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.

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function f,

We give a short argument that yields a new lower bound on the number of subsampled rows from a bounded, orthonormal matrix necessary to form a matrix with the restricted isometry property. We show that a matrix formed by uniformly subsampling rows of an N × N Hadamard matrix contains a K-sparse vector in the kernel, unless the number of subsampled rows is Ω(K log K log (N/K)) --- our lower bound applies whenever min(K, N/K) > logCN. Containing a sparse vector in the kernel precludes not only the restricted isometry property, but more generally the application of those matrices for uniform sparse recovery.

Two classes of distributions that are widely used in the analysis of Bayesian auctions are the Monotone Hazard Rate (MHR) and Regular distributions. They can both be characterized in terms of the rate of change of the associated virtual value functions: for MHR distributions the condition is that for values $$v < v'$$ , $$\phi (v') - \phi (v) \ge v' - v$$ , and for regular distributions, $$\phi (v') - \phi (v) \ge 0$$ . Cole and Roughgarden introduced the interpolating class of $$\alpha $$ -Strongly Regular distributions ( $$\alpha $$ -SR distributions for short), for which $$\phi (v') - \phi (v) \ge \alpha (v' - v)$$ , for $$0 \le \alpha \le 1$$ . In this paper, we investigate five distinct auction settings for which good expected revenue bounds are known when the bidders’ valuations are given by MHR distributions. In every case, we show that these bounds degrade gracefully when extended to $$\alpha $$ -SR distributions. For four of these settings, the auction mechanism requires knowledge of these distribution(s) (in the other setting, the distributions are needed only to ensure good bounds on the expected revenue). In these cases we also investigate what happens when the distributions are known only approximately via samples, specifically how to modify the mechanisms so that they remain effective and how the expected revenue depends on the number of samples.