A Novel Gaussian Min-Max Theorem and its Applications
This incremental advance improves theoretical tools for high-dimensional statistics and machine learning, benefiting researchers in fields like signal processing and optimization.
The authors extended the classical Gaussian min-max and convex Gaussian min-max theorems to handle Gaussian matrices with independent but non-identically-distributed rows, and applied this to multi-source Gaussian regression and binary classification of Gaussian mixture models, achieving results like exact asymptotic error characterization.
A celebrated result by Gordon allows one to compare the min-max behavior of two Gaussian processes if certain inequality conditions are met. The consequences of this result include the Gaussian min-max (GMT) and convex Gaussian min-max (CGMT) theorems which have had far-reaching implications in high-dimensional statistics, machine learning, non-smooth optimization, and signal processing. Both theorems rely on a pair of Gaussian processes, first identified by Slepian, that satisfy Gordon's comparison inequalities. In this paper, we identify such a new pair. The resulting theorems extend the classical GMT and CGMT Theorems from the case where the underlying Gaussian matrix in the primary process has iid rows to where it has independent but non-identically-distributed ones. The new CGMT is applied to the problems of multi-source Gaussian regression, as well as to binary classification of general Gaussian mixture models.