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.