Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales

Abstract

ABSTRACT:The Alon-Roichman theorem states that for every ε > 0 there is a constant c(ε), such that the Cayley graph of a finite group G with respect to c(ε) log |G| elements of G, chosen independently and uniformly at random, has expected second largest eigenvalue less than ε. In particular, such a graph is an expander with high probability.Landau and Russell, and independently Loh and Schulman, improved the bounds of the theorem. Following Landau and Russell we give a new proof of the result, improving the bounds even further. When considered for a general group G, our bounds are in a sense best possible. We also give a generalization of the Alon-Roichman theorem to random coset graphs.Our proof uses a Hoeffding-type result for operator valued random variables, which we believe can be of independent interest.

User-Friendly Tail Bounds for Sums of Random Matrices

“…As a consequence, their approach generally leads to tail bounds that depend on a scale parameter involving "the sum of eigenvalues." See, for example, the bound (1.3) or the matrix probability inequalities presented in the papers [3,10,20,45]. In contrast, our result on the subadditivity of cumulants, Lemma 3.4, implies that…”

“…Several other authors have exploited their technique to obtain matrix extensions of classical probability inequalities. Christofides and Markström establish a matrix version of the Azuma and Hoeffding inequalities [10]. Gross [20,Theorem 6] and Recht [45,Theorem 3.2] develop two different matrix extensions of Bernstein's inequality.…”

Recovering Low-Rank Matrices From Few Coefficients in Any Basis

“…It is obvious that Y is in the range of R and that its expectation value (equal to sgn ρ) fulfills the conditions in (35). What is more, the operator Chernoff bound can be used to control the deviation of Y from that expected value -so there is hope that we have found a solution.…”

“…Note added: After the pre-print version of this paper was published, the author was made aware of a related matrixvalued martingale bound in [35]. The derivations used in [35] are very similar in spirit to ours (however, their results cannot be applied directly to the problem treated here, because no variance information is incorporated). A few months after our pre-print appeared, more sophisticated matrix-valued martingale bounds were established in [36].…”

The spectrum of random $k$-lifts of large graphs (with possibly large $k$)

“…Given this bound, Theorem 1.1 follows quite easily, while other proof techniques for bounding spectra of random matrices (such as the trace method [15,13,14,16] and the discrepancybased ideas of Feige and Ofek [11]) can be quite technical. In our setting, Theorem 1.2 is also an improvement over other general concentration bounds for random matrices, most notably the operator Chernoff bound of Ahlswede and Winter [1] and the matrix Hoeffding bound of Christofides and Markström [6]. A key advantage of Theorem 1.2 over related results is that its "variance" term can be much smaller, especially in the graph-theoretical setting; this is discussed in more detail in Remark 7.1 of [18].…”