ignored since no definitive values for ET-UT are available.  However, following Stephenson (1997) and using approximate values for (ET - UT) of +330	s for 1385 and +220 s for 1457, the residuals amount to about 48$''$ and 12$''$, and perihelion time corrections of only about -0.002 and +0.005 day, respectively.
\\
\begin{verbatim}

	Epoch = 1385 Nov.  8.0 TT
	T = 1385 Nov.  6.327 TT          Peri. = 200.036
	e = 0.95505                      Node  = 255.125   2000.0
	q = 0.78362 AU                   Incl. =  73.829
	a = 17.431967 AU     n = 0.013542     P =  72.78 years

	Epoch = 1457 Jan. 14.0 TT
	T = 1457 Jan. 30.1002 TT         Peri. = 199.9041
	e = 0.954800                     Node  = 255.2502  2000.0
	q = 0.778438 AU                  Incl. =  74.0399
	a = 17.22216 AU    n = 0.0137903    P =  71.47 years

	Epoch = 1812 Aug. 30.0 TT
	T = 1812 Sep. 15.82612 TT        Peri. = 199.29022
	e = 0.9553274                    Node  = 255.63879 2000.0
	q = 0.7771051 AU                 Incl. =  73.95643
	a = 17.3955643 AU   n = 0.01358458   P =  72.55 years

	Epoch = 1884 Jan. 25.0 TT
	T = 1884 Jan. 26.21681 TT        Peri. = 199.17679
	e = 0.9550368                    Node  = 255.77454 2000.0
	q = 0.7757320 AU                 Incl. =  74.04048
	a = 17.2526163 AU   n = 0.01375377   P =  71.66 years

	Epoch = 1954 May  18.0 TT
	T = 1954 May  22.88058 TT        Peri. = 199.02746
	e = 0.9548317                    Node  = 255.89097 2000.0
	q = 0.7736564 AU                 Incl. =  74.17689
	a = 17.1283021 AU   n = 0.01390377   P =  70.89 years

	Epoch = 2024 May  10.0 TT
	T = 2024 Apr. 20.99698 TT        Peri. = 198.98718
	e = 0.9545914                    Node  = 255.85595 2000.0
	q = 0.7807641 AU                 Incl. =  74.19138
	a = 17.1941867 AU   n = 0.01382393   P =  71.30 years
\end{verbatim}

Figures 3 and 4 show the apparent paths for both apparitions based on Celoria's	and Hasegawa's orbits, respectively.  They also show the paths based on the linked orbit by Kobayashi.  It can be seen that the apparent paths from the linked orbit by Kobayashi are quite similar to the apparent paths from orbits	by both Celoria and Hasegwawa.\\

\begin{center}
	\includegraphics[width=0.9\textwidth]{"1457_chart_new.jpg"}\\
\end{center}

Figure 3: Comparison of the paths of comet C/1457 A1 based on Celoria's orbit and comet 12P/Pons-Brooks based on the linked orbit.\\

\begin{center}
	\includegraphics[width=0.9\textwidth]{"1385_chart_new.jpg"}\\
\end{center}

Figure 4: Comparison of the paths of comet C/1385 U1 based on Hasegawa's orbit and comet 12P/Pons-Brooks based on the linked orbit.\\

\section{Discussion of sightings at other apparitions}

On the basis of the orbits given in Table 2 below, a search was conducted in historical comet reports for other sightings of comet 12P/Pons-Brooks. It is reasonable to assume that non-gravitational forces, which are also present for this comet, should not change the predicted orbits before	1385 by a large amount, since this would require a substantial change in these forces that were quite constant between 1385 and now.

It should be stressed that for a comet to be noticed by chance without a telescope, it needs to be placed in dark skies at solar elongations > $40^{\circ}-50^{\circ}$, and at a certain brightness (say, brighter than visual magnitude 3-4 in a moonless sky).  For most of the apparitions discussed in the text below,	this is not the case.  The apparitions of 1457 and 1385 were very	favorable, where the comet was close to the earth and bright enough to be easily seen.  From the orbital circumstances, a perihelion occurring between August and January provides the most favorable viewing conditions. However, one has to take into account that 12P is prone to outbursts. This is why it seems nevertheless useful to look at each apparition and see whether other historic candidates are available.\\

	Table 2:  Orbital elements by T. Kobayashi for 12P/Pons-Brooks, based on observations from 1385-2020.

\begin{small}
\begin{verbatim}
	0012     33 03 24.392  0.783058  0.955528 201.801  253.835   73.443    330331
	0012    105 08 27.621  0.777178  0.955319 201.719  253.956   73.656   1050820
	0012    176 05 17.343  0.774058  0.954814 201.597  254.078   73.899   1760519
	0012    245 09 15.466  0.772882  0.954562 201.409  254.157   74.028   2450914
	0012    315 03 12.893  0.785689  0.954230 201.456  254.037   73.992   3150331
	0012    386 09 08.388  0.787368  0.954884 201.372  254.025   73.741   3860825
	0012    459 07 16.245  0.787975  0.955172 201.303  254.095   73.647   4590802
	0012    532 06 28.502  0.783021  0.955135 201.252  254.266   73.771   5320709
	0012    603 07 03.510  0.777660  0.954684 201.134  254.378   74.057   6030627
	0012    673 01 08.923  0.777703  0.954382 200.940  254.435   74.145   6730110
	0012    742 10 28.225  0.779050  0.954594 200.797  254.473   74.053   7421015
	0012    813 08 06.236  0.780271  0.955070 200.671  254.407   73.909   8130823
	0012    886 04 02.499  0.782316  0.955414 200.607  254.504   73.748   8860402
	0012    959 01 24.841  0.777944  0.955474 200.548  254.627   73.733   9590129
	0012   1030 07 12.737  0.772739  0.955157 200.430  254.763   73.951  10300625
	0012   1100 06 12.277  0.770363  0.954827 200.273  254.872   74.143  11000617
	0012   1170 02 27.511  0.781288  0.954490 200.288  254.843   74.039  11700210
	0012   1241 04 11.464  0.783938  0.954852 200.192  254.899   73.886  12410418
	0012   1313 05 01.651  0.784435  0.955108 200.093  254.962   73.812  13130510
	0012   1385 11 06.327  0.783616  0.955047 200.036  255.125   73.829  13851108
	0012   1457 01 30.100  0.778438  0.954800 199.904  255.250   74.040  14570114
	0012   1527 03 12.625  0.776586  0.954673 199.720  255.326   74.106  15270328
	0012   1597 07 03.752  0.777688  0.954791 199.582  255.410   74.060  15970618
	0012   1668 04 17.408  0.777538  0.955174 199.441  255.389   73.979  16680426
	0012   1740 07 14.017  0.779445  0.955366 199.371  255.513   73.900  17400628
	0012   1812 09 15.826  0.777105  0.955327 199.290  255.639   73.956  18120830
	0012   1884 01 26.217  0.775732  0.955037 199.177  255.775   74.040  18840125
	0012   1954 05 22.881  0.773656  0.954832 199.027  255.891   74.177  19540518
	0012   2024 04 20.997  0.780764  0.954591 198.987  255.856   74.191  20240510

\end{verbatim}
\end{small}
In the following we discuss earlier returns to perihelion that were missed by observers.  Returns to perihelion prior 1385 are only discussed if there was a promising candidate identified or the comet experienced a particularly favorable apparition.
	\\

	\textbf{Perihelion 1740 July 14}
	\\
This apparition was not favorable concerning the observing geometry.  The comet	might have become brighter than magnitude 10 in April, but already at an elongation below $50^{\circ}$. In May the brightness may have attained magnitude 7, but the elongation was then below $40^{\circ}$.  Perihelion was reached almost behind the sun, the elongation being then around $16{\circ}$, the brightness perhaps magnitude 4. The comet then moved quickly southward and remained at elongations below $45^{\circ}$. There is also no promising candidate in any known records to be found.
	\\

	\textbf{Perihelion 1668 April 17}
	\\
This return to perihelion was also not favorable, with the observing geometry being very poor.  In January the comet might have been at magnitude 9 at elongations of just below $60^{\circ}$. Perihelion was reached at only $22^{\circ}$ elongation with a magnitude of perhaps 4.

There was one comet observed in 1668 from March 3 to 30, which is known as C/1668 E1.  Its orbit is in no way compatible with 12P and can clearly be excluded.  This was a very bright comet, a sunskirter with a perihelion distance of only 0.066 AU that had a long tail and was brighter than Venus.

There are other records of a comet seen earlier in 1668. In a paper by Park and Chae (2007), a comet is mentioned that was seen by Korean observers from March 11. While Park and Chae attribute this object to 12P, it is more likely that it is another description of C/1668 E1.  It would have been strange to see another bright comet in the same general region of the sky, as only one bright known comet was then observed widely throughout the world.	And it again has to be stressed that 12P was likely near magnitude 11
on 1668 March 11!  It would have taken a very large and long-lived outburst	to bring it to a brightness level to be seen with the unaided eye by the Koreans (and then only by the Koreans).
	\\

	\textbf{Perihelion 1597 July 3}
	\\
The year 1597 saw another unfavorable return to perihelion for comet 12P, similar to or even worse than the one of 1740.  There is also no candidate record in historic sources.
\pagebreak
	\\

	\textbf{Perihelion 1527 March 12}
	\\
In November 1526, the comet might have become brighter than magnitude 10 at an elongation of just below $70^{\circ}$.  It continued to perihelion in March	1527, which was reached at an elongation of just above $30^{\circ}$ and with a brightness of perhaps magnitude 4.  There are historic records of comets in 1523 and 1529, but the descriptions do not fit.
	\\

	\textbf{Perihelion 1313 May 1}
	\\
The next perihelion before 1385 occurred in 1313, and it was again an unfavorable apparition for earth-based viewers (comparable to 1668). The comet remained at low elongations, and perihelion was attained with magnitude perhaps 4 at elongations below $15^{\circ}$.

There was a comet seen on 1313 April 13, about 1.5 months prior to perihelion passage for 12P (when it would have been perhaps at magnitude 5 and at $15^{\circ}$ elongation).  Park and Chae have suggested this comet as a candidate for 12P, too.  Unfortunately, the indicated position in Gemini is not consistent with the position in Aries given by our orbit. So this object can be clearly ruled out.

Pingre (p.\ 425) gives a description of the same or another object following the Asian account based on the manuscripts of the historian Mussati (1727, p.\ 554) who lived from 1261-1329:

``In Europe, on April 16, Jupiter and Venus were in conjunction in the sign of Gemini. Four days later a comet was seen in Italy towards the place in the sky where the Sun appeared, when it was about to enter the waters of the ocean: its hairy tail, similar to a whitish smoke, extended to the distance of twenty feet on the west side (it should be read, on the east side). After gradually weakening for a fortnight, this Comet finally vanished. Other Historians similarly testify that the Comet was seen from the west side; therefore his tail could not look to the West."

