1,368 research outputs found
User-friendly Tail Bounds for Matrix Martingales
This report presents probability inequalities for sums of adapted sequences of random,
self-adjoint matrices. The results frame simple, easily verifiable hypotheses on the summands, and
they yield strong conclusions about the large-deviation behavior of the maximum eigenvalue of the
sum. The methods also specialize to sums of independent random matrices
Small-Deviation Inequalities for Sums of Random Matrices
Random matrices have played an important role in many fields including
machine learning, quantum information theory and optimization. One of the main
research focuses is on the deviation inequalities for eigenvalues of random
matrices. Although there are intensive studies on the large-deviation
inequalities for random matrices, only a few of works discuss the
small-deviation behavior of random matrices. In this paper, we present the
small-deviation inequalities for the largest eigenvalues of sums of random
matrices. Since the resulting inequalities are independent of the matrix
dimension, they are applicable to the high-dimensional and even the
infinite-dimensional cases
Dimension-free tail inequalities for sums of random matrices
We derive exponential tail inequalities for sums of random matrices with no
dependence on the explicit matrix dimensions. These are similar to the matrix
versions of the Chernoff bound and Bernstein inequality except with the
explicit matrix dimensions replaced by a trace quantity that can be small even
when the dimension is large or infinite. Some applications to principal
component analysis and approximate matrix multiplication are given to
illustrate the utility of the new bounds
An asymptotically Gaussian bound on the Rademacher tails
An explicit upper bound on the tail probabilities for the normalized
Rademacher sums is given. This bound, which is best possible in a certain
sense, is asymptotically equivalent to the corresponding tail probability of
the standard normal distribution, thus affirming a longstanding conjecture by
Efron. Applications to sums of general centered uniformly bounded independent
random variables and to the Student test are presented.Comment: The discussion and references are expanded; the proofs of Lemmas 2.2
and 2.3 are simplifie
Deriving Matrix Concentration Inequalities from Kernel Couplings
This paper derives exponential tail bounds and polynomial moment inequalities
for the spectral norm deviation of a random matrix from its mean value. The
argument depends on a matrix extension of Stein's method of exchangeable pairs
for concentration of measure, as introduced by Chatterjee. Recent work of
Mackey et al. uses these techniques to analyze random matrices with additive
structure, while the enhancements in this paper cover a wider class of
matrix-valued random elements. In particular, these ideas lead to a bounded
differences inequality that applies to random matrices constructed from weakly
dependent random variables. The proofs require novel trace inequalities that
may be of independent interest.Comment: 29 page
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