Adaptive Sparse M\"obius Transforms for Learning Polynomials

2026-02-05

AI Summary

This work addresses the challenge of exactly learning an $s$-sparse, degree-$d$ real-valued Boolean polynomial over $\{0,1\}^n$—equivalently, recovering its Möbius transform in the AND basis—where high coherence among basis vectors renders conventional compressed sensing methods ineffective. The authors introduce adaptive group testing to this setting for the first time, proposing two novel algorithms for Möbius inversion: Fully Adaptive Sparse Möbius Transform (FASMT) and Partially Adaptive Sparse Möbius Transform (PASMT). FASMT achieves a near-optimal query complexity of $O(sd \log(n/d))$, while PASMT reduces the number of adaptive rounds to $O(d^2 \log(n/d))$, independent of $s$. Both methods synergistically combine sparse recovery with combinatorial group testing, significantly outperforming existing approaches in hypergraph reconstruction and overcoming the theoretical and algorithmic barriers imposed by basis coherence.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We consider the problem of exactly learning an $s$-sparse real-valued Boolean polynomial of degree $d$ of the form $f:\{ 0,1\}^n \rightarrow \mathbb{R}$. This problem corresponds to decomposing functions in the AND basis and is known as taking a M\"obius transform. While the analogous problem for the parity basis (Fourier transform) $f: \{-1,1 \}^n \rightarrow \mathbb{R}$ is well-understood, the AND basis presents a unique challenge: the basis vectors are coherent, precluding standard compressed sensing methods. We overcome this challenge by identifying that we can exploit adaptive group testing to provide a constructive, query-efficient implementation of the M\"obius transform (also known as M\"obius inversion) for sparse functions. We present two algorithms based on this insight. The Fully-Adaptive Sparse M\"obius Transform (FASMT) uses $O(sd \log(n/d))$ adaptive queries in $O((sd + n) sd \log(n/d))$ time, which we show is near-optimal in query complexity. Furthermore, we also present the Partially-Adaptive Sparse M\"obius Transform (PASMT), which uses $O(sd^2\log(n/d))$ queries, trading a factor of $d$ to reduce the number of adaptive rounds to $O(d^2\log(n/d))$, with no dependence on $s$. When applied to hypergraph reconstruction from edge-count queries, our results improve upon baselines by avoiding the combinatorial explosion in the rank $d$. We demonstrate the practical utility of our method for hypergraph reconstruction by applying it to learning real hypergraphs in simulations.

Problem

• Sparse polynomial learning
• Möbius transform
• AND basis
• Boolean functions
• Hypergraph reconstruction

Innovation

• Adaptive Group Testing
• Sparse Möbius Transform
• Boolean Polynomial Learning
• Hypergraph Reconstruction
• Query Complexity