On that date - April 17 - 12P/Pons-Brooks would have been at an elongation of only $13^{\circ}$	and visible low above the western horizon in twilight with a magnitude of maybe 4.5. If the comet would have experienced an outburst around that time it might have been visible even under such conditions but the descritption of a long tail seems to contradict a recent outburst. Probably this account also relates to the Asian Gemini object.
	\\

	\textbf{Perihelion 1241 April 11}
	\\
Another unfavourable apparition with the comet becoming brighter than magnitude 6 already at a small elongation of below $35^{\circ}$. Maximum brightness of about magnitude 4.5 was attained at an elongation of $20^{\circ}$. In May the comet had 	traveled southward and became fainter than magnitude 6 at an elongation of 	about $45^{\circ}$. The Japanese text {\it Dai Nihon shi} reports that on February 17, ``a broom star was seen".(Pankenier 2008, p.\ 149). At that time the comet might have been as bright as magnitude 6.5-7 and at an elongation of $43^{\circ}$. It should have been visible only if there was an outburst.
	\\

	\textbf{Perihelion 959 January 24}
	\\
The 959 return to perihelion of 12P is similar to that of 1457, when Toscanelli saw the comet from Italy.  There is one comet in historical records in 959, but the details are very uncertain.  They come from a Byzantine text dated 990 and provide no observational details, but rather relate it to the death of Constantine VII Porphyrogenitus (who died on 959 November 9; Kronk 1999).  The comet would then be expected to be bright in January.  Hasegawa (1980) gives a date of 959 Oct. 17 for this	comet and lists another for 959 May, seen from Arabia.

Stryuck (1740, p.\ 217) wrote:  ``In the Year 959, a Comet was seen as a dim [literally ``sad and dark"] light.  ({\it Constantin.\ Porphyr.\ incerti Continuat.}, p.\ 289 [e.g., cf.\ Niebuhr 1838]; {\it Symeon Magist \& Logoth. Annal.}, p.\ 496).  When the Comet was seen at the death of the mentioned Emperor, than it	must have appeared in the middle of November."  Struyck also suggested	identify with the 1652 comet:  ``This was the Comet	that was seen in the Year 1652."  Chambers (1889, p.\ 572) cites two sources for a comet in 959, one saying ``a gloomy and obscure star"	(citing the extension of the {\it Chronographia} of Theophanes the	Confessor by Constantine VII, likely taking his citation directly from Struyck) and the other saying that it appeared from Oct.\ 17 to Nov.\ 1 [but a careful reading of the second source, Tackio (1653), doesn't appear to mention either the 959 comet of these specific dates].
	\\

	\textbf{Perihelion 886 April 2}
	\\
This apparition is comparable to that of 1241. The comet was already at an elongation of below $40^{\circ}$ when it became brighter than magnitude 6 at the	end of February. Maximum magnitude of 4.5 was attained at an elongation of $22^{\circ}$. It then moves southward and became fainter than magnitude 6 in mid-May at an elongation of about $45^{\circ}$. Three Chinese texts mention a comet seen between June 6 and July 5 Pankenier 2008, p.\ 102). The  {\it Xin Tang shu: Xizong ji}, the {\it Xin Tang shu: tianwen zhi} and the {\it Jiu Tang shu: Xizong ji} say that a ``star became fuzzy" in JI (lunar mansion 7, near $\gamma$ Sgr) and WEI (lunar mansion 6, near $\alpha$ Peg). It then passed through BEIDOU (near $\alpha$ UMa) and SHETI (near $\o$ and $\eta$ Boo). This can not be 12P since it was already situated far south.
	\\

	\textbf{Perihelion 813 August 6}
	\\
The 813 return to perihelion was not perfect concerning the geometrical	conditions.  The comet may have become brighter than magnitude 6 in July	at an elongation of about $30^{\circ}$.  Maximum brightness with magnitude perhaps 4 was attained at the beginning of August with a similar elongation.  The comet then moved southward and became fainter than magnitude 6 in September.

Interestingly, Pingre (1783, pp.\ 337-338) lists a comet for 813 August 4, but his description (based on the medieval author Theophanes the Confessor) leaves great doubt whether this was indeed a comet:  ``On August 4 a comet was seen, which resembled two moons joined together; they separated, and having taken different forms, at length appeared like a man without a head" (translation from Chambers 1889, p.\ 568). The description sounds more like that of a short-lived transient such as a bright meteor/fireball.  This object of Theophanes was not included in the catalogues of Williams (1871), Ho (1962), and Hasegawa (1980), and is probably
not related to 12P.
	\\

	\textbf{Perihelion 742 October 28}
	\\
The 742 return to perihelion was quite favorable. At the beginning of September, comet 12P would be expected to have become brighter than magnitude 6, while	being at an elongation of $65{\circ}$ and a declination of about $+62^{\circ}$. It then moved southward and attained a maximum magnitude of perhaps 1.5 in October, then with an elongation of $50^{\circ}$-$55^{\circ}$ and moving in declination	from $0^{\circ}$ to $-20^{\circ}$ around its closest approach to the earth.

Despite these favorable observing conditions, no historic object can be	identified unambiguously from historical records.  Pingre (1783, p.\ 336)	cites several sources for objects around this year.	For 742 and 743, Cedreni (1647, pp.\ 460-461) mentions ``a sign in the sky" and ``a sign in the sky appearing towards the North, which fell down to the ground like dust", respectively. For 743, Hoyland (2011, p.\ 242) gives four different chronicles describing a ``sign in the sky" which more or less agree with each other probably due to copying. This sign is said to have appeared in June and looked like three ``columns of fire that flickered and then remained constant". This sounds very much like an aurora. In June comet 12P would have been at perhaps magnitude 10. An identity, if real at all, is very unlikely.

Two of the chronicles go on to say that another such sign was seen in September. Here the month would be matching with the visibility of 12P but details are too scarce to suspect it is a misdated description of a comet.

Pingre, Cedreni and Hoyland mention another comet seen from Syria in 744 or 745, possibly in January. This object may have also been seen from Asia (Ho, p.\ 171) Lubienietz says that in 745 a comet was seen in Cancer according to an anonymous report from the German city of Nuremberg. The comet was seen for 39 days. All these objects - if real - have probably no relation to comet 12P.
	\\

	\textbf{Perihelion 673 January 8}
	\\
The 673 return to perihelion of comet 12P geometrically falls between the favorable apparitions of 1385 and 1497.  The comet would be predicted to have become brighter than magnitude 6 at the beginning of November 672,	at an elongation of around $75^{\circ}$.  Being at almost $+50^{\circ}$ declination, it then moved southerly, reaching maximum brightness of perhaps magnitude 2 at the end of the year. The elongation was then around $50^{\circ}$-$55^{\circ}$, and the declination around $0^{\circ}$. The comet then continued to move southward and should have become fainter than magnitude 6 in March 673.

Pingre (1783, p.\ 331) lists a comet for this year -- however, with no details that help to decide on any identity with comet 12P/Pons-Brooks.  At first,	he cites two sentences from ancient chronicles:  ``In the first year of	Thierry's reign, we saw a Comet. A fire appeared in the sky for ten days. An extraordinary iris caused so much fright, that it was believed that the last day was near."  The king mentioned was Theoderic III, who became king of Neustria in 673 and king of Austrasia (and thus of all 	Franks) in 679.  Pingre then concludes:  ``All this may be reduced to an aurora borealis.  Of the Authors quoted [...], only one calls it a comet; while he is contemporary, the word comet is sometimes very ambiguous."

Stryuck (1740, p.\ 209) wrote:  ``In the Year 673, in the Month of March, a Fire shined 10 days in the Sky. ({\it Centuria. Magdeburg.}, cent.\ 7, cap.\ 13, p.\ 564)"; he also suggested identity of the comet	of 673 with the 1337 and 1558 comets Stryuck (1753, pp.\ 19-20). Hevelius (1668, p.\ 812) and Funccius (p.\ 124) also mention a comet seen for 10 days in 673.

Lubinietz (1667, p.\ 116) gives a comet for 674 and refers to Alstedius  (1650, p.\ 506) and Berckringeri (1665, p.\ 32). The two latter sources are based on	de Cesarea (1483) who also gives the phrase with the fire in the skies for ten days.

Asian sources do not help in this case.  For the period 672 September 27 to October 25, the Korean \textit{Samguk sagi} and \textit{Jeungbo munheon bigo} speak of a ``broom star" that ``emerged seven times in the north" (Pankenier et al. 2008, p.\ 74; Ho 1962).  This could have been comet 12P if it had been unusually	bright and experiencing an outburst.
	\\

	\textbf{Perihelion 386 September 8}
	\\
The return to perihelion in 386 would have produced a quite favorable apparition.  With the end of July, the comet would have appeared in	the morning sky at about magnitude 6 at an elongation of $46^{\circ}$.At perihelion, the comet would have reached a maximum magnitude of about 3.5 at an elongation of $44^{\circ}$; it then started to move southward and would be presumably fainter than magnitude 6 by the end of October at an elongation of $45^{\circ}$.

Pingre (1783, p.\ 303) reports a comet seen in this year in Sagittarius but	says that it was a misdated account of comet C/390 Q1.  Ho (1962) reports Asian records that state that there was a comet seen from April/May and disappeared in July/August and situated in Sagittarius.  Biot (1843b) noted a comet seen in China in Sgr in April that was visible until July (see also Carl 1864, p.\ 18).  Hasegawa (1980)	considered this to be a nova.  Since the time and position do not match, an identity with 12P is impossible.
	\\

	\textbf{Perihelion 245 September 15}
	\\
A comet observed in the year 245 is probably an earlier sighting of comet 12P/Pons-Brooks.  Pankenier et al. (2008, p.\ 40) provides:  ``6th year of the Zhengshui reign period of King Qi of Wei, 8th month, day wuwu [55]; a white broom star 2 chi long appeared in QIXING [LM 25]. It advanced as far as ZHANG [LM 26] for 23 days in total then was extinguished.	[\textit{Song shu: tianwen zhi}] ch. 23".

Other authors (Kronk 1999; Ho 1962; Williams 1871; Pingre 1783) use a slightly different wording as shown as an example in the report by Ho: ``On a wu-wu day in the eighth month of the sixth year of the Cheng-Shih reign-period a white (hui) comet measuring 2 ft (chhih) appeared at the Chhi-Hsing (25th lunar mansion) moving towards the Chang (26th lunar	mansion) and disappeared after 23 days."

This means that a comet appeared on 245 Sept. 18 (probably 17.9 UT)	close to $\alpha$, $\iota$, and $\tau$ Hya (QIXING or Chhi-Hsing), and moved towards $\kappa$, $\lambda$, $\mu$, and $\nu$ Hya (ZHANG or Chang).  The tail was about $3^{\circ}$ long.

The following derived position assumes that the comet was situated within QIXING.
	\\

	245 Sep 17.9: 09h 35m $-05^{\circ}$ 00' (2000.0).
	\\

Using Kobayashi's orbit for that epoch, the comet is situated about $8^{\circ}$ from the above position on 245 Sept. 17.9.  Adjusting the perihelion time by +2.9 day QIXING would be around $6.5^{\circ}$ (cf. fig. 5). The brightness would have been at	magnitude 2-3 around that time.\\

\includegraphics[width=0.9\textwidth]{"245_chart_new.jpg"}\\

Figure 5: Position of comet 12P/Pons-Brooks in relation to the Chinese constellation QIXING where the comet was seen on 245 Sep. 17.9 UT.  Shown are the position based on Kobayashi's nominal linked orbit and after an adjustment of t + 2.9 days\\

The general direction of movement then carries the comet indeed in the direction of $\kappa$, $\lambda$, $\mu$, and $\nu$ Hya (also around $7^{\circ}$ distance).  The problem here is that Pankenier et al. say that the comet disappeared after 23 days in the region of ZHANG.  This should not have been the case.  After 23 days (Oct. 10), it would already be in Cen, some $23^{\circ}$ away from ZHANG (or even farther, when using the perihelion date of Sept. 9).  All other sources do not connect ZHANG with the date of the last sighting and can be understood	as a scenario in which the comet was moving in the direction of ZHANG and disappeared after 23 days, which would agree with the expected path of comet	12P/Pons-Brooks.

