Learning a mixture of two subspaces over finite fields
Learning a mixture of two subspaces over finite fields
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发表时间：
2020-10
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通讯作者：
Aidao Chen;Anindya De;Aravindan Vijayaraghavan
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作者：
Aidao Chen;Anindya De;Aravindan Vijayaraghavan
我们研究了在$\mathbb{F}_2^n$上学习两个子空间混合的问题。目标是从两个子空间$A_0$和$A_1$中均匀抽取的样本的（加权）混合物中恢复单个子空间。这个问题在计算上具有挑战性，因为它抓住了退化设置中“带噪声的学习奇偶”的臭名昭著的问题，当$A_1 \subseteq A_0$。这与可以在多项式时间内解决的实数上的类似问题（Vidal'03）形成对比。这就引出了一个自然的问题：带噪声的学习奇偶是在$\mathbb{F}_2^n$上获得学习混合子空间的有效算法的唯一计算障碍吗？本文的主要成果是对上述问题的肯定回答。即，我们显示了以下结果：当子空间$A_0$和$A_1$不具有可比性，即$A_0$和$A_1$不包含在彼此内部时，则使用多项式时间算法来恢复子空间$A_0$和$A_1$。2. 如果$A_1$是$A_0$的子空间，并且维度上有很大的差距，即$dim(A_1) \le \alpha dim(A_1)$对应$\alpha<1$，则有一个$n^{O(1/(1-\alpha))}$时间算法来恢复子空间$A_0$和$A_1$。因此，我们的算法暗示了两个子空间的学习混合问题的计算可跟踪性，除了在由带噪声的学习偶捕获的退化设置中。
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., $dim(A_1) \le \alpha dim(A_1)$ 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.