Basis Learning as an Algorithmic Primitive

A number of important problems in theoretical computer science and machine learning can be interpreted as recovering a certain basis. These include symmetric matrix eigendecomposition, certain tensor decompositions, Independent Component Analysis (ICA), spectral clustering and Gaussian mixture learning. Each of these problems reduces to an instance of our general model, which we call a “Basis Encoding Function" (BEF). We show that learning a basis within this model can then be provably and efficiently achieved using a first order iteration algorithm (gradient iteration). Our algorithm goes beyond tensor methods while generalizing a number of existing algorithms—e.g., the power method for symmetric matrices, the tensor power iteration for orthogonal decomposable tensors, and cumulant-based FastICA—all within a broader function-based dynamical systems framework. Our framework also unifies the unusual phenomenon observed in these domains that they can be solved using efficient non-convex optimization. Specifically, we describe a class of BEFs such that their local maxima on the unit sphere are in one-to-one correspondence with the basis elements. This description relies on a certain “hidden convexity" property of these functions. We provide a complete theoretical analysis of the gradient iteration even when the BEF is perturbed. We show convergence and complexity bounds polynomial in dimension and other relevant parameters, such as perturbation size. Our perturbation results can be considered as a non-linear version of the classical Davis-Kahan theorem for perturbations of eigenvectors of symmetric matrices. In addition we show that our algorithm exhibits fast (superlinear) convergence and relate the speed of convergence to the properties of the BEF. Moreover, the gradient iteration algorithm can be easily and efficiently implemented in practice.