Apparently the original text contains some ambiguity in interpretation, and it can indeed be put in both ways.  Upon the request from the authors of this paper,	a word-by-word translation by Ye (2020b) gives:  ``...advanced and arrived Zhang and settled for 23 days and then extinguished."  From a linguistical point-of-view,	it is not fully clear whether the word `settled' refers to the apparition or to the position of the comet with respect to ZHANG.  We nevertheless think that the identification of 12P/Pons-Brooks with the comet of 245 is highly probable and would make it the comet with the third-longest observational arc after 1P/Halley and 109P/Swift-Tuttle.

\section{The apparition of 2024}

The impending apparition of 2024 is not very favorable, but will be	better than the last one in 1954.  An analysis by renowned visual observer	Max Beyer (1958) included 76 observations made by himself using the 26-cm equatorial of the Hamburg-Bergedorf observatory (Germany) and shows	numerous outbursts; he notes that the amplitude of these outbursts decreases with decreasing distance from the sun.  At least five outbursts with amplitudes of at least $1^{mag}$ magnitude can be seen in his combined lightcurve
of his visual observations and of photographic ones by G. van Biesbroeck. He finally gives lightcurve parameters of H0 =$4.66^{mag}$ and n = 4.33.

This generally agrees with parameters derived by Green (2020) from observations	in the database of the ICQ (H0 = $4.0^{mag}$, n = 3.2), which also roughly agree with the limited brightness information for the apparitions of the 19th century.  An analysis of historic brightness information in Kronk (2003)	confirmed not only the tendency for outbursts but also showed that the comet shows a rather steep decline in brightness after perihelion, hinting to a lightcurve asymmetry.  It should also be taken into account that the	comet has never been observed at distances farther way than 4.5 AU from the	sun pre-perihelion and 2.2 AU post-perihelion. Especially for the pre-perihelion behavior, predictions are very complicated.

A new analysis of the apparition of 1954 using data from the ICQ database and including the observations from Beyer show a clear difference between the pre- and the post-perihelion parts of the lightcurve (fig. 6). \\
\begin{verbatim}
	pre-perihelion: 	H0 = 4.5, n = 5.2
	post-perihelion:	H0 = 5.2, n = 5.1

\end{verbatim}
It should be noted, however, that all post-perihelion estimates come from one observer only (A. Jones) and it is possible that the difference between pre- and	post-perihelion is due to observer bias. The lightcurve also shows that the tendency for small outbursts is much more apparent prior perihelion.\\

	\includegraphics[width=1.0\textwidth]{"figure6.png"}\\

Figure 6: Lightcurve of comet 12P/Pons-Brooks based on the ICQ database and Max Beyer's	visual observations (1958). (Lightcurve prepared with \textit{``Comet for Windows"}, URL {\tt http://http://www.aerith.net/project/comet.html})\\

The ephemeris below uses the values of Green (labelled Mag(a)) and the pre-perihelion parameters from the new analysis (labelled Mag(b)).
\\
\begin{small}
\begin{verbatim}
	Date    TT    R. A. (2000) Decl.     Delta      r     Elong.  Phase  Mag(a) Mag(b)
	2020 03 12    18 17.81   +16 22.8   12.557   12.411    79.3     4.5  18.2   24.2
	2020 04 21    18 17.16   +18 44.6   11.841   12.200   108.7     4.5  18.1   24.0
	2020 05 31    18 07.69   +20 37.1   11.290   11.987   131.5     3.6  17.9   23.8
	2020 07 10    17 53.70   +21 05.0   11.064   11.770   132.1     3.7  17.8   23.6
	2020 08 19    17 42.81   +19 59.5   11.163   11.550   110.1     4.7  17.7   23.6
	2020 09 28    17 40.62   +18 06.7   11.430   11.328    81.6     5.0  17.7   23.5
	2020 11 07    17 47.72   +16 29.6   11.643   11.102    54.9     4.2  17.7   23.4
	2020 12 17    18 01.02   +15 55.5   11.613   10.873    39.6     3.3  17.6   23.3
	2021 01 26    18 15.65   +16 47.2   11.259   10.640    49.1     4.0  17.5   23.1
	2021 03 07    18 26.20   +19 01.2   10.636   10.404    73.9     5.3  17.3   22.9
	2021 04 16    18 27.77   +22 03.7    9.913   10.164   101.7     5.5  17.0   22.6
	2021 05 26    18 18.08   +24 46.4    9.308    9.920   124.7     4.8  16.8   22.3
	2021 07 05    18 00.88   +25 49.0    8.993    9.672   129.6     4.6  16.7   22.1
	2021 08 14    17 45.81   +24 42.2    8.994    9.420   112.0     5.7  16.6   21.9
	2021 09 23    17 41.22   +22 19.7    9.183    9.163    85.7     6.3  16.5   21.8
	2021 11 02    17 48.69   +20 07.2    9.352    8.902    60.3     5.6  16.5   21.7
	2021 12 12    18 04.79   +19 10.3    9.318    8.635    43.9     4.5  16.3   21.5
	2022 01 21    18 23.98   +20 03.5    8.991    8.364    47.9     5.0  16.1   21.3
	2022 03 02    18 39.94   +22 52.2    8.402    8.086    68.2     6.5  15.9   20.9
	2022 04 11    18 46.04   +27 07.4    7.685    7.803    93.1     7.4  15.6   20.5
	2022 05 21    18 36.94   +31 29.9    7.035    7.514   114.7     7.0  15.2   20.1
	2022 06 30    18 13.92   +33 53.8    6.618    7.218   122.7     6.8  15.0   19.8
	2022 08 09    17 50.09   +32 56.6    6.487    6.915   110.9     7.9  14.8   19.5
	2022 09 18    17 40.46   +29 38.9    6.546    6.604    88.9     8.8  14.6   19.2
	2022 10 28    17 48.98   +26 16.1    6.614    6.284    66.5     8.3  14.5   19.0
	2022 12 07    18 11.31   +24 33.7    6.530    5.956    50.8     7.4  14.3   18.6
	2023 01 16    18 40.85   +25 30.4    6.208    5.618    49.4     7.6  14.0   18.2
	2023 02 25    19 10.55   +29 32.8    5.663    5.268    61.8     9.5  13.5   17.6
	2023 04 06    19 32.17   +36 38.0    4.991    4.907    79.4    11.6  13.0   17.0
	2023 05 16    19 33.79   +45 46.6    4.332    4.532    94.9    12.8  12.4   16.2
	2023 06 25    18 59.15   +53 48.6    3.807    4.141   102.1    13.9  11.8   15.4
	2023 08 04    17 55.68   +55 27.5    3.456    3.734    97.9    15.6  11.3   14.6
	2023 09 13    17 19.54   +50 07.6    3.211    3.306    86.5    17.7  10.7   13.8
	2023 10 23    17 33.16   +43 09.6    2.952    2.854    74.6    19.6  10.0   12.8
	2023 12 02    18 27.92   +38 36.5    2.591    2.376    66.4    22.4   9.1   11.5
	2024 01 11    20 06.47   +37 53.3    2.143    1.868    60.6    27.3   7.8    9.7
	2024 02 20    22 48.67   +37 21.1    1.751    1.339    49.5    34.1   6.2    7.4
	2024 03 31    02 05.11   +23 33.2    1.611    0.875    28.6    33.1   4.6    4.8
	2024 05 10    04 31.52   -03 41.3    1.577    0.859    29.6    35.5   4.5    4.6
	2024 06 19    07 00.37   -29 55.0    1.578    1.313    55.9    39.9   5.9    7.0
	2024 07 29    10 07.58   -44 45.7    1.983    1.842    67.0    30.5   7.6    9.4
	2024 09 07    12 40.01   -47 29.2    2.730    2.351    57.7    21.2   9.2   11.5
	2024 10 17    14 23.46   -47 30.8    3.520    2.831    40.0    13.1  10.3   13.1
	2024 11 26    15 39.66   -47 42.0    4.128    3.283    27.4     7.9  11.2   14.3
	2025 01 05    16 36.61   -48 20.5    4.428    3.712    38.8     9.5  11.8   15.1
\end{verbatim}
\end{small}
\section{References}

