Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Learning a mixture of two subspaces over finite fields   We study the problem of learning a mixture of two subspaces over
$\mathbb{F}_2^n$. The goal is to recover the individual subspaces, given
samples from a (weighted) mixture of samples drawn uniformly from the two
subspaces $A_0$ and $A_1$.
  This problem is computationally challenging, as it captures the notorious
problem of "learning parities with noise" in the degenerate setting when $A_1
\subseteq A_0$. This is in contrast to the analogous problem over the reals
that can be solved in polynomial time (Vidal'03). This leads to the following
natural question: is Learning Parities with Noise the only computational
barrier in obtaining efficient algorithms for learning mixtures of subspaces
over $\mathbb{F}_2^n$?
  The main result of this paper is an affirmative answer to the above question.
Namely, we show the following results: 1. When the subspaces $A_0$ and $A_1$
are incomparable, i.e., $A_0$ and $A_1$ are not contained inside each other,
then there is a polynomial time algorithm to recover the subspaces $A_0$ and
$A_1$. 2. In the case when $A_1$ is a subspace of $A_0$ with a significant gap
in the dimension i.e., $dim(A_1) \le \alpha dim(A_0)$ for $\alpha<1$, there is
a $n^{O(1/(1-\alpha))}$ time algorithm to recover the subspaces $A_0$ and
$A_1$.
  Thus, our algorithms imply computational tractability of the problem of
learning mixtures of two subspaces, except in the degenerate setting captured
by learning parities with noise.

26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
2
The categories of this paper are: math
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol