On a group of invariances in a class of functions

TL;DR

This paper investigates invariances in a specific class of parametric functions composed of polynomial and monomial maps, providing a detailed characterization that can aid in sparse representations, obfuscation, and optimization.

Contribution

It identifies and characterizes nontrivial invariances in a class of functions formed by compositions of polynomials and monomials, offering a constructive description of their structure.

Findings

Identified nontrivial invariances in the function class.

Provided a constructive description of invariance structure.

Implications for sparse representations and optimization.

Abstract

A class of parametric functions formed by alternating compositions of multivariate polynomials and rectification style monomial maps is studied (the layer-wise exponents are treated as fixed hyperparameters and are not optimized). For this family, nontrivial parametric invariances are identified and characterized, i.e., distinct parameter settings that induce identical input-output maps. A constructive description of the invariance structure is provided, enabling sparse function representations, parameter obfuscation, and potential dimensionality reduction for optimization.