	Alstedius, J. H. (1650).  {\it Thesaurus Chronologiae} (Herbornae).\\
	Berckringeri, D. (1665).  {\it Dissertatio Histrorico-Politica de Cometis} (Ultrajectum: Meinard).\\
	Beyer, M. (1958).  ``Physische Beobachtungen von Kometen. X",
	{\it A.N.}\  {\bf 284}, 112-128.\\
	Biot, E. C. (1843a).  ``Catalogue Des Com\`etes observ\'ees en
	Chine depuis l'an 1230 jusqu'\`a l'an 1640 de notre \`ere", in
	{\it Connaissance des Temps ... Pour L'An 1846} (Paris:  Bachelier,
	Imprimeur-Libraire du Bureau des Longitudes), Additions, p.\ 57.\\
	Biot, E. C. (1843b).  ``Catalogue Des \'Etoiles extraordinaires
	observ\'ens en Chine depuis les temps anciens jusqu'\`a l'an 1203 de
	notre \`ere", in {\it Connaissance des Temps ... Pour L'An 1846}
	(Paris:  Bachelier, Imprimeur-Libraire du Bureau des Longitudes),
	Additions, p.\ 64.\\
	Carl, P. (1864).  {\it Repertorium der Cometen-Astronomie}
	(Muenchen:  M. Rieger'sche Universitaets-Buchhandlung).\\
	Cedreni, G. (1647).  {\it Compendium Historiarum. Tomus II} (Paris).\\
	Celoria, G. (1884).  ``Comete del 1457", {\it A.N.}\ {\bf 110}, 174.\\
	Celoria, G. (1894).  ``Con un capitolo sui lavori astronomici del Toscanelli",
	in {\it Raccolta di documenti e studi pubblicati dalla R.\ Commissione
		colombiana} (Roma:  Ministero della pubblica istruzione), part.\ 5, vol.\ 2.\\
	Celoria, G. (1921).  ``Sulle osservazioni di comete fatte da Paolo Dal Pozzo
	Toscanelli e sui lavori astronomici in generale", {\it Pubblicazioni del
		Reale Osservatorio Astronomico di Brera in Milano}, 55.\\
	Chambers, G. F. (1889).  {\it A Handbook of Descriptive and Practical
		Astronomy}, 4th ed. (Oxford:  Clarendon Press), Vol.\ 1.\\
	de Cesarea, E. (1483).  {\it Chronicon} (venezia: Erhard Ratdolt).\\
	Festou, M. C.; B. Morando; and P. Rocher (1985).  ``The orbit of periodic
	comet Crommelin between the years 1000 and 2100",
	{\it Astron. Ap.}\ {\bf 142}, 421-429.\\
	Funccio, J. (1578). {\it Chronologia} (Witebergae).\\
	Galle, J. C. (1894).  {\it Verzeichniss der Elemente der bisher
		berechneten Cometenbahnen} (Leipzig:  Verlag von Wilhelm Kugelmann).\\
	Gould, G. P., transl.\ (1977).  {\it Manilius:  Astronomica}
	(Cambridge, MA:  Harvard University Press), p.\ 77.\\
	Green, D. W. E. (2020a). {\it CBET} No. 4727.\\
	Green, D. W. E. (2020b). {\it CBET} No. 4805.\\
	Hasegawa, I. (1979).  ``Orbits of ancient and medieval comets",
	{\it Publ.\ Astron.\ Soc.\ Japan} {\bf 31}, 257-270.\\
	Hasegawa, I. (1980).  ``Catalogue of Ancient and Naked-Eye Comets",
	{\it Vistas Astron.}\ {\bf 24}, 59-102.\\
	Hevelii, J. (1668).  {\it Cometographia, totam naturam cometarum} (Gedani, Simon Reininger).\\
	Hind, J. R. (1846). ``Schreiben des Herrn J. R. Hind an den Herausgeber",
	{\it A.N.}\ {\bf 23}, 177.\\
	Ho, P. Y. (1962).  ``Ancient and mediaeval observations of comets and
	novae in Chinese sources", {\it Vistas Astron.}\ {\bf 5}, 127-225.\\
	Hoyland, R. G. (2011).  {\it Theophilus of Edessa's Chronicle},
	Translated Texts for Historians, vol. 57 (Liverpool Univ. Press).\\
	Jervis, J. L. (1985).  {\it Cometary Theory in Fifteenth-Century Europe},
	Studia Copernicana (Wroclaw).\\
	Kobayashi, T. (2020). {\it Nakano Note}, No. 4136 (2020 June 28); posted
	at website URL\\
	 {\tt http://www.oaa.gr.jp/\textasciitilde oaacs/nk/nk4048.htm}.\\
	Kronk, G. W. (1999).  {\it Cometography}, Vol.\ 1 (Cambridge University
	Press).\\
	Kronk, G. W. (2003).  {\it Cometography}, Vol.\ 2 (Cambridge University
	Press).\\
	Kronk, G. W. (2007).  {\it Cometography}, Vol.\ 4 (Cambridge University
	Press).\\
	Lubienietz, S. (1667).  {\it Systematis Cometici Tomus Secundus . . . Theatri
		Cometici pars posterior . . . Historia Cometarum, . . .} (Amsterdam:
	Francisco Cuyper).\\
	Marsden, B. G. (1975).  {\it Catalogue of Cometary Orbits}, 2nd ed.
	(Cambridge, MA:  Smithsonian Astrophysical Observatory).\\
	Meyerus Baliolanus, J. M. (1561). {\it Commentarii sive annales rerum
		Flandricarum} (Anvers: Steelsius). \\
	Mussati, A. (1723). ``Historia Augusta", {\it Rerum Italicarum Scriptores}.
	Tomus Decimus. (Mediolanum. Flandricarum)\\
    Nakano, S. (2020a). {\it Nakano Note}, No. 4048 (2020 Mar. 3); posted
    at website URL \\
    {\tt http://www.oaa.gr.jp/\textasciitilde oaacs/nk/nk4048.htm}.\\
    Nakano, S. (2020b). {\it Nakano Note}, No. 4136 (2020 Jun 28); posted
    at website URL\\
     {\tt http://www.oaa.gr.jp/\textasciitilde oaacs/nk/nk4136.htm}.\\
    Niebuhr, B. G., ed.\ (1838).  {\it Corpus Scriptorum Historiae Byzantinae.
	Theophanes Continuatus} (Bonn:  Impensis Ed.\ Weberi).\\
    Pankenier, D. W.; Z. Xu; and Y. Jiang (2008).  {\it Archaeoastronomy in
	East Asia} (Amherst, NY:  Cambria Press).\\
    Park, S.-Y.; and J. Chae (2007).  ``Analysis of Korean Historical Comet
    Records", {\it Publ.\ Korean Astron. Soc.}\ {\bf 22}, 151-168.\\
    Peirce, B. (1846). {\it American Almanac for 1847}, p.\ 83.\\
    Pingre, A. G. (1783).  {\it Cometographie ou traite historique et
	theoretique des cometes. Tome premier} (Imprimerie Royale, Paris).\\
    Procter, M.; and A. C. D. Crommelin (1937).  {\it Comets:  Their Nature,
	Origin, and Place in the Science of Astronomy} (The Technical Press, London).\\
    Ramsey, J. T.; and A. L. Licht (1997).  {\it The Comet of 44 B.C.\ and
	Caesar's Funeral Games} (Atlanta, GA:  Scholar's Press), p.\ 94.\\
    Schulhof, L. (1885).  ``Ueber muthmassliche frühere Erscheinungen des
    Cometen 1873 VII", {\it A.N.}\ {\bf 113}, 143-144.\\
    Stephenson, F. R. (1997).  {\it Historical Eclipses and Earth's Rotation}
    (Cambridge University Press), p.\ 515.\\
    Struyck, N. (1740).  ``Inleiding tot de Algemeene Kennis der Comeeten,
    of Staarsterren", in {\it Inleiding tot de Algemeene Geographie, benevens
	eenige Sterrekundige en andere Verhandelingen}  (Amsterdam:  Isaak Tirion).\\
    Struyck, N. (1753).  ``Vervolg van de Beschryving der Comeeten of
    Staartsterren", in {\it Vervolg van de Beschryving der Staartsterren,
	en Nader Ontdekkingen Omtrent den Staat van 't Menschelyk Geslagt}
    (Amsterdam:  Isaak Tirion).\\
    Sueyro, E. (1624). {\it Anales de Flandes, Segunda Parte} (Anvers: Pedro
    y Iuan Belleros).\\
    Tackio, J. (1653).  {\it Coeli Anomalon, id est, De Cometis, sive Stellis
	Crinitis ...} (Giessen, Germany:  Ex officina Typographica Chemliniana).\\
    Werlauff, E. C. (1847). {\it Íslenzkir annálar, sive Annales Islandici ab
	anno Christi 803 ad annum 1430}  (sumptibus Legati Arnæ-Magnæani).\\
    Williams, J. (1871).  {\it Observations of Comets from BC 611 to
	AD 1640} (London:  Stangeways and Walden).\\
    Wyttenbach, J. H., Müller, M. F. J. (1838). {\it Gesta Trevirorum integra
	lectionis varietate et animadversionibus illustrata ac indice duplici
	instructa. Vol. II} (Trier: Lintz).\\
    Ye, Q. et. al. (2020a).  ``Recovery of Returning Halley-Type Comet 12P/Pons-Brooks with the Lowell Discovery Telescope", {\it RNAAS}\ {\bf 4}, No. 7, 2020 July 7.\\
    Ye, Q. (2020b).  Personal communication.\\

\end{document}


\section{Introduction}\label{sec:introduction}}
\else
\section{Introduction}
\label{sec:introduction}
\fi

\IEEEPARstart{A}{dvanced} data collection techniques in today's world require
researchers to work with large volumes of nonlinear data, such as global climate
patterns \cite{daley1993atmospheric, jones2009high}, satellite signals
\cite{manjunath1996texture, zumberge1997precise},  social and mobile networks
\cite{carrington2005models, becker2013human}, the human genome
\cite{huang2009systematic,schafer2005empirical}, and patterns in collective
motion \cite{gajamannage2015identifying,gajamannage2015detecting}. Studying,
analyzing, and predicting such large datasets is challenging, and many such
tasks might be implausible without the presence of Nonlinear Dimensionality
Reduction (NDR) techniques. NDR interprets high-dimensional data using a reduced
dimensional representation that corresponds to the intrinsic nonlinear
dimensionality of the data \cite{van2009dimensionality}. Manifolds are often
thought of as being smooth, however many existing NDR methods do not directly
leverage this important feature. Sometimes, ignoring the underlying smoothness
of the manifold can lead to inaccurate embeddings, especially when the data has
been contaminated by noise.

Many NDR methods have been developed over the last two decades due to the lack
of accuracy and applicability of classic Linear Dimensionality Reduction (LDR)
methods such as Principal Component Analysis (PCA) \cite{jolliffe2002principal},
which finds directions of maximum variance, or Multi-Dimensional Scaling (MDS)
\cite{cox2000multidimensional}, which attempts to preserve the squared Euclidean
distance between pairs of points.  As the Euclidean distance used in MDS
quantifies the distance between points in the high-dimensional space, rather the
actual distance along the manifold, it can have difficulty inferring a faithful
low-dimensional embedding. On the other hand, Isometric Mapping (Isomap)
represents the pairwise distance between points using \emph{geodesic distances}
and is an NDR method that successfully resolves the aforesaid problem in MDS
\cite{tenenbaum2000global}. Although Isomap has been successfully used to
analyze low-dimensional embedding of data from several domains, such as
collective motion \cite{delellis2014collective}, face recognition
\cite{yang2002face}, and hand-writing digit classification
\cite{yang2002extended}, this method can suffer from short-circuiting
\cite{balasubramanian2002isomap}, low-density of the data
\cite{lee2002curvilinear}, and non-convexity \cite{zha2003isometric}, all of
which can be magnified in the presence of noisy measurements.  It is therefore
our goal here to propose a new method which ameliorates some of
these issues as compared to Isomap.

Generally speaking, NDR approaches reveal a smooth low-dimensional, nonlinear
manifold representation of high-dimensional data. While there are many unique
capabilities provided by current NDR methods, most of them encounter poor
performance in specific instances. In particular, many current NDR methods are not
adept at preserving the smoothness of the embedded manifold in the presence of
noise. Specifically, Isomap closely mimics the underlying manifold's geometry
using a graph structure that it makes using a neighborhood search
\cite{friedman1977algorithm, agarwal1999geometric}, over the high-dimensional
data. Geodesics are generally \emph{piecewise linear}, thus, the manifold constructed using geodesics in this method is not actually smooth at each node, as demonstrated in
Fig.~\ref{fig:sphere_with_network}(a). Moreover, the length of such a geodesic
between two points is not necessarily the manifold distance.  In fact,
given a sufficiently smooth manifold, the Isomap generated geodesic distance
will generally be an \emph{over-estimate} of the true manifold distance, again
as demonstrated in Fig.~\ref{fig:sphere_with_network}(b). Of course, such issues are intensive in the presence of noisy measurements. Accordingly, herein we propose to \emph{replace the segments of the piecewise linear Isomap geodesic by a smoothing spline} as shown by the black curve in
Fig.~\ref{fig:sphere_with_network}(b) and \emph{consider the length of the
spline as the estimation of the manifold distance between points}.

\begin{figure}[htp]
    \centering
    \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width= 2.2in]{figure1a.pdf}
        \caption{}
    \end{subfigure}%
    ~
    \begin{subfigure}{0.2\textwidth}
        \centering
	\vspace{2mm}
        	\includegraphics[height= 1.7in]{figure1b.pdf}
	\vspace{4mm}
        \caption{}
    \end{subfigure}
    \caption{This figure demonstrates the lack of smoothness of the geodesics generated by Isomap. (a) Three nearest neighbors for each point (blue dots) of a spherical dataset of 300 points are found and joined by line segments (shown in blue) to create a graph structure.  The Isomap manifold distance between two arbitrary points $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ is estimated as the length of the geodesic (red path), that is defined as the shortest path between two points, computed by using, for example, Floyd's algorithm \cite{floyd1962algorithm}. (b) However, our approach creates a \emph{smoothing spline}, shown by the black curve, that is fitted through the points in the geodesic as a better approximation of the distances on the smooth manifold than the geodesic distance.}
\label{fig:sphere_with_network}
\end{figure}

There are few NDR methods found in the literature that utilize smoothing splines
for embedding like our approach.  For example, Local Spline Embedding (LSE) also
uses a smoothing spline to perform the embedding \cite{xiang2009nonlinear}.
However, this method minimizes the reconstruction error of the objective
function and embeds the data using smoothing splines that map local coordinates
of the underlying manifold to global coordinates. Specifically, LSE assumes the
existence of a smooth low-dimensional underlying manifold and the embedding is
based on an eigenvalue decomposition that is used to project the data onto a
tangent plane. However, differing from our approach, LSE assumes that the data
is noise free and unaffected by anomalies. The Principal Manifold Finding
Algorithm (PMFA) is another NDR method that also uses cubic smoothing splines to
represent the manifold and then quantifies the intrinsic distances of the points
on the manifold as lengths of the splines \cite{gajamannage2015dimensionality}.
However, this approach embeds high-dimensional data by reducing the
reconstruction error over a two-dimensional space. As this method only performs
two-dimensional embeddings, its applicability is limited for problems with
larger intrinsic dimensionality.  As we will demonstrate in the sequel, our
proposed methods overcome the limitations of these methods.

