1,491 research outputs found
A Matrix Expander Chernoff Bound
We prove a Chernoff-type bound for sums of matrix-valued random variables
sampled via a random walk on an expander, confirming a conjecture due to
Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the
Golden-Thompson inequality which improves in some ways the inequality of
Sutter, Berta, and Tomamichel, and may be of independent interest, as well as
an adaptation of an argument for the scalar case due to Healy. Secondarily, we
also provide a generic reduction showing that any concentration inequality for
vector-valued martingales implies a concentration inequality for the
corresponding expander walk, with a weakening of parameters proportional to the
squared mixing time.Comment: Fixed a minor bug in the proof of Theorem 3.
Scalable iterative methods for sampling from massive Gaussian random vectors
Sampling from Gaussian Markov random fields (GMRFs), that is multivariate
Gaussian ran- dom vectors that are parameterised by the inverse of their
covariance matrix, is a fundamental problem in computational statistics. In
this paper, we show how we can exploit arbitrarily accu- rate approximations to
a GMRF to speed up Krylov subspace sampling methods. We also show that these
methods can be used when computing the normalising constant of a large
multivariate Gaussian distribution, which is needed for both any
likelihood-based inference method. The method we derive is also applicable to
other structured Gaussian random vectors and, in particu- lar, we show that
when the precision matrix is a perturbation of a (block) circulant matrix, it
is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure
Sampling of min-entropy relative to quantum knowledge
Let X_1, ..., X_n be a sequence of n classical random variables and consider
a sample of r positions selected at random. Then, except with (exponentially in
r) small probability, the min-entropy of the sample is not smaller than,
roughly, a fraction r/n of the total min-entropy of all positions X_1, ...,
X_n, which is optimal. Here, we show that this statement, originally proven by
Vadhan [LNCS, vol. 2729, Springer, 2003] for the purely classical case, is
still true if the min-entropy is measured relative to a quantum system. Because
min-entropy quantifies the amount of randomness that can be extracted from a
given random variable, our result can be used to prove the soundness of locally
computable extractors in a context where side information might be
quantum-mechanical. In particular, it implies that key agreement in the
bounded-storage model (using a standard sample-and-hash protocol) is fully
secure against quantum adversaries, thus solving a long-standing open problem.Comment: 48 pages, late
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