arXiv subjects

Shravas Rao

Publications and source records attributed to Shravas Rao.

15 recordsLinked to original sources

arXiv2025

Towards a Banach Space Chernoff Bound for Markov Chains via Chaining Arguments

Let $\{Y_i\}_{i=1}^{\infty}$ be a stationary reversible Markov chain with state space $[N]$, let $(X, \| \cdot \|)$ be a real-valued Banach space and let $f_1, \ldots, f_n: [N] \rightarrow X$ be functions with mean $0$ such that $\|f_i(v)\| \leq 1$ for all $i$ and $v$. We prove bounds on the expected value of and deviation bounds for the random variable $\|f_1(Y_1)+\cdots+f_n(Y_n)\|$. For large enough $n$ that depends on the Banach space (and not $N$), these bounds behave similarly as known bounds for independent random variables. When the Banach space in question is the set of matrices equipped with the $\ell_2 \rightarrow \ell_2$ operator norm, for large enough $n$, our bounds on the expected value improve upon known bounds and match what is known for independent random variables up to a factor in the spectral gap.

math.PR↗

arXiv2024

Satisfying the Restricted Isometry Property with the Optimal Number of Rows and Slightly Less Randomness

A matrix $Φ\in \mathbb{R}^{Q \times N}$ satisfies the restricted isometry property if $\|Φx\|_2^2$ is approximately equal to $\|x\|_2^2$ for all $k$-sparse vectors $x$. We give a construction of RIP matrices with the optimal $Q = O(k \log(N/k))$ rows using $O(k\log(N/k)\log(k))$ bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to $ε$-biased distributions.

cs.IT↗

arXiv2024

Expanderizing Higher Order Random Walks

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.

cs.DS↗

arXiv2023

An Improved Lower Bound for Sparse Reconstruction from Subsampled Walsh Matrices

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 $N \times N$ Walsh 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 $\min(K, N/K) > \log^C N$. 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.

cs.IT↗

arXiv2021

The Fourier Transform of Restrictions of Functions on the Slice

This paper considers the Fourier transform over the slice of the Boolean hypercube. We prove a relationship between the Fourier coefficients of a function over the slice, and the Fourier coefficients of its restrictions. As an application, we prove a Goldreich-Levin theorem for functions on the slice based on the Kushilevitz-Mansour algorithm for the Boolean hypercube.

math.CO↗

arXiv2020

Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, $\bullet \quad \mathrm{deg}(f) = O(\widetilde{\mathrm{deg}}(f)^2)$: The degree of $f$ is at most quadratic in the approximate degree of $f$. This is optimal as witnessed by the OR function. $\bullet \quad \mathrm{D}(f) = O(\mathrm{Q}(f)^4)$: The deterministic query complexity of $f$ is at most quartic in the quantum query complexity of $f$. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We apply these results to resolve the quantum analogue of the Aanderaa--Karp--Rosenberg conjecture. We show that if $f$ is a nontrivial monotone graph property of an $n$-vertex graph specified by its adjacency matrix, then $\mathrm{Q}(f)=Ω(n)$, which is also optimal. We also show that the approximate degree of any read-once formula on $n$ variables is $Θ(\sqrt{n})$.

quant-ph↗

arXiv2019

Concentration of Markov chains with bounded moments

Let $\{W_t\}_{t=1}^{\infty}$ be a finite state stationary Markov chain, and suppose that $f$ is a real-valued function on the state space. If $f$ is bounded, then Gillman's expander Chernoff bound (1993) provides concentration estimates for the random variable $f(W_1)+\cdots+f(W_n)$ that depend on the spectral gap of the Markov chain and the assumed bound on $f$. Here we obtain analogous inequalities assuming only that the $q$'th moment of $f$ is bounded for some $q \geq 2$. Our proof relies on reasoning that differs substantially from the proofs of Gillman's theorem that are available in the literature, and it generalizes to yield dimension-independent bounds for mappings $f$ that take values in an $L_p(μ)$ for some $p\ge 2$, thus answering (even in the Hilbertian special case $p=2$) a question of Kargin (2007).

math.PR↗

arXiv2019

The Littlewood-Offord Problem for Markov Chains

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable $ε_1 v_1 + \cdots + ε_n v_n$ lies in the Euclidean unit ball, where $ε_1, \ldots, ε_n \in \{-1, 1\}$ are independent Rademacher random variables and $v_1, \ldots, v_n \in \mathbb{R}^d$ are fixed vectors of at least unit length.We extend many known results to the case that the $ε_i$ are obtained from a Markov chain, including the general bounds first shown by Erdős in the scalar case and Kleitman in the vector case, and also under the restriction that the $v_i$ are distinct integers due to Sárközy and Szemeredi. In all extensions, the upper bound includes an extra factor depending on the spectral gap. We also construct a pseudorandom generator for the Littlewood-Offord problem using similar techniques.

math.CO↗

arXiv2019

A Hoeffding inequality for Markov chains

We prove deviation bounds for the random variable $\sum_{i=1}^{n} f_i(Y_i)$ in which $\{Y_i\}_{i=1}^{\infty}$ is a Markov chain with stationary distribution and state space $[N]$, and $f_i: [N] \rightarrow [-a_i, a_i]$. Our bound improves upon previously known bounds in that the dependence is on $\sqrt{a_1^2+\cdots+a_n^2}$ rather than $\max_{i}\{a_i\}\sqrt{n}.$ We also prove deviation bounds for certain types of sums of vector--valued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten $\infty$-norm of a random matrix whose entries are obtained from a Markov chain.

