Year: 2014
Type: preprint
Abstract: Let $f:\{-1,1\}^n$ be a polynomial with at most $s$ non-zero real coefficients. We give an algorithm for exactly reconstructing f given random examples from the uniform distribution on $\{-1,1\}^n$ that runs in time polynomial in $n$ and $2s$ and succeeds if the function satisfies the unique sign property: there is one output value which corresponds to a unique set of values of the participating parities. This sufficient condition is satisfied when every coefficient of f is perturbed by a small random noise, or satisfied with high probability when s parity functions are chosen randomly or when all the coefficients are positive. Learning sparse polynomials over the Boolean domain in time polynomial in $n$ and $2s$ is considered notoriously hard in the worst-case. Our result shows that the problem is tractable for almost all sparse polynomials. Then, we show an application of this result to hypergraph sketching which is the problem of learning a sparse (both in the number of hyperedges and the size of the hyperedges) hypergraph from uniformly drawn random cuts. We also provide experimental results on a real world dataset.
Source: arXiv (Cornell University)
Authors: Kocaoglu, Murat, Shanmugam, Karthikeyan, Dimakis, Alexandros G., Klivans, Adam
Institution: The University of Texas at Austin
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2108013602