Variational approximation methods have become a mainstay of contemporary Machine Learning methodology, but currently have little presence in Statistics. We devise an effective variational approximation strategy for fitting generalized linear mixed models (GLMM) appropriate for grouped data. It involves Gaussian approximation to the distributions of random effects vectors, conditional on the res...

In the phase retrieval problem, an unknown vector is to be recovered given quadratic measurements. This problem has received considerable attention in recent times. In this paper, we present an algorithm to solve a nonconvex formulation of the phase retrieval problem, that we call Incremental Truncated Wirtinger Flow. Given random Gaussian sensing vectors, we prove that it converges linearly to...

Let Fn = (F1,n, ...., Fd,n), n > 1, be a sequence of random vectors such that, for every j = 1, ..., d, the random variable Fj,n belongs to a fixed Wiener chaos of a Gaussian field. We show that, as n → ∞, the components of Fn are asymptotically independent if and only if Cov(F 2 i,n, F 2 j,n) → 0 for every i 6= j. Our findings are based on a novel inequality for vectors of multiple Wiener-Itô ...

This paper develops and compares the MAP and MMSE estimators for spherically-contoured multivariate Laplace random vectors in additive white Gaussian noise. The MMSE estimator is expressed in closed-form using the generalized incomplete gamma function. We also find a computationally efficient yet accurate approximation for the MMSE estimator. In addition, this paper develops an expression for t...

We examine the problem of estimating the trace of a matrix A when given access to an oracle which computes x†Ax for an input vector x. We make use of the basis vectors from a set of mutually unbiased bases, widely studied in the field of quantum information processing, in the selection of probing vectors x. This approach offers a new state of the art single shot sampling variance while requirin...

In the context of mod-Gaussian convergence, as defined previously in our work with J. Jacod, we obtain asymptotic formulas and lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian in this sense, with increasing covariance matrix. This is motivated by the conjecture concerning the density of the set of values of the Riemann zeta function on the c...

Abstract. We study random vectors of the form (Tr(AV ), . . . ,Tr(AV )), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et ...

We obtain an invariance principle for non-stationary vector-valued stochastic processes. It is shown that, under mild conditions, the partial sums of non-stationary processes can be approximated on a richer probability space by sums of independent Gaussian random vectors with nearly optimal bounds. The latter Gaussian approximation result has a wide range of applications in the study of multipl...

In this tutorial article we give a Bayesian derivation of a basic state estimation result for discrete-time Markov process models with independent process and measurement noise and measurements not a ecting the state. We then list some properties of Gaussian random vectors and show how the Kalman ltering algorithm follows from the general state estimation result and a linear-Gaussian model de n...