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## Tuesday, December 02, 2014

### Sparse Polynomial Learning and Graph Sketching

Very interesting connection between solving a CS problem  and sketching graph from random queries. I wonder it could help in devising biochemical networks.

Sparse Polynomial Learning and Graph Sketching by Murat Kocaoglu, Karthikeyan Shanmugam,  Alexandros G. Dimakis, Adam Klivans

Let f: \{-1,1\}^n \rightarrow \mathbb{R} 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 2^{s} and succeeds if the function satisfies the \textit{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 2^{s} 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.
earlier: Learning Fourier Sparse Set Functions by Peter Stobbe Andreas Krause

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