Adaptive Sparse Möbius Transforms for Learning Polynomials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908816760635392 |
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| author | Erginbas, Yigit Efe Kang, Justin Singh Polito, Elizabeth Ramchandran, Kannan |
| author_facet | Erginbas, Yigit Efe Kang, Justin Singh Polito, Elizabeth Ramchandran, Kannan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06246 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Adaptive Sparse Möbius Transforms for Learning Polynomials Erginbas, Yigit Efe Kang, Justin Singh Polito, Elizabeth Ramchandran, Kannan Machine Learning 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. |
| title | Adaptive Sparse Möbius Transforms for Learning Polynomials |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.06246 |