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The gauge function, closely related to the atomic norm, measures thecomplexity of a statistical model, and has found broad applications in machinelearning and statistical signal processing. In a high-dimensional learningproblem, the gauge function attempts to safeguard against overfitting bypromoting a sparse (concise) representation within the learning alphabet.  In this work, within the context of linear inverse problems, we pinpoint thesource of its success, but also argue that the applicability of the gaugefunction is inherently limited by its convexity, and showcase several learningproblems where the classical gauge function theory fails. We then introduce anew notion of statistical complexity, gauge$_p$ function, which overcomes thelimitations of the gauge function. The gauge$_p$ function is a simplegeneralization of the gauge function that can tightly control the sparsity of astatistical model within the learning alphabet and, perhaps surprisingly, drawsfurther inspiration from the Burer-Monteiro factorization in computationalmathematics.  We also propose a new learning machine, with the building block of gauge$_p$function, and arm this machine with a number of statistical guarantees. Thepotential of the proposed gauge$_p$ function theory is then studied for twostylized applications. Finally, we discuss the computational aspects and, inparticular, suggest a tractable numerical algorithm for implementing the newlearning machine.

The Nonconvex Geometry of Linear Inverse Problems