This paper is structured as follows.  In Section~\ref{sec:mds_isomap}, we will
detail the MDS and Isomap algorithms and describe the evolution of our NDR
method from these methods. Section~\ref{sec:sge} presents our NDR method, Smooth
Geodesic Embedding (SGE), that fits geodesics, as in Isomap, by smoothing
splines. We analyze the performance of the SGE method in
Section~\ref{sec:per_analysis} using three representative examples: a
semi-spherical dataset; images of faces; and images of hand written digits.
Finally, we provide discussion and conclusions in Section~\ref{sec:conclusion}.

\section{Multidimensional scaling and Isomap}\label{sec:mds_isomap}
We begin our analysis by deriving the mathematical details of the LDR method MDS. Then, we proceed to discuss Isomap which replaces the Euclidean distance in MDS by a geodesic distance. Next, we derive our method, SGE, as an extension of Isomap that fits geodesics by smoothing splines.

\subsection{Multidimensional scaling}\label{sec:mds}
Multidimensional scaling is a classic LDR algorithm that leverages the squared Euclidean distance matrix $\boldsymbol{D}=[d_{ij}^2]_{n\times n}$; where $d_{ij} = \|\boldsymbol{y}_i - \boldsymbol{y}_j\|_2$ and $n$ is the order of the high-dimensional space. Here, $\boldsymbol{y}_i$, $\boldsymbol{y}_j\in \mathbb{R}^{n\times 1}$, are two points on the high-dimensional dataset $\boldsymbol{Y}=[\boldsymbol{y}_1; \dots; \boldsymbol{y}_i; \dots; \boldsymbol{y}_j; \dots; \boldsymbol{y}_n]$. This method first transforms the distance matrix $\boldsymbol{D}$ into a Gram matrix $\boldsymbol{S}=[s_{ij}]_{n\times n}$, which is derived by \emph{double-centering} \cite{lee2007nonlinear} the data using
\begin{equation}\label{eqn:double_centering}
s_{ij}=-\frac{1}{2}\big[d^2_{ij}-\mu_i(d^2_{ij}) -\mu_j(d^2_{ij})+\mu_{ij}(d^2_{ij})\big].
\end{equation}
Here, $\mu_i(d^2_{ij})$ and $\mu_j(d^2_{ij})$ are the means of the $i$-th row and $j$-th column, respectively, of the squared distance matrix, and $\mu_{ij}(d^2_{ij})$ is the mean of the entire matrix $D$. MDS then computes the Eigenvalue Decomposition (EVD) of $\boldsymbol{S}$ as
\begin{equation}\label{eqn:evd}
\boldsymbol{S}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{U}^T,
\end{equation}
where $\boldsymbol{U}$ is a unitary matrix ($\boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I}$) providing the singular vectors and $\boldsymbol{\Sigma}$ is a diagonal matrix of singular-values. The Gram matrix $\boldsymbol{S}$, that is made from the Euclidean distance matrix $\boldsymbol{D}$, is symmetric and positive semidefinite\footnote{A symmetric $n\times n$ matrix $\boldsymbol{M}$ is said to be positive semidefinite, if $\boldsymbol{z}^T \boldsymbol{M} \boldsymbol{z}\ge 0$ for all non-zero $\boldsymbol{z}\in\mathbb{R}^{n \times 1}$.}.  Thus, all the eigenvalues of $\boldsymbol{S}$ are non-negative, and both the Singular-Value Decomposition (SVD) and the EVD of $\boldsymbol{S}$ are the same~\cite{lee2007nonlinear}. $\boldsymbol{\Sigma}$ and $\boldsymbol{U}$ are arranged such that the diagonal of $\boldsymbol{\Sigma}$ contains the eigenvalues of $S$ in descending order, and the columns of $\boldsymbol{U}$ represent the corresponding eigenvectors in the same order. We estimate $p$-dimensional latent variables of the high-dimensional dataset by
\begin{equation}\label{eqn:latent_var}
\hat{\boldsymbol{X}}=\boldsymbol{I}_{p\times n}\boldsymbol{\Sigma}^{1/2}\boldsymbol{U}^T.
\end{equation}
Here, $\hat{\boldsymbol{X}}$ is the $d$-dimensional embedding of the input data $\boldsymbol{Y}$.
Note, in the case of a matrix $\boldsymbol{S}$ which is not symmetric positive semidefinite, the EVD has negative eigenvalues which then violate Eqn.~(\ref{eqn:latent_var})~\cite{lee2007nonlinear}.  Accordingly, as we discuss in Section~\ref{sec:isomap}, we replace the EVD computation on $\boldsymbol{S}$ by SVD.

Multidimensional scaling has limited applicability as it is inherently a linear method (when $D$ is a squared Euclidean distance matrix). However, the NDR scheme Isomap overcomes this problem by employing geodesic distances instead of Euclidean distances.

\subsection{Isomap}\label{sec:isomap}
Isomap works by creating a graph structure, based upon high-dimensional data, that estimates the intrinsic geometry of the manifold. The graph structure used by Isomap can be parameterized in multiple ways, but herein we
focus on the parameter $\delta$ which measures the number of \emph{nearest neighbors}
to a given point \cite{agarwal1999geometric}.  The nearest neighbor collection for each point is transformed into a graph structure by treating points as graph nodes and connecting each pair of nearest neighbors by an edge having the weight equal to the Euclidean distance between the two points. Given such a graph, the distance between any two points is measured as the \emph{shortest path distance in the graph}, which is commonly called the \emph{geodesic distance}.

The geodesic distance between any two points in the data can be computed in many ways, including Dijkstra's algorithm \cite{dijkstra1959note}.  We employ Floyd's algorithm \cite{floyd1962algorithm} for this task as it computes the shortest paths between all pairs of nodes in one batch, and is more efficient than Dijkstra's algorithm in this case.

As in MDS, we first formulate the doubly centered matrix $\boldsymbol{S}$ from the squared geodesic distance matrix using Eqn.~(\ref{eqn:double_centering}). Here, the doubly centered matrix is not necessarily positive semidefinite as we \emph{approximate} the true geodesic distance matrix by the shortest graph distance \cite{lee2007nonlinear}.  Thus, the eigenvalue decomposition of the $\boldsymbol{S}$ matrix might produce negative eigenvalues and  Eqn.~(\ref{eqn:latent_var}) does not hold in this case. To overcome this problem, it is standard to perform the SVD over the Gram matrix $\boldsymbol{S}$ as
\begin{equation}\label{eqn:svd}
\boldsymbol{S}=\boldsymbol{V}\boldsymbol{\Sigma} \boldsymbol{U}^T,
\end{equation}
where $\Sigma$ is a diagonal matrix of singular values (non negative), and $U$ and $V$ are unitary matrices. The latent variables of the higher dimensional input data are revealed by Eqn.~(\ref{eqn:latent_var}) with $\boldsymbol{\Sigma}$ and $\boldsymbol{U}^T$ obtained from Eqn.~(\ref{eqn:svd}).

Isomap emphasizes nonlinear features of the manifold, however the lengths measured using geodesics might not faithfully reflect the correct manifold distance, as we demonstrate in Fig.~\ref{fig:sphere_with_network}.  Accordingly, we propose to  overcome this drawback in Isomap by utilizing a smoothing approach for geodesics.

\section{Smoothing geodesics embedding}\label{sec:sge}
Our goal is to fit the geodesics computed in Isomap with smoothing splines to more closely mimic the manifold and preserve the geometry of the embedding. Classic smoothing spline constructions \cite{de1972calculating} require one input parameter, denoted by $s$, that controls the smoothness of the spline fitted through the points in a geodesic.  Our proposed method, SGE, has five parameters:
\begin{itemize}
\item $\delta$ (inherent from Isomap) for the number of nearest neighbors,
\item$\mu_s$ controls the smoothness of the splines,
\item $\nu$ controls the threshold of the length of splines before reducing the order of the spline to the next level,
\item $h$ controls the number of discretizations that the method uses to evaluate the length of a spline, and
\item finally, $p$ prescribes the number of embedding dimensions.
\end{itemize}

Here, we demonstrate our approach by constructing a spline on an arbitrary geodesic $\mathcal{G}$, having $m\ge2$ points, in the graph created by a neighborhood search algorithm.  For an index $k$ we have that $d$-dimensional points in $\mathcal{G}$ are given by
\begin{equation}\label{eqn:geo_pts}
\big\{\boldsymbol{y}_k=[y_{1k}, \dots, y_{dk}]^T\vert k=1 \dots, m\big\}.
\end{equation}
For each dimension $l\in \big\{1, \dots, d\big\}$, we fit $\big\{y_{lk}\vert k=1 \dots, m\big\}$ using one dimensional smoothing splines $\hat{f}_l(z)$ of order $\theta+1$ that are parameterized in $z \in [0,1]$ by minimizing
\begin{equation}\label{eqn:spline}
\sum^m_{k=1}\big[y_{lk}-\hat{f}_l(z_k)\big] + s\int^1_0 \big[\hat{f}_l^{(\theta)}(z)\big]^2dz
\end{equation}
as in \cite{de1972calculating}. Here, $(\theta)$ represents the order of the derivative of $\hat{f}_l$, and $z_k$ is a discretization of the interval $[0,1]$ such that $z_1=0$, $z_k=(k-1)/(m-1)$, and $z_m=1$. Minimizing of Eqn.~(\ref{eqn:spline}) yields $d$ one-dimensional smoothing splines $\{\hat{f}_l(z)|l=1,\dots,d\}$. We combine these one dimensional splines and obtain a $d$-dimensional smoothing spline of the points $\big\{\boldsymbol{y}_k\vert k=1 \dots, m\big\}$ in $\mathcal{G}$,
\begin{equation}\label{eqn:dspline}
\hat{\boldsymbol{f}}(z)=[\hat{f}_1(z), \dots, \hat{f}_d(z)]^T,
\end{equation}
that estimates the \emph{smooth geodesic}. In numerical implementations, the order $\theta+1$ of the spline $\hat{f}$ should be less than number of points $m$ in the geodesic \cite{de1972calculating}.

