Learning Set Functions that are Sparse in Non-Orthogonal Fourier Bases

TL;DR

This paper introduces new algorithms for learning set functions that are sparse in non-orthogonal Fourier bases, applicable to real-world problems like recommender systems, with improved query efficiency.

Contribution

It presents a novel family of algorithms for learning Fourier-sparse set functions using non-orthogonal Fourier transforms, extending prior work focused on orthogonal bases.

Findings

Algorithms require at most nk - k log2 k + k queries

Effective in modeling substitutes and complements in item sets

Demonstrated success on real-world applications

Abstract

Many applications of machine learning on discrete domains, such as learning preference functions in recommender systems or auctions, can be reduced to estimating a set function that is sparse in the Fourier domain. In this work, we present a new family of algorithms for learning Fourier-sparse set functions. They require at most $nk - k \log_2 k + k$ queries (set function evaluations), under mild conditions on the Fourier coefficients, where $n$ is the size of the ground set and $k$ the number of non-zero Fourier coefficients. In contrast to other work that focused on the orthogonal Walsh-Hadamard transform, our novel algorithms operate with recently introduced non-orthogonal Fourier transforms that offer different notions of Fourier-sparsity. These naturally arise when modeling, e.g., sets of items forming substitutes and complements. We demonstrate effectiveness on several real-world applications.