We study the greedy independent set algorithm on sparse Erdős-Rényi random graphs ${\mathcal G}(n,c/n)$. This range of $p$ is of interest due to the threshold at $c=e$, beyond which it appears that greedy algorithms are affected by a sudden change in the independent set landscape. A large deviation principle was recently established by Bermolen et al. (2020), however, the proof and rate function are somewhat involved. Upper bounds for the rate function were obtained earlier by Pittel (1982). By discrete calculus, we identify the optimal trajectory realizing a given large deviation and obtain the rate function in a simple closed form. In particular, we show that Pittel's bounds are sharp. The proof is brief and elementary. We think the methods presented here will be useful in analyzing the tail behavior of other random growth and exploration processes.
We prove a large deviation principle for a greedy exploration process on an Erdös-Rényi (ER) graph when the number of nodes goes to infinity. To prove our main result, we use the general strategy to study large deviations of processes proposed by Feng and Kurtz, based on the convergence of non-linear semigroups. The rate function can be expressed in a closed-form formula, and associated optimization problems can be solved explicitly, providing the large deviation trajectory. Also, we derive an LDP for the size of the maximum independent set discovered by such an algorithm and analyze the proba…
We prove a large deviation principle for the greedy exploration of configuration models, building on a time-discretized version of the method proposed by Bermolen et al. and Brightwell et al. for jointly constructing a random graph from a given degree sequence and its exploration. The proof of this result follows the general strategy to study large deviations of processes proposed by Feng and Kurtz, based on the convergence of non-linear semigroups. We provide an intuitive interpretation of the LD cost function using Cramer's theorem for the average of random variables with appropriate distrib…
We study here sparse recovery problems in the presence of additive noise. We analyze a thresholding version of the CoSaMP algorithm, named Thresholding Greedy Pursuit (TGP). We demonstrate that an appropriate choice of thresholding parameter, even without the knowledge of sparsity level of the signal and strength of the noise, can result in exact recovery with no false discoveries as the dimension of the data increases to infinity.
Finding independent sets of maximum size in fixed graphs is well known to be an NP-hard task. Using scaling limits, we characterise the asymptotics of sequential degree-greedy explorations and provide sufficient conditions for this algorithm to find an independent set of asymptotically optimal size in large sparse random graphs with given degree sequences. In the special case of sparse Erdös-Rényi graphs, our results allow to give a simple proof of the so-called $e$-phenomenon identified by Karp and Sipser for matchings and to give an alternative characterisation of the asymptotic independence…
In this paper we introduce a new notion of convergence of sparse graphs which we call Large Deviations or LD-convergence and which is based on the theory of large deviations. The notion is introduced by "decorating" the nodes of the graph with random uniform i.i.d. weights and constructing random measures on $[0,1]$ and $[0,1]^2$ based on the decoration of nodes and edges. A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on $[0,1]^d, d=1,2$.…
We consider the model of hashing with linear probing and we establish the moderate and large deviations for the total displacement in sparse tables. In this context, Weibull-like-tailed random variables appear. Deviations for sums of such heavy-tailed random variables are studied in \cite{Nagaev69-1,Nagaev69-2}. Here we adapt the proofs therein to deal with conditioned sums of such variables and solve the open question in \cite{TFC12}. By the way, we establish the deviations of the total displacement in full tables, which can be derived from the deviations of empirical processes of i.i.d.\ ran…
This paper concerns the large deviations of a system of interacting particles on a random graph. There is no stochasticity, and the only sources of disorder are the random graph connections, and the initial condition. The average number of afferent edges on any particular vertex must diverge to infinity as $N\to \infty$, but can do so at an arbitrarily slow rate. These results are thus accurate for both sparse and dense random graphs. A particular application to sparse Erdos-Renyi graphs is provided. The theorem is proved by pushing forward a Large Deviation Principle for a `nested empirical m…
In the field of uncertainty quantification, sparse polynomial chaos (PC) expansions are commonly used by researchers for a variety of purposes, such as surrogate modeling. Ideas from compressed sensing may be employed to exploit this sparsity in order to reduce computational costs. A class of greedy compressed sensing algorithms use least squares minimization to approximate PC coefficients. This least squares problem lends itself to the theory of optimal design of experiments (ODE). Our work focuses on selecting an experimental design that improves the accuracy of sparse PC approximations for …
Computing the size of maximum independent sets is a NP-hard problem for fixed graphs. Characterizing and designing efficient algorithms to estimate this independence number for random graphs are notoriously difficult and still largely open issues. In a companion paper, we showed that a low complexity degree-greedy exploration is actually asymptotically optimal on a large class of sparse random graphs. Encouraged by this result, we present and study two variants of sequential exploration algorithms: static and dynamic degree-aware explorations. We derive hydrodynamic limits for both of them, wh…
This article provides a new toolbox to derive sparse recovery guarantees from small deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics as these constants finely assess how a linear operator is conditioned on the set of sparse vectors and hence how it performs in SRSR. While it is an open problem to construct deterministic matrices with apposite RICs, one can prove that such matrices exist using random matrices mo…
Consider $β> 1$ and $\lfloor β\rfloor$ its integer part. It is widely known that any real number $α\in \Bigl[0, \frac{\lfloor β\rfloor}{β- 1}\Bigr]$ can be represented in base $β$ using a development in series of the form $α= \sum_{n = 1}^\infty x_nβ^{-n}$, where $x = (x_n)_{n \geq 1}$ is a sequence taking values into the alphabet $\{0,\; ...\; ,\; \lfloor β\rfloor\}$. The so called $β$-shift, denoted by $Σ_β$, is given as the set of sequences such that all their iterates by the shift map are less than or equal to the quasi-greedy $β$-expansion of $1$. Fixing a Hölder continuous potential $A$,…
In this paper we consider random linear under-determined systems with block-sparse solutions. A standard subvariant of such systems, namely, precisely the same type of systems without additional block structuring requirement, gained a lot of popularity over the last decade. This is of course in first place due to the success in mathematical characterization of an $\ell_1$ optimization technique typically used for solving such systems, initially achieved in \cite{CRT,DOnoho06CS} and later on perfected in \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}. The success that we achieve…
