Linear classifiers are a basic model in supervised learning. We study the problem of learning a mixture of linear classifiers over Gaussian marginals. Despite significant interest in this problem, including in the context of neural networks, basic questions like efficient learnability and identifiability of the model remained open. In this paper, we design algorithms for recovering the parameters of the mixture of $k$ linear classifiers. We obtain two algorithms which both have polynomial dependence on the ambient dimension $n$, and incur an exponential dependence either on the number of the components $k$ or a natural separation parameter $\Delta>0$. These algorithmic results in particular settle the identifiability question under provably minimal assumptions.

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.

Smoothed analysis is a powerful paradigm in overcoming worst-case intractability in unsupervised learning and high-dimensional data analysis. While polynomial time smoothed analysis guarantees have been obtained for worst-case intractable problems like tensor decompositions and learning mixtures of Gaussians, such guarantees have been hard to obtain for several other important problems in unsupervised learning. A core technical challenge in analyzing algorithms is obtaining lower bounds on the least singular value for random matrix ensembles with dependent entries, that are given by low-degree polynomials of a few base underlying random variables. In this work, we address this challenge by obtaining high-confidence lower bounds on the least singular value of new classes of structured random matrix ensembles of the above kind. We then use these bounds to design algorithms with polynomial time smoothed analysis guarantees for the following three important problems in unsupervised learning: (1) Robust subspace recovery, when the fraction of inliers in the d-dimensional subspace T of the n-dimensional Euclidean space is at least (d/n)tfor any positive integer t. This contrasts with the known worst-case intractability when the fraction of inliers is at most d/n, and the previous smoothed analysis result (Hardt and Moitra, 2013). (2) Learning overcomplete hidden markov models, where the size of the state space is any polynomial in the dimension of the observations. This gives the first polynomial time guarantees for learning overcomplete HMMs in the smoothed analysis model. (3) Higher order tensor decompositions, where we generalize and analyze the so-called FOOBI algorithm of Cardoso to find order-t rank-one tensors in a subspace. This gives polynomially robust decomposition algorithms for order-2t tensors with rank nt.