Abstract

We consider the question of estimating a solution to a system of equations that involve convex nonlinearities, a problem that is common in machine learning and signal processing. Because of these nonlinearities, conventional estimators based on empirical risk minimization generally involve solving a non-convex optimization program. We propose a method (called “anchored regression”) that is based on convex programming and amounts to maximizing a linear functional (perhaps augmented by a regularizer) over a convex set. The proposed convex program is formulated in the natural space of the problem, and avoids the introduction of auxiliary variables, making it computationally favorable. Working in the native space also provides us with the flexibility to incorporate structural priors (e.g., sparsity) on the solution. For our analysis, we model the equations as being drawn from a fixed set according to a probability law. Our main results provide guarantees on the accuracy of the estimator in terms of the number of equations weare solving, the amount of noise present, a measure of statistical complexity of the random equations, and thegeometry of the regularizer at the true solution. We also provide recipes for constructing the anchor vector (that determines the linear functional to maximize) directly from the observed data. We will discuss applications of this technique to nonlinear problems including phase retrieval, blind deconvolution, and inverting the action of a neural network. This is joint work with Sohail Bahmani.