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Isotropy and Log-Concave Polynomials: Accelerated Sampling and High-Precision Counting of Matroid Bases [article]

Nima Anari, Michał Dereziński
2020 arXiv   pre-print
We design a new approximate sampling algorithm that leverages isotropy for the class of distributions μ that have a log-concave generating polynomial; this class includes determinantal point processes,  ...  As an application of our results, we show how to approximately count bases of a matroid of rank k over a ground set of n elements to within a factor of 1+ϵ in time O((n+1/ϵ^2)· poly(k, log n)).  ...  We then use µ −i (U) as our estimate for Z −i and as before estimate the ratios Z −(i−1) /Z −i and so on using empirical means of unbiased estimators.  ... 
arXiv:2004.09079v1 fatcat:uoa57mwnxjgiha6zi37cwbu33q

Domain Sparsification of Discrete Distributions using Entropic Independence [article]

Nima Anari, Michał Dereziński, Thuy-Duong Vuong, Elizabeth Yang
2021 arXiv   pre-print
Our work significantly extends the reach of prior work of Anari and Dereziński who obtained domain sparsification for distributions with a log-concave generating polynomial (corresponding to α=1).  ...  As a corollary of our new analysis techniques, we also obtain a less stringent requirement on the accuracy of marginal estimates even for the case of log-concave polynomials; roughly speaking, we show  ...  Since we have a uniform distribution over matroid bases, µD k→(d+1) is log-concave, and thus it satisfies 1-entropic independence.  ... 
arXiv:2109.06442v2 fatcat:bpblgidim5brrebiyqmd6iktpm