Choosing the order of the spline is challenging, since while a spline with some specified order might perfectly fit the data (and thereby over-fit the noise in the data), another spline with a different order might only fit the data weakly. The length of the fitted spline between two points is defined as the manifold distance between those two points, thus either an over-fitted or under-fitted spline might provide an incorrect manifold distance. To overcome this problem, we introduce the spline threshold (in percentage) which allows the maximum length of a spline that can yield beyond the length of the geodesic. We treat the geodesic distance as the default manifold distance between two points. If the length of a spline with a specific order exceeds this limit, SGE reduces the order of the spline to the next level until the length of the spline satisfies the threshold or reaches the default length of the geodesic distance. It is worthwhile to try a spline fit with a lower order when a higher order spline fails numerically.

We present below our procedure of choosing the order of a given spline under three main cases (1, 2, and 3) and sub-cases (a, b, \dots):

\begin{itemize}
\item \textbf{Case--1} If $m\ge4$:
\begin{itemize}
\item \textbf{Case--a:} we first fit the points in the geodesic with a cubic smoothing spline $\hat{\boldsymbol{f}}(z)$ where $z\in[0,1]$ according to Eqn.~(\ref{eqn:spline}) and  Eqn.~(\ref{eqn:dspline}). Note that, a cubic smoothing spline is represented by $\theta=2$  in Eqn.~(\ref{eqn:spline}). We discretize this spline into $h$ segments $z_{k_1}=(k_1-1)/(h-1); k_1=1,\dots, h$ and compute the length,
\begin{equation}\label{eqn:len_spl}
d_{\hat{\boldsymbol{f}}}=\sum^{h-1}_{k_1=1}\|\hat{\boldsymbol{f}}(z_{k_1+1})-\hat{\boldsymbol{f}}(z_{k_1})\|.
\end{equation}
 Then, the length $d_{\hat{\boldsymbol{f}}}$ is compared with the corresponding geodesic distance
\begin{equation}\label{eqn:len_geo}
d_\mathcal{G}=\sum_{k=1}^{m-1}\|\boldsymbol{y}_{k+1}-\boldsymbol{y}_k\|.
\end{equation}
If $d_{\hat{\boldsymbol{f}}}<d_\mathcal{G}(100+\nu)/100$ (so that $\nu$ is thought of as a percentage), then we accept $d_{\hat{\boldsymbol{f}}}$ as the length of the smooth geodesic, otherwise we proceed to Case--b. The parameter $\nu$ (in percentage) defines the threshold (the upper bound) of the length of the spline $\hat{\boldsymbol{f}}$ that is allowed to exceed from the length of the corresponding geodesic.\\
\item \textbf{Case--b:} we fit the data with a quadratic (i.e., $\theta=1$ ) spline $\hat{\boldsymbol{f}}$ according to Eqn.~(\ref{eqn:spline}) and  Eqn.~(\ref{eqn:dspline}) and compute the length of the quadratic spline using Eqn.~(\ref{eqn:len_spl}). If $d_{\hat{\boldsymbol{f}}}<d_\mathcal{G}(100+\nu)/100$, then we accept $d_{\hat{\boldsymbol{f}}}$ as the  length of the smooth geodesic, otherwise move to the next case. \\
\item \textbf{Case--c:} we make a linear (i.e., $\theta=0$) fit $\hat{\boldsymbol{f}}$ according to Eqn.~(\ref{eqn:spline}) and Eqn.~(\ref{eqn:dspline}), and measure the length using Eqn.~(\ref{eqn:len_spl}). If $d_{\hat{\boldsymbol{f}}}<d_\mathcal{G}(100+\nu)/100$ in the linear fit, then we accept $d_{\hat{\boldsymbol{f}}}$, otherwise we move to Case--d. \\
\item\textbf{Case--d:} instead of fitting a spline, we consider the original geodesic itself as the fit and treat $d_\mathcal{G}$ as the length of the smooth geodesic.
\end{itemize}
\item \textbf{Case--2} If $m=3$:\\
The spline fitting process here is started from fitting a quadratic spline as only three points are in the geodesic. Thus, we carry-out all the Cases b--d as in Case--1.
\item \textbf{Case--3} If $m=2$:\\
We have only two points in the geodesic, thus, we perform Cases c--d as in Case--1.
\end{itemize}

We use the smoothing parameter $s$ to offset the spline fit between no fitting error (when $s=0$) and the best smoothness (when $s\rightarrow \infty$). The parameter $s$ controls the sum of square errors between the training points and the fitted function. The best value for $s$ ensuring the least error while having a sufficient smoothness is bounded by a function of the number of points in the geodesic as
\begin{equation}\label{eqn:spl_interval}
m-\sqrt{m}\le s \le m+\sqrt{m},
\end{equation}
\cite{reinsch1967smoothing}. Since the number of points in geodesics vary, we are unable to input a onetime value as the smoothing parameter into the method that satisfies the inequality (\ref{eqn:spl_interval}). In order to control this, here we introduce a new parameter called the smoothing multiplier $\mu_s\ge0$ such that $s=\mu_s m$. Now, given $\mu_s$, SGE uses different smoothing parameters for different smooth geodesics depending on $m$.

For each pair of point indices $i,j$ in the dataset, we execute the aforesaid procedure and approximate the length of the smooth geodesic $d_{ij}$.  Then, we square the entries $d_{ij}$ and create the matrix $\boldsymbol{D}=[d^2_{ij}]_{n\times n}$. We perform double centering on $\boldsymbol{D}$ using Eqn.~(\ref{eqn:double_centering}) to obtain the doubly centered matrix $\boldsymbol{S}$. Then, we compute SVD as in Eqn.~(\ref{eqn:svd}) followed by computing $p$-dimensional latent variables $\hat{\boldsymbol{X}}$ according to Eqn.~(\ref{eqn:latent_var}). A summary of the method SGE is presented in Algorithm \ref{alg:algorithm}.

\begin{algorithm*}[!htp]
\caption{ \textit{Smooth Geodesics Embedding (SGE).
\\ Inputs: Data ($\boldsymbol{Y}$), number of nearest neighbors ($\delta$), smoothing multiplier ($\mu_s$), spline threshold percentage ($\nu$), number of discretizations ($h$), and embedding dimension ($p$).
\\Outputs: List of $p$ largest singular values ($\lambda_l;l=1,\dots,p$) and $p$-dimensional embedding ($\hat{\boldsymbol{X}}$). }}\label{alg:detecting_transitions}.
\begin{algorithmic}[1]
\State For each point in $\boldsymbol{Y}$, choose $\delta$ nearest points as neighbors \cite{friedman1977algorithm}.
\State Consider all the point in $\boldsymbol{Y}$ as nodes and if any two nodes are chosen to be neighbors in 1, then join them  by an edge having the length equal to the squared Euclidean distance between them. This step converts the dataset into a graph.
\State For each pair of nodes in the graph, find the points $\mathcal{G}=\big\{\boldsymbol{y}_k\vert k=1 \dots, m\big\}$ in the shortest path using Floyd's algorithm \cite{floyd1962algorithm}. Here, $m=|\mathcal{G}|\ge2$.
\State The points in $\mathcal{G}$ are fitted with a smoothing spline and its length is computed:
\newline
Case--1 ($m\ge4$):
\begin{addmargin}[1em]{2em}
Case--a:
\begin{addmargin}[1em]{2em}
fit $\mathcal{G}$ with a cubic smoothing spline using Eqn.~(\ref{eqn:spline}) and Eqn.~(\ref{eqn:dspline}), then approximate the length $d_{\hat{\boldsymbol{f}}}$ of that using Eqn.~(\ref{eqn:len_spl}). Let, the length of the geodesic is $d_\mathcal{G}$ [Eqn.~(\ref{eqn:len_geo})].  If $d_{\hat{\boldsymbol{f}}}<d_\mathcal{G}(100+\nu)/100$, then accept $d_{\hat{\boldsymbol{f}}}$ as the length of the smooth geodesic, otherwise proceed to Case--b.
\end{addmargin}
Case--b:
\begin{addmargin}[1em]{2em}
fit $\mathcal{G}$ with a quadratic smoothing spline using Eqn.~(\ref{eqn:spline}) and Eqn.~(\ref{eqn:dspline}).  Approximate the length $d_{\hat{\boldsymbol{f}}}$ of that spline using Eqn.~(\ref{eqn:len_spl}). If $d_{\hat{\boldsymbol{f}}}<d_\mathcal{G}(100+\nu)/100$, then accept $d_{\hat{\boldsymbol{f}}}$ as the length of the smooth geodesic, otherwise proceed to Case--c.
\end{addmargin}
Case--c:
\begin{addmargin}[1em]{2em}fit $\mathcal{G}$ with a linear smoothing spline using Eqn.~(\ref{eqn:spline}) and Eqn.~(\ref{eqn:dspline}). Approximate the length $d_{\hat{\boldsymbol{f}}}$ of that spline using Eqn.~(\ref{eqn:len_spl}). If $d_{\hat{\boldsymbol{f}}}<d_\mathcal{G}(100+\nu)/100$, then accept $d_{\hat{\boldsymbol{f}}}$ as the length of the smooth geodesic, otherwise proceed to Case--d. \end{addmargin}
Case--d:
\begin{addmargin}[1em]{2em}
Consider $d_\mathcal{G}$ as the approximated length of the smooth geodesic.
\end{addmargin}
\end{addmargin}
Case--2 ($m=3$):
Perform Cases b--d similarly as in Case--1.
\newline
Case--3 ($m=2$):
Perform Cases c--d similarly as in Case--1.
\State Fill the distance matrix $\boldsymbol{D}=[d^2_{ij}]_{n\times n}$ where $d_{ij}$ is the length of the smooth geodesic between nodes $i$ and $j$ computed in 3-4. Double center $\boldsymbol{D}$ and convert it to a Gramian matrix $\boldsymbol{S}$ using the Eqn.~(\ref{eqn:double_centering}).
\State Perform the SVD on $\boldsymbol{S}$ using Eqn.~(\ref{eqn:svd}) and extract $p$ largest singular values $\lambda_l;l=1,\dots,p$ along with the latent variable $\hat{\boldsymbol{X}}$ as given by Eqn.~(\ref{eqn:latent_var}).
\end{algorithmic}\label{alg:algorithm}
\end{algorithm*}

\section{Performance analysis}\label{sec:per_analysis}
In this section we demonstrate the effectiveness of our proposed NDR approach using three representative examples. As the first example, we use a synthetic dataset of a semi-sphere to analyze the performance of SGE with respect to neighborhood size, smoothness, sparsity, and noise. Then, we study the performance of SGE using two standard benchmark datasets: 1) face images \cite{deeplearning}; and 2) images of handwritten digits (2's, 4's, 6's, and 8's) \cite{mnistdatabase}.

\subsection{Embedding of a semi-sphere}
We begin by embedding a synthetic dataset sampled from a semi-spherical manifold using SGE and Isomap to demonstrate the key concepts of our proposed SGE technique since, in this case, we can analytically compute the manifold distance on the semi-sphere and then compare with the embedding distances computed by SGE and Isomap. We sample 600 points from the manifold defined by
\begin{align}\label{eqn:sphere}\nonumber
y_1 &= r\cos(\gamma_1)\cos(\gamma_2), \\
y_2 & = r\cos(\gamma_1)\sin(\gamma_2), \\ \nonumber
y_3 & = r\sin(\gamma_1),\\ \nonumber
\end{align}
for $\gamma_1=\mathcal{U}[-\pi/2, \pi/2]$ and $\gamma_2=\mathcal{U}[0, \pi]$ where $\mathcal{U}[a,b]$ denotes a uniform distribution between $a$ and $b$. Here, $r$ is the radius of the semi-sphere which is set to $20+\mathcal{N}[0,3^2]$, where $\mathcal{N}[0,3^2]$ is a random variable sampled from a Gaussian distribution with mean 0 and variance $3^2$. Here, we impose a high noise into the dataset as we are going to investigate the robustness of our method of embedding noisy datasets.

