Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Too many secants: a hierarchical approach to secant-based dimensionality
  reduction on large data sets   A fundamental question in many data analysis settings is the problem of
discerning the "natural" dimension of a data set. That is, when a data set is
drawn from a manifold (possibly with noise), a meaningful aspect of the data is
the dimension of that manifold. Various approaches exist for estimating this
dimension, such as the method of Secant-Avoidance Projection (SAP).
Intuitively, the SAP algorithm seeks to determine a projection which best
preserves the lengths of all secants between points in a data set; by applying
the algorithm to find the best projections to vector spaces of various
dimensions, one may infer the dimension of the manifold of origination. That
is, one may learn the dimension at which it is possible to construct a
diffeomorphic copy of the data in a lower-dimensional Euclidean space. Using
Whitney's embedding theorem, we can relate this information to the natural
dimension of the data. A drawback of the SAP algorithm is that a data set with
$T$ points has $O(T^2)$ secants, making the computation and storage of all
secants infeasible for very large data sets. In this paper, we propose a novel
algorithm that generalizes the SAP algorithm with an emphasis on addressing
this issue. That is, we propose a hierarchical secant-based
dimensionality-reduction method, which can be employed for data sets where
explicitly calculating all secants is not feasible.

36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph