Seminars
View all Seminars | Download ICal for this eventMoments, Random Walks, and Limits for Spectrum Approximation
Series: Bangalore Theory Seminars
Speaker: Apoorv Vikram Singh Ph.D student NYU
Date/Time: Jul 17 14:00:00
Location: CSA Seminar Hall (Room No. 254, First Floor)
Abstract:
We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments. We show that there are distributions on [??1, 1] that cannot be approximated to accuracy ε in Wasserstein-1 distance even if we know all of their moments to multiplicative accuracy (1 ± 2??Ω(1/ε)); this result matches an upper bound of Kong and Valiant [Annals of Statistics, 2017]. To obtain our result, we provide a hard instance involving distributions induced by the eigenvalue spectra of carefully constructed graph adjacency matrices. Efficiently approximating such spectra in Wasserstein-1 distance is a well-studied algorithmic problem, and a recent result of Cohen-Steiner et al. [KDD 2018] gives a method based on accurately approximating spectral moments using 2O(1/ε) random walks initiated at uniformly random nodes in the graph.
As a strengthening of our main result, we show that improving the dependence on 1/ε in this result would require a new algorithmic approach. Specifically, no algorithm can compute an ε-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of 2Ω(1/ε) random walks of length 2Ω(1/ε) started at random nodes.
Based on joint work with Yujia Jin, Christopher Musco, and Aaron Sidford.
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We are grateful to the Kirani family for generously supporting the theory seminar series
Hosts: Rameesh Paul, Rachana Gusain, Rahul Madhavan, KVN Sreenivas