## Approximating the Permanent with Deep Rejection Sampling

21 May 2021, 20:42 (modified: 15 Jan 2022, 18:47)NeurIPS 2021 PosterReaders: Everyone
Keywords: approximation scheme, matrix permanent, practice, rejection sampling
TL;DR: We present a technique for boosting the acceptance–rejection method with superior empirical performance and analytic bounds.
Abstract: We present a randomized approximation scheme for the permanent of a matrix with nonnegative entries. Our scheme extends a recursive rejection sampling method of Huber and Law (SODA 2008) by replacing the permanent upper bound with a linear combination of the subproblem bounds at a moderately large depth of the recursion tree. This method, we call deep rejection sampling, is empirically shown to outperform the basic, depth-zero variant, as well as a related method by Kuck et al. (NeurIPS 2019). We analyze the expected running time of the scheme on random \$(0, 1)\$-matrices where each entry is independently \$1\$ with probability \$p\$. Our bound is superior to a previous one for \$p\$ less than \$1/5\$, matching another bound that was only known to hold when every row and column has density exactly \$p\$.
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Code: https://github.com/Kalakuh/deepar
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