We set the spline threshold $\nu$ and spline discretization $h$ to be $10\%$ and 100, respectively. Then, we run the SGE algorithm repeatedly over the spherical dataset with $\delta=2, 3, \dots, 8$; $\mu_s=0, 0.1, \dots, 1.0$ and obtain two-dimensional embeddings. Here, we have 77 different pairs of $\delta$'s and $\mu_s$'s, those then produce 77 two-dimensional embeddings. Now, we asses the performance of the methods in terms of distance preserving ability between the original data and the embedding. For each such embedding (77 in total), we compute distances  between points in the embedding space using the Euclidean metric and denote the distance matrix by $\boldsymbol{D}_S$. Now, we run Isomap with the same sequence of $\delta$'s above and obtain its two-dimensional embeddings. The distance matrix for the embedding of Isomap is denoted by $\boldsymbol{D}_I$. Now, we compute true manifold distances between points of the dataset using cosine distances. If $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ are two points on a semi-sphere with radius $r$, the manifold distance $d$ is given by
\begin{equation}\label{eqn:mani_dist}
d = r\gamma; \ \gamma=\cos^{-1}\begin{pmatrix}\frac{\boldsymbol{\alpha}\boldsymbol{\beta}}{|\boldsymbol{\alpha}||\boldsymbol{\beta}|}\end{pmatrix},
\end{equation}
\cite{stewart2012essential}. We compute all the pairwise distances using the above equation and form the distance matrix $\boldsymbol{D}_M$ for the manifold.

The embedding error of SGE, denoted by $\mathcal{E}_S$, is computed as the Mean Absolute Deviation (MAD) between embedding and data \cite{petruccelli1999applied}. Since the distance matrices are symmetric and have zeros on the diagonal, MAD
can then be computed using
\begin{equation} \label{eqn:mad}
\mathcal{E}_S=\frac{2}{n(n-1)}\sum^n_{i=2}\sum^n_{j=i}\big|(\boldsymbol{D}_R)_{ij}-(\boldsymbol{D}_S)_{ij}\big|.
\end{equation}
Similarly, for each pair of $\delta$ and $\mu_s$, we also compute MAD between the embedding of Isomap and the original data that we denote by $\mathcal{E}_I$. Fig.~\ref{fig:comp_embedding} illustrates MADs for Isomap ($\mathcal{E}_I$), SGE ($\mathcal{E}_E$), and their differences ($\mathcal{E}_I-\mathcal{E}_S$), versus $\delta$ and $\mu_s$. Fig.~\ref{fig:comp_embedding}(a) and \ref{fig:comp_embedding}(b) show that both methods display decreasing errors for increasing $\delta$'s (i.e., increasing neighbors), while SGE has decreasing error as $\mu_s$ increases. Fig.~\ref{fig:comp_embedding}(c) also indicates that SGE performs better than Isomap for larger smoothing multipliers \emph{for all $\delta$'s}. Moreover, this plot shows SGE performs worst when $\delta=2$ and $\mu_s=0$, and performs best when $\delta=2$ and $\mu_s=1$.
\begin{figure}[htp]
        	\centering
        	\includegraphics[width=3in]{figure2.pdf}
        	\caption{Analyzing the performance of Isomap and SGE embeddings using Mean Absolute Deviation (MAD). Herein, we compute, (a) MAD between Isomap embedding and data, denoted by $\mathcal{E}_I$, for different neighborhood sizes ($\delta$'s), and (b) MAD between SGE embedding and data, dented by $\mathcal{E}_S$,  for different neighborhood sizes and smoothing multiplier ($\mu$'s). (c) The difference of errors between these two methods ($\mathcal{E}_I-\mathcal{E}_S$) which is computed in the variable space $\delta$ and $\mu_s$.  \emph{The green cells denote that the performance of SGE is superior to that of Isomap.}}
	\label{fig:comp_embedding}
\end{figure}

Next, we analyze the influence of data sparsity of the manifold for embedding with SGE and compare that with Isomap. For that task, we produce a sequence of spherical datasets with an increasing number of points.  We create the first dataset of 200 points using Eqn.~(\ref{eqn:sphere}) with $r=20+\mathcal{N}[0,2^2]$, then add another 100 points, generated using the same equation, into the first dataset to produce the second dataset. Similarly, we generate the last dataset of 1200 points. Then, we embed these datasets in two-dimensions using both  Isomap with $\delta=3$ and SGE with $\delta=3$, $\mu_s=1$, and $\nu=10\%$. We compute the embedding errors $\mathcal{E}_S$ and $\mathcal{E}_I$ using MAD for each dataset as explained before. Since a significantly higher noise is used for the datasets, we run both methods over each dataset for 16 realizations to allow us to compute averages. Fig.~\ref{fig:analysis_n}(a) shows the mean of embedding errors of 16 realizations and error bars for the errors of Isomap and SGE embeddings. We observe that the error associated with SGE embedding is smaller than that of Isomap for \emph{all values of $n$.}

Finally, we study the embedding error in terms of the size of the noise present in the data. For that task we formulate a latticed semi-sphere of 600 points using Eqn.~(\ref{eqn:sphere}) with equally appropriately discretized $\gamma_1 \in[-\pi/2,\pi/2]$ and $\gamma_2 \in [0, \pi]$. Then, we impose increasing uniform noise levels into the parameter representing the radius as $r=20+\eta\text{U}[-1,1]$; $\eta=0, 0.3, 0.9, \dots, 3$ and produce 11 datasets. We embed each dataset 25 times (25 realizations) using Isomap with $\delta=3$ and SGE with $\delta=3$, $\nu=10\%$, and $\mu_s=1$. Fig.~\ref{fig:analysis_n}(b) presents embedding errors for both methods computed using Eqn.~(\ref{eqn:mad}). We observe that, while $\mathcal{E}_S$ slowly increases with increasing $\eta$, $\mathcal{E}_I$ increases more quickly with increasing $\eta$. \emph{Note that, the errorbars for $\mathcal{E}_S$ are significantly smaller compared to that of $\mathcal{E}_I$ at any given $\eta$.}

\begin{figure}[htp]
        	\centering
        	\includegraphics[width=3in]{figure3.pdf}
        	\caption{Mean embedding error of Isomap, denoted by $\mathcal{E}_S$, (in red) and that of SGE, denoted by $\mathcal{E}_S$, (in blue) versus, (a) sparcity and (b) noise. Error bars represent standard deviations of errors computed over realizations.  \emph{Note that, SGE has both lower average error and lower variance in the error across trials.}}
	\label{fig:analysis_n}
\end{figure}

\subsection{Embedding of face images}
In this section, we validate the SGE method using a real-world dataset of face images available in \cite{deeplearning}. This dataset consists 698 images of dimension $64\times 64$ each with a varying pose and direction of lighting, as shown by a sample of 16 snapshots in Fig.~\ref{fig:face_images}(a). We randomly choose 400 images as our baseline dataset and generate three other datasets of 400 images from the baseline dataset by imposing Gaussian noise with standard deviations ($\sigma's$) 0.1, 0.2, and 0.3 [Fig.~\ref{fig:face_images}(b)]. We set $\delta=4$, $\nu=10\%$, $h=100$ in SGE and run this algorithm over each dataset (4 in total) 5 times for $\mu_s= $0, 0.3, 0.6, 0.9, 1.2. Then, we embed these four datasets in two dimensions using Isomap with $\delta=4$.

We use the ability to preserve distances between the original and the embedding data to analyze the performance of the method \cite{shaw2009structure}.   In particular,
we view the distances in the original (noise free) imagery as the ``true'' distances and judge the algorithm's ability to recover those distances after the imagery has been corrupted by noise.
For both the data and the embedding, we first search $\delta$ nearest neighbors for each point and then produce a weighted graph by treating points in the dataset as nodes and connecting each two neighbors by an edge having the length equal to their Euclidean distance. The weighted graph constructed through the nearest neighbor search is a simple graph\footnote{A simple graph is an undirected graph that does not contain loops (edges connected at both ends to the same vertex ) and multiple edges (more than one edge between any two different vertices) \cite{balakrishnan2012textbook}.} that does not contain self-loops or multiple edges. We compute the $ij$-th entry of the adjacency distance matrix $A$ for the data as
\begin{equation}\label{eqn:adj1}
A_{ij}= \begin{cases}
      	d(i,j) 	& : \text{if} \ \exists \ \text{an edge} \  ij \ \text{in the graph} \\
           	&  \ \ \text{of the original data,}\\
      	0 	& : \text{otherwise,}
   \end{cases}
\end{equation}
and the $ij$-th entry of the adjacency distance matrix $\tilde{A}$ for the embedding data as
\begin{equation}\label{eqn:adj2}
\tilde{A}_{ij}= \begin{cases}
      	d(i,j) 	& : \text{if} \ \exists \ \text{an edge} \  ij \ \text{in the graph} \\
		&  \ \ \text{of the embedding data,}\\
      	0 	& : \text{otherwise.}
   \end{cases}
 \end{equation}
Here, $d(i,j)$ is the Euclidean distance between nodes $i$ and $j$. In this paper, we impose Gaussian noise into real-world datasets such as face images and images of handwritten digits. \emph{Thus, we think of our original data as the uncorrupted data before we impose the noise.}

For $n$ points in the dataset, the error associated with the neighbors' distance is computed as the normalized sum of pairwise absolute differences between entries of the adjacency distance matrices,
\begin{equation}\label{eqn:adj_error}
\text{error}=\frac{1}{n\delta}\sum_{i,j}\big\vert A_{ij}-\tilde{A}_{ij}\big\vert,
\end{equation}
where $\delta$ is the neighbor parameter \cite{shaw2009structure}.

