Anchored Regression: Solving Random Convex Equations via Convex Programming

We consider the problem of estimating a solution to (random) systems of equations that involve convex nonlinearities which has applications in machine learning and signal processing. Conventional estimators based on empirical risk minimization generally lead to non-convex programs that are often computationally intractable. We propose anchored regression, a new approach that utilizes an anchor vector to formulate an estimator based on a simple convex program. We analyze accuracy of this method and specify the required sample complexity. The proposed convex program is formulated in the natural space of the problem rather than a lifted domain, which makes it computationally favorable. This feature of anchored regression also provides great flexibility as structural priors (e.g., sparsity) can be seamlessly incorporated through convex regularization. We also provide recipes for constructing the anchor vector from the data.