Improved analysis of higher order random walks and applications

The motivation of this work is to extend the techniques of higher order random walks on simplicial complexes to analyze mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second eigenvalue of the down-up walk on a pure simplicial complex, in terms of the second eigenvalues of its links. We show some applications of this result in analyzing mixing times of Markov chains, including sampling independent sets of a graph and sampling common independent sets of two partition matroids.

[1]  Dror Weitz,et al.  Counting independent sets up to the tree threshold , 2006, STOC '06.

[2]  R. Meshulam,et al.  Homological connectivity of random k-dimensional complexes , 2009, Random Struct. Algorithms.

[3]  Eric Vigoda,et al.  Improved bounds for sampling colorings , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[4]  Tomás Feder,et al.  Balanced matroids , 1992, STOC '92.

[5]  Roy Meshulam,et al.  The Clique Complex and Hypergraph Matching , 2001, Comb..

[6]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[7]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[8]  Tali Kaufman,et al.  Bounded degree cosystolic expanders of every dimension , 2015, STOC.

[9]  Ravi Montenegro,et al.  Mathematical Aspects of Mixing Times in Markov Chains , 2006, Found. Trends Theor. Comput. Sci..

[10]  Irit Dinur,et al.  High Dimensional Expanders Imply Agreement Expanders , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[11]  Nathan Linial,et al.  Homological Connectivity Of Random 2-Complexes , 2006, Comb..

[12]  Tali Kaufman,et al.  High Order Random Walks: Beyond Spectral Gap , 2017, APPROX-RANDOM.

[13]  Nima Anari,et al.  Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[14]  Nima Anari,et al.  Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid , 2018, STOC.

[15]  Karim A. Adiprasito,et al.  Hodge theory for combinatorial geometries , 2015, Annals of Mathematics.

[16]  S. Bobkov,et al.  Modified Logarithmic Sobolev Inequalities in Discrete Settings , 2006 .

[17]  Noga Alon,et al.  lambda1, Isoperimetric inequalities for graphs, and superconcentrators , 1985, J. Comb. Theory, Ser. B.

[18]  Elizabeth L. Wilmer,et al.  Markov Chains and Mixing Times , 2008 .

[19]  Tali Kaufman,et al.  Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders , 2014, ArXiv.

[20]  Eli Upfal,et al.  Probability and Computing: Randomized Algorithms and Probabilistic Analysis , 2005 .

[21]  Tali Kaufman,et al.  High dimensional expanders and property testing , 2014, ITCS.

[22]  Izhar Oppenheim,et al.  Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps , 2014, Discret. Comput. Geom..

[23]  Mark Jerrum,et al.  A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph , 1995, Random Struct. Algorithms.

[24]  Heng Guo,et al.  Modified log-Sobolev Inequalities for Strongly Log-Concave Distributions , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[25]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[26]  Madhur Tulsiani,et al.  Approximating Constraint Satisfaction Problems on High-Dimensional Expanders , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[27]  Yotam Dikstein,et al.  Boolean function analysis on high-dimensional expanders , 2018, Electron. Colloquium Comput. Complex..

[28]  Yotam Dikstein,et al.  Agreement Testing Theorems on Layered Set Systems , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[29]  Ron Aharoni,et al.  The intersection of a matroid and a simplicial complex , 2006 .

[30]  P. Diaconis,et al.  LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS , 1996 .

[31]  Tali Kaufman,et al.  High Dimensional Random Walks and Colorful Expansion , 2016, ITCS.

[32]  Tali Kaufman,et al.  Walking on the Edge and Cosystolic Expansion , 2016, ArXiv.

[33]  V. Climenhaga.  Markov chains and mixing times , 2013 .

[34]  Tali Kaufman,et al.  On Expansion and Topological Overlap , 2016, Symposium on Computational Geometry.

[35]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[36]  Alexander Lubotzky,et al.  Explicit constructions of Ramanujan complexes of type , 2005, Eur. J. Comb..

[37]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[38]  Madhur Tulsiani,et al.  List Decoding of Direct Sum Codes , 2020, SODA.

[39]  Eric Vigoda,et al.  Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains , 2004, math/0503537.

[40]  Alistair Sinclair,et al.  Algorithms for Random Generation and Counting: A Markov Chain Approach , 1993, Progress in Theoretical Computer Science.

[41]  Martin E. Dyer,et al.  Path coupling: A technique for proving rapid mixing in Markov chains , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[42]  Botong Wang,et al.  Enumeration of points, lines, planes, etc , 2016, 1609.05484.

[43]  János Pach,et al.  Overlap properties of geometric expanders , 2011, SODA '11.

[44]  M. Gromov.  Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry , 2010 .

[45]  D. Aldous.  Random walks on finite groups and rapidly mixing markov chains , 1983 .

[46]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries , 2004, JACM.

[47]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[48]  Sidhanth Mohanty,et al.  High-Dimensional Expanders from Expanders , 2019, ITCS.

[49]  J. Cheeger.  A lower bound for the smallest eigenvalue of the Laplacian , 1969 .

[50]  Santosh S. Vempala,et al.  Simulated annealing in convex bodies and an O*(n4) volume algorithm , 2006, J. Comput. Syst. Sci..

[51]  Amnon Ta-Shma,et al.  List Decoding with Double Samplers , 2018, Electron. Colloquium Comput. Complex..

[52]  Ori Parzanchevski,et al.  Isoperimetric inequalities in simplicial complexes , 2012, Comb..

[53]  Thomas P. Hayes.  A simple condition implying rapid mixing of single-site dynamics on spin systems , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[54]  Martin E. Dyer,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991, JACM.