Fig.~\ref{fig:face_images}(c) illustrates the embedding errors of Isomap, denoted by $\mathcal{E}_I$, and SGE, denoted by $\mathcal{E}_S$, for $\sigma= $ 0, 0.1, 0.2, 0.3 and $\mu_s= $ 0, 0.3, 0.6, 0.9, 1.2.  We observe that the error increases in both methods when the noise in the data increases. However, the error of embedding noisy data can be reduced significantly by choosing appropriate non-zero smoothing multipliers in SGE as shown here. Fig.~\ref{fig:face_images}(d) showing the difference of errors ($\mathcal{E}_S$-$\mathcal{E}_S$) demonstrates that SGE performs better in terms of error than Isomap for all the noise levels with $\mu_s \ge 0.3$.
\begin{figure}[hpt]
	\vspace{20pt}
	\begin{subfigure}{0.5\textwidth}
        		\centering
        		\includegraphics[width=3.4in]{figure4a.pdf}
    	\end{subfigure}
	~
	\begin{subfigure}{0.5\textwidth}
        		\centering
        		\includegraphics[width=3.5in]{figure4b.pdf}
    	\end{subfigure}
	~
	\begin{subfigure}{0.5\textwidth}
        		\centering
        		\includegraphics[width=3.5in]{figure4c.pdf}
	\end{subfigure}
	\caption{Embedding of face images ($64\times 64$ dimensional), distorted with different noise levels, using Isomap and SGE with different smoothing levels. (a) A sample of 16 face images \cite{deeplearning}; where the snapshots in the first, second, and third rows, represent left-right light change, left-right pose change, and up-down pose change, respectively. (b) Face images are distorted by imposing three levels of Gaussian noise $\sigma=$ 0.1, 0.2, and 0.3. The datasets (four in total) are embedded using Isomap and then using SGE with smoothing multipliers $\mu_s=$ 0, 0.3 0.6, 0.9, 1.2. Then, (c) the embedding errors of Isomap ($\mathcal{E}_I$) and SGE ($\mathcal{E}_S$), and (c) their error difference are computed.  \emph{The green cells denote that the performance of SGE is superior to that of Isomap.}}\label{fig:face_images}
\end{figure}
\subsection{Embedding of handwritten digits}
Next, we embed handwritten digits available from the Mixed National Institute of Standards and Technology (MNIST) database \cite{mnistdatabase} using SGE and study the performance of the method on this dataset.  This dataset contains 60,000 $28 \times 28$ dimensional images of handwritten digits from 0 to 9. We sample two arbitrary datasets for our study, each with 400 images, such that one dataset has only the digit 2 and the other dataset has the digits 2, 4, 6, and 8.

We run Isomap over the dataset having the digit 2 with $\delta=4$. We run SGE two times: first with $\delta=4$, $\mu_s=0$, $\nu=10\%$, and $h=100$; and second with $\delta=4$, $\mu_s=0.6$, $\nu=10\%$, and $h=100$. Thus, aforesaid procedure yields three two-dimensional embeddings. We formulate the adjacency distance matrices for the data and embedding using Eqns.~(\ref{eqn:adj1}) and (\ref{eqn:adj2}), respectively, and compute the error of embedding using Eqn.~(\ref{eqn:adj_error}). Then, we distort the dataset of 400 images with Gaussian noise having $\sigma=0.2$ and run Isomap with $\delta=4$. We run the noisy dataset two times in SGE: first with parameters $\delta=4$, $\mu_s=0$, $\nu=10\%$, and $h=100$; and second with $\delta=4$, $\mu_s=0.6$, $\nu=10\%$, and $h=100$. The embedding errors for Isomap, SGE with $\mu_s=0$, and SGE with $\mu_s=0.6$, are given in Table~\ref{tab:mnist}(a). We see in this table that, regardless of the noise present in the data, the error associated with SGE without smoothing is greater than that of Isomap, while that of SGE with smoothing is smaller than that of Isomap. Moreover, the error of embedding is increased when moving from the noisy dataset to the noise free dataset by .87 for Isomap, while that is only increased by .24 for SGE with $\mu_s=0.6$.  This is due to the fact that setting the smoothing parameter $\mu$ to $\mu_s=0.6$ allows SGE to recover the manifold corrupted by noisy measurements.

Next, we embed a sample of 400 digits, consisting of 2's, 4's, 6's, and 8's, into two dimensions using Isomap and SGE. We run Isomap over this dataset with $\delta=4$. Then, run SGE two times: first with $\delta=4$, $\nu=10\%$, $\mu_s=0$, and $h=100$; and second with $\delta=4$, $\nu=10\%$, $\mu_s=0$, and $h=100$. Thereafter, we distort the dataset with a Gaussian noise having $\sigma=0.3$ and then run Isomap with $\delta=4$ followed by running SGE with the same two parameter sets that we used before. Then, we compute the Isomap and SGE errors associated with embedding of noise free and noisy datasets using Eqn.~(\ref{eqn:adj_error}) that we present in  Table~\ref{tab:mnist}(b). Similarly to the embedding of the digit 2, regardless of the error in the data, here we also note that the embedding error of SGE \emph{with no smoothing} is greater than that of Isomap, while that of SGE \emph{with smoothing} is smaller than that of Isomap. Moreover, moving from embedding of noise free data to embedding of noisy data, while the error associated with Isomap is increased by .88, that of SGE with $\mu_s=0.9$ is increased only by .21.

Finally, we compare the classification ability of both methods with the presence of high noise. In this dataset, clear clustering of similar digits allows for better classification accuracy.  To demonstrate the desired clustering we present two dimensional Isomap and SGE embeddings of the noisy dataset ($\sigma=0.3$) of digits 2, 4, 6, and 8, that we present in Fig.~\ref{fig:handwring_2468}. Therein, we observe that while Isomap is unable to achieve a clear clustering of digits, SGE with $\mu_s=0.9$ achieves qualitatively better clustering, even under the high noise present in the data.

\begin{table}[htp]
\begin{center}
\begin{adjustbox}{width=.48\textwidth}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{(a)} & \multirow{2}{*}{Noise} & \multirow{2}{*}{Isomap} & \multicolumn{2}{|c|}{SGE}\\
\cline{4-5}
& &  & $\mu_s=0$ & $\mu_s=0.6$\\
\hline
\multirow{2}{*}{Digit ``2"} & $\sigma=0$ &  7.02 & 7.13 & 5.86\\
\cline{2-5}
& $\sigma=0.2$ & 7.89 & 8.19 & 6.10\\
\hline \hline
\multirow{2}{*}{(b)} & \multirow{2}{*}{Noise} & \multirow{2}{*}{Isomap} & \multicolumn{2}{|c|}{SGE}\\
\cline{4-5}
& & & $\mu_s=0$ & $\mu_s=0.9$\\
\hline
Digits ``2", ``4", & $\sigma=0$ & 7.38 & 7.43 & 6.09\\
\cline{2-5}
``6", and ``8" & $\sigma=0.3$ & 8.20 & 8.36 & 6.30\\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\caption {Errors of Isomap and SGE Embeddings of, (a) a sample of 400 handwritten 2's; and (b) a sample of 400 handwritten digits having number 2's, 4's, 6's, and 8's. The first row of (a) shows the error when the  dataset of digit 2 is embedded using both Isomap, and SGE with two smoothing coefficients $\mu_s=0$ and $\mu_s=0.6$. Then, the dataset is imposed with a Gaussian noise of $\sigma=0.2$ and embedded using both Isomap, and SGE with $\mu_s=0$ and $\mu_s=0.6$ that you see in the second row of (a). The first row of (b) represents the errors of Isomap embedding, and SGE embeddings with $\mu_s=0$ and $\mu_s=0.9$, of the noise free version of the sample of digits having the numbers 2, 4, 6, and 8. The second row of (b) represents the errors of Isomap embedding, and SGE embeddings with $\mu_s=0$ and $\mu_s=0.9$, of the noisy version of the dataset created by imposing a Gaussian noise with $\sigma=0.3$.} \label{tab:mnist}
\end{table}

\begin{figure}[htp]
        	\centering
        	\includegraphics[width=3.6in]{figure5.pdf}
        	\caption{Isomap and SGE Embeddings of handwritten numbers 2, 4, 6, and 8. Different digits are shown in different colors (2's in green, 4's in orange, 6's in brown, and 8's in gray) in the two-dimensional embedding space and the embedded snapshots illustrate the appearance of arbitrarily chosen handwritten digits. The left panel shows the Isomap embedding of the noisy dataset ($\sigma=0.3$) and the right panel shows the SGE embedding of the same dataset with $\mu_s=0.9$. The embedding error of each case is indicated in the title of the corresponding panel. Note that, qualitatively speaking, the SGE embedding appears to have better clustering of similar digits than that of Isomap.}
	\label{fig:handwring_2468}
\end{figure}

\section{Conclusion}\label{sec:conclusion}
Nonlinear dimensionality reduction methods can recover unfaithful embeddings due to presence of  high noise in the data.  In order to obtain a faithful embedding for noisy data, some smoothing procedure should be performed in the embedding. With this idea in mind, herein we introduced a novel nonlinear dimensionality reduction framework using smooth geodesics that emphasizes the underlying smoothness of the manifold. Our method begins by first searching for nearest neighbors for each point using a $\delta$-nearest neighbor search \cite{friedman1977algorithm}. Then, we create a weighted graph by representing all of the points as nodes and joining neighboring nodes with edges having their Euclidean distances as weights. For each pair of nodes in the graph, we create a geodesic \cite{tenenbaum2000global}, that is defined as the shortest path between the given nodes generated using Floyd's algorithm \cite{cormen2009introduction}. We fit each such geodesic with a smoothing spline (called a smooth geodesic) with smoothing multiplier ($\mu_s$) and spline threshold ($\nu$) \cite{reinsch1967smoothing, de1972calculating}. The length of these splines are treated as manifold distances between corresponding points. Finally, we use the classic MDS method to find the dimension of the distance matrix of smooth geodesics and perform the embedding.

In SGE, the order of the spline fit is set to either three, two, or one, depending on the spline threshold. Since a sufficient smoothness and a low fitting error can be obtained by cubic smoothing splines, we first rely on a spline fit of  order three. However, we observed that the smoothing spline routing in \cite{de1972calculating} fits very long cubic splines for some specific smoothing multipliers. Thus, if the length of a cubic smoothing spline doesn't satisfy the threshold, we reduce the order of the spline to the next level.

We first demonstrated the effectiveness of our technique on a synthetic dataset representing a section of a semi-sphere.  We observed that smoothing approach provides better performance than what is achieved by standard Isomap when embedding a noisy dataset. We also observed that the errors in both methods decrease as the neighborhood size increases [Fig.~\ref{fig:comp_embedding}(a) and (b)].  However, when the neighborhood size is small, say $\delta=2$, SGE has clear performance advantages over Isomap for noisy data when a sufficient smoothness is employed [Fig.~\ref{fig:comp_embedding}(c)]. The spherical dataset also demonstrated that SGE is more robust to sparse sampling than Isomap [Fig.~\ref{fig:analysis_n}(a)]. Moreover, while increasing noise in the data always appears to reduce the performance of the embedding, irrespective to the method that is used, we see that Isomap is highly effected by increasing noise while SGE, with a judicious choice of smoothing parameter, is more robust [Fig.~\ref{fig:analysis_n}(b)].

We also studied two standard benchmark data sets, face images and handwritten digit images, and found that SGE provided similar superior performance on noisy versions of those data sets.  In particular, for the digit classification task, we observed that SGE provides qualitatively superior clustering of similar digits in the presence of noise. As future work, we will quantify the classification performance of the low dimensional nonlinear embedding using a variety of standard supervised machine learning techniques.

The NDR method that we introduced here ensures better performance and preserves the topology of the manifold by emphasizing the smoothness of the manifold when embedding noisy data. This method is an extension of famous NDR method Isomap where we replaced the geodesics with smoothing splines.  In the future, we plan to examine such techniques in more generality.  For example, one can imagine generalizing Isomap to the case where geodesics are not a good approximation of long manifold distances.  In such a case, one can attempt to treat the long manifold distances as unknown, and employ matrix completion techniques on distance matrices where some entries are not observed.

\ifCLASSOPTIONcompsoc

  \section*{Acknowledgments}
\else

  \section*{Acknowledgment}
\fi
The author's would like to thank Chen Zou for the support given in coding and
also would like to thank the NSF XSEDE Jetstream
\cite{stewart2015jetstream,towns2014xsede} under allocation TG-DMS160019 for
support of this work.
\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

\bibliographystyle{IEEEtran}