On the method of typical bounded differences

Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that random functions are near their means. Of particular importance is the case where f(X) is a function of independent random variables X=(X_1, ..., X_n). Here the well known bounded differences inequality (also called McDiarmid's or Hoeffding-Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |f(X)-f(X')| \leq c_k whenever X,X' differ only in X_k. While this is easy to check, the main disadvantage is that it considers worst-case changes c_k, which often makes the resulting bounds too weak to be useful. In this paper we prove a variant of the bounded differences inequality which can be used to establish concentration of functions f(X) where (i) the typical changes are small although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations why concentration should hold. Indeed, given an event \Gamma that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where \Gamma occurs. The point is that the resulting typical changes c_k are often much smaller than the worst case ones. To illustrate its application we consider the reverse H-free process, where H is 2-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollob\'as and Erd\H{o}s.Comment: 25 page

Upper Tail Estimates with Combinatorial Proofs

We study generalisations of a simple, combinatorial proof of a Chernoff bound similar to the one by Impagliazzo and Kabanets (RANDOM, 2010). In particular, we prove a randomized version of the hitting property of expander random walks and apply it to obtain a concentration bound for expander random walks which is essentially optimal for small deviations and a large number of steps. At the same time, we present a simpler proof that still yields a "right" bound settling a question asked by Impagliazzo and Kabanets. Next, we obtain a simple upper tail bound for polynomials with input variables in $[0, 1]$ which are not necessarily independent, but obey a certain condition inspired by Impagliazzo and Kabanets. The resulting bound is used by Holenstein and Sinha (FOCS, 2012) in the proof of a lower bound for the number of calls in a black-box construction of a pseudorandom generator from a one-way function. We then show that the same technique yields the upper tail bound for the number of copies of a fixed graph in an Erd\H{o}s-R\'enyi random graph, matching the one given by Janson, Oleszkiewicz and Ruci\'nski (Israel J. Math, 2002).Comment: Full version of the paper from STACS 201

Spectral Methods from Tensor Networks

A tensor network is a diagram that specifies a way to "multiply" a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks. In this work we leverage the full power of this abstraction to design new algorithms for certain continuous tensor decomposition problems. An important and challenging family of tensor problems comes from orbit recovery, a class of inference problems involving group actions (inspired by applications such as cryo-electron microscopy). Orbit recovery problems over finite groups can often be solved via standard tensor methods. However, for infinite groups, no general algorithms are known. We give a new spectral algorithm based on tensor networks for one such problem: continuous multi-reference alignment over the infinite group SO(2). Our algorithm extends to the more general heterogeneous case.Comment: 30 pages, 8 figure