16,660 research outputs found
Mixing times of lozenge tiling and card shuffling Markov chains
We show how to combine Fourier analysis with coupling arguments to bound the
mixing times of a variety of Markov chains. The mixing time is the number of
steps a Markov chain takes to approach its equilibrium distribution. One
application is to a class of Markov chains introduced by Luby, Randall, and
Sinclair to generate random tilings of regions by lozenges. For an L X L region
we bound the mixing time by O(L^4 log L), which improves on the previous bound
of O(L^7), and we show the new bound to be essentially tight. In another
application we resolve a few questions raised by Diaconis and Saloff-Coste, by
lower bounding the mixing time of various card-shuffling Markov chains. Our
lower bounds are within a constant factor of their upper bounds. When we use
our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an
O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov
chain for linear extensions.Comment: 39 pages, 8 figure
Random Walks on Hypergraphs with Edge-Dependent Vertex Weights
Hypergraphs are used in machine learning to model higher-order relationships
in data. While spectral methods for graphs are well-established, spectral
theory for hypergraphs remains an active area of research. In this paper, we
use random walks to develop a spectral theory for hypergraphs with
edge-dependent vertex weights: hypergraphs where every vertex has a weight
for each incident hyperedge that describes the contribution
of to the hyperedge . We derive a random walk-based hypergraph
Laplacian, and bound the mixing time of random walks on such hypergraphs.
Moreover, we give conditions under which random walks on such hypergraphs are
equivalent to random walks on graphs. As a corollary, we show that current
machine learning methods that rely on Laplacians derived from random walks on
hypergraphs with edge-independent vertex weights do not utilize higher-order
relationships in the data. Finally, we demonstrate the advantages of
hypergraphs with edge-dependent vertex weights on ranking applications using
real-world datasets.Comment: Accepted to ICML 201
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