math.PR↗

arXiv2019

Improved Lower Bounds for the Restricted Isometry Property of Subsampled Fourier Matrices

Let $A$ be an $N \times N$ Fourier matrix over $\mathbb{F}_p^{\log{N}/\log{p}}$ for some prime $p$. We improve upon known lower bounds for the number of rows of $A$ that must be sampled so that the resulting matrix $M$ satisfies the restricted isometry property for $k$-sparse vectors. This property states that $\|Mv\|_2^2$ is approximately $\|v\|_2^2$ for all $k$-sparse vectors $v$. In particular, if $k = Ω( \log^2{N})$, we show that $Ω(k\log{k}\log{N}/\log{p})$ rows must be sampled to satisfy the restricted isometry property with constant probability.

cs.IT↗

arXiv2017

A Sharp Tail Bound for the Expander Random Sampler

Consider an expander graph in which a $μ$ fraction of the vertices are marked. A random walk starts at a uniform vertex and at each step continues to a random neighbor. Gillman showed in 1993 that the number of marked vertices seen in a random walk of length $n$ is concentrated around its expectation, $Φ:= μn$, independent of the size of the graph. Here we provide a new and sharp tail bound, improving on the existing bounds whenever $μ$ is not too large.

math.PR↗

arXiv2016

Arithmetic expanders and deviation bounds for random tensors

We prove hypergraph variants of the celebrated Alon-Roichman theorem on spectral expansion of sparse random Cayley graphs. One of these variants implies that for every prime $p\geq 3$ and any $\varepsilon > 0$, there exists a set of directions $D\subseteq \mathbb{F}_p^n$ of size $O_{p,\varepsilon}(p^{(1-1/p +o(1))n})$ such that for every set $A\subseteq \mathbb{F}_p^n$ of density $α$, the fraction of lines in $A$ with direction in $D$ is within $\varepsilonα$ of the fraction of all lines in $A$. Our proof uses new deviation bounds for sums of independent random multi-linear forms taking values in a generalization of the Birkhoff polytope. The proof of our deviation bound is based on Dudley's integral inequality and a probabilistic construction of $\varepsilon$-nets. Using the polynomial method we prove that a Cayley hypergraph with edges generated by a set~$D$ as above requires $|D| \geq Ω_p(n^{p-1})$ for (our notion of) spectral expansion for hypergraphs.

math.CO↗

arXiv2016

Applications of $α$-strongly regular distributions to Bayesian auctions

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 $v < v'$, $ϕ(v') - ϕ(v) \ge v' - v$, and for regular distributions, $ϕ(v') - ϕ(v) \ge 0$. Cole and Roughgarden introduced the interpolating class of $α$-Strongly Regular distributions ($α$-SR distributions for short), for which $ϕ(v') - ϕ(v) \ge α(v' - v)$, for $0 \le α\le 1$. 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 $α$-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.

cs.GT↗

arXiv2015

On Lipschitz Bijections between Boolean Functions

For two functions $f,g:\{0,1\}^n\to\{0,1\}$ a mapping $ψ:\{0,1\}^n\to\{0,1\}^n$ is said to be a $\textit{mapping from $f$ to $g$}$ if it is a bijection and $f(z)=g(ψ(z))$ for every $z\in\{0,1\}^n$. In this paper we study Lipschitz mappings between boolean functions. Our first result gives a construction of a $C$-Lipschitz mapping from the ${\sf Majority}$ function to the ${\sf Dictator}$ function for some universal constant $C$. On the other hand, there is no $n/2$-Lipschitz mapping in the other direction, namely from the ${\sf Dictator}$ function to the ${\sf Majority}$ function. This answers an open problem posed by Daniel Varga in the paper of Benjamini et al. (FOCS 2014). We also show a mapping from ${\sf Dictator}$ to ${\sf XOR}$ that is 3-local, 2-Lipschitz, and its inverse is $O(\log(n))$-Lipschitz, where by $L$-local mapping we mean that each of its output bits depends on at most $L$ input bits. Next, we consider the problem of finding functions such that any mapping between them must have large \emph{average stretch}, where the average stretch of a mapping $ϕ$ is defined as ${\sf avgStretch}(ϕ) = {\mathbb E}_{x,i}[dist(ϕ(x),ϕ(x+e_i)]$. We show that any mapping $ϕ$ from ${\sf XOR}$ to ${\sf Majority}$ must satisfy ${\sf avgStretch}(ϕ) \geq Ω(\sqrt{n})$. In some sense, this gives a "function analogue" to the question of Benjamini et al. (FOCS 2014), who asked whether there exists a set $A \subset \{0,1\}^n$ of density 0.5 such that any bijection from $\{0,1\}^{n-1}$ to $A$ has large average stretch. Finally, we show that for a random balanced function $f:\{0,1\}^n\to\{0,1\}^n$ with high probability there is a mapping $ϕ$ from ${\sf Dictator}$ to $f$ such that both $ϕ$ and $ϕ^{-1}$ have constant average stretch. In particular, this implies that one cannot obtain lower bounds on average stretch by taking uniformly random functions.

cs.DM↗

arXiv2012

Finding hitting times in various graphs

The hitting time, h_uv, of a random walk on a finite graph G, is the expected time for the walk to reach vertex v given that it started at vertex u. We present two methods of calculating the hitting time between vertices of finite graphs, along with applications to specific classes of graphs, including grids, trees, and the 'tadpole' graphs.

math.PR↗