Adaptive Sparse Möbius Transforms for Learning Polynomials
This work addresses a fundamental challenge in learning sparse functions for researchers in computational learning theory and related fields, offering incremental improvements over existing methods for specific applications like hypergraph reconstruction.
The paper tackles the problem of exactly learning sparse real-valued Boolean polynomials in the AND basis, which is challenging due to coherent basis vectors, by developing adaptive algorithms that exploit group testing. The results include the FASMT algorithm with O(sd log(n/d)) adaptive queries and near-optimal query complexity, and the PASMT algorithm with O(sd^2 log(n/d)) queries, improving hypergraph reconstruction by avoiding combinatorial explosion in rank d.
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 Möbius 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 Möbius transform (also known as Möbius inversion) for sparse functions. We present two algorithms based on this insight. The Fully-Adaptive Sparse Möbius 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 Möbius 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.