Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tight bounds for learning a mixture of two gaussians   We consider the problem of identifying the parameters of an unknown mixture
of two arbitrary $d$-dimensional gaussians from a sequence of independent
random samples. Our main results are upper and lower bounds giving a
computationally efficient moment-based estimator with an optimal convergence
rate, thus resolving a problem introduced by Pearson (1894). Denoting by
$\sigma^2$ the variance of the unknown mixture, we prove that
$\Theta(\sigma^{12})$ samples are necessary and sufficient to estimate each
parameter up to constant additive error when $d=1.$ Our upper bound extends to
arbitrary dimension $d>1$ up to a (provably necessary) logarithmic loss in $d$
using a novel---yet simple---dimensionality reduction technique. We further
identify several interesting special cases where the sample complexity is
notably smaller than our optimal worst-case bound. For instance, if the means
of the two components are separated by $\Omega(\sigma)$ the sample complexity
reduces to $O(\sigma^2)$ and this is again optimal.
  Our results also apply to learning each component of the mixture up to small
error in total variation distance, where our algorithm gives strong
improvements in sample complexity over previous work. We also extend our lower
bound to mixtures of $k$ Gaussians, showing that $\Omega(\sigma^{6k-2})$
samples are necessary to estimate each parameter up to constant additive error.

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The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
2
The categories of this paper are: math
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol