Expanderizing Higher Order Random Walks
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arXiv
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| Format: | Preprint |
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2024
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| author | Alev, Vedat Levi Rao, Shravas |
| author_facet | Alev, Vedat Levi Rao, Shravas |
| contents | We study a variant of the down-up and up-down walks over an $n$-partite simplicial complex, which we call expanderized higher order random walks -- where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph $H$. When $H$ is the clique, this random walk reduces to the usual down-up walk and when $H$ is the directed cycle, this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincaré inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph $H$. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan. We show that when initiated with an expander graph our expanderized random walks have mixing time $O(n \log n)$ for sampling a uniformly random list colorings of a graph $G$ of maximum degree $Δ= O(1)$ where each vertex has at least $(11/6 - ε) Δ$ and at most $O(Δ)$ colors and $O\left( \frac{n \log n}{(1 - \| J\|)^2}\right)$ for sampling the Ising model with a PSD interaction matrix $J \in R^{n \times n}$ satisfying $\| J \| \le 1$ and the external field $h \in R^n$-- here the $O(\bullet)$ notation hides a constant that depends linearly on the largest entry of $h$. As expander graphs can be very sparse, this decreases the amount of randomness required to simulate the down-up walks by a logarithmic factor. We also prove some simple results which enable us to argue about log-Sobolev constants of higher order random walks and provide a simple and self-contained analysis of local-to-global $Φ$-entropy contraction in simplicial complexes -- giving simpler proofs for many pre-existing results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_08927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Expanderizing Higher Order Random Walks Alev, Vedat Levi Rao, Shravas Data Structures and Algorithms We study a variant of the down-up and up-down walks over an $n$-partite simplicial complex, which we call expanderized higher order random walks -- where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph $H$. When $H$ is the clique, this random walk reduces to the usual down-up walk and when $H$ is the directed cycle, this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincaré inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph $H$. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan. We show that when initiated with an expander graph our expanderized random walks have mixing time $O(n \log n)$ for sampling a uniformly random list colorings of a graph $G$ of maximum degree $Δ= O(1)$ where each vertex has at least $(11/6 - ε) Δ$ and at most $O(Δ)$ colors and $O\left( \frac{n \log n}{(1 - \| J\|)^2}\right)$ for sampling the Ising model with a PSD interaction matrix $J \in R^{n \times n}$ satisfying $\| J \| \le 1$ and the external field $h \in R^n$-- here the $O(\bullet)$ notation hides a constant that depends linearly on the largest entry of $h$. As expander graphs can be very sparse, this decreases the amount of randomness required to simulate the down-up walks by a logarithmic factor. We also prove some simple results which enable us to argue about log-Sobolev constants of higher order random walks and provide a simple and self-contained analysis of local-to-global $Φ$-entropy contraction in simplicial complexes -- giving simpler proofs for many pre-existing results. |
| title | Expanderizing Higher Order Random Walks |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2405.08927 |