A Novel Convex Gaussian Min Max Theorem for Repeated Features

The Convex Gaussian Min-Max Theorem (CGMT) allows for the study of min-max optimization problems over bilinear Gaussian forms by instead considering an alternative optimization problem whose statistical properties are tied to that of the primary optimization. We prove a generalization of the CGMT to a family of problems in machine learning (ML) with correlated entries in the data matrix. This family includes various familiar examples of problems with shared weights or repeated features. In particular, we make use of our theorem to obtain asymptotically exact learning curves for regression with vector valued labels, regression with complex variables, and regression with convolution.