In many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data. More specifically, if we have a set of N independent, identically distributed measurements of an M dimensional random vector the maximum likelihood estimate is the sample covariance matrix. Here we consider the case where N < M such that this estimate is singular (non-invertible) and therefore fundamentally bad. We present a radically new approach to deal with this situation. Let X be the M × N data matrix, where the columns are the N independent realizations of the random vector with covariance matrix Σ. Without loss of generality, and for simplicity, we can assume that the random variables have zero mean. We would like to estimate Σ from X. Let K be the classical sample covariance matrix. Fix a parameter 1 ≤ L ≤ N and consider an ensemble of L × M random unitary matrices, {Φ}, having Haar probability measure (isotropically random). Pre-and post-multiply K by Φ, and by the conjugate transpose of Φ respectively, to produce a nonsingular L × L reduced dimension covariance estimate. A new estimate for Σ, denoted by covL(K), is obtained by a) projecting the reduced covariance estimate out (to M × M ) through preand post-multiplication by the conjugate transpose of Φ, and by Φ respectively, and b) taking the expectation over the unitary ensemble. Another new estimate (this time for Σ −1 ), invcovL(K), is obtained by a) inverting the reduced covariance estimate, b) projecting the inverse out (to M × M ) through pre-and post-multiplication by the conjugate transpose of Φ, and by Φ respectively, and c) taking the expectation over the unitary ensemble. We show that the estimate cov is equivalent to diagonal loading. Both estimates invcov and cov retain the original eigenvectors and make nonzero the formerly zero eigenvalues. We have a closed form analytical expression for invcov in terms of its eigenvector and eigenvalue decomposition. We motivate the use of invcov through applications to linear estimation, supervised learning, and high-resolution spectral estimation. We also compare the performance of the estimator invcov with respect to diagonal loading.

This paper centers on the limit eigenvalue distribution for random Vandermonde matrices with unit magnitude complex entries. The phases of the entries are chosen independently and identically distributed from the interval [−π, π]. Various types of distribution for the phase are considered and we establish the existence of the empirical eigenvalue distribution in the large matrix limit on a wide range of cases. The rate of growth of the maximum eigenvalue is examined and shown to be no greater than O(log N ) and no slower than O(log N/ log log N ) where N is the dimension of the matrix. Additional results include the existence of the capacity of the Vandermonde channel (limit integral for the expected log determinant).

Let {T k } ∞ k=1 be a family of * -free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the free central limit Theorem. More precisely, let Bn = T * 1 T * 2 . . . T * n Tn . . . T2T1 then Bn is a positive operator and B 1/2n n converges in distribution to an operator Λ. We completely determine the probability distribution ν of Λ from the distribution µ of |T | 2 . This gives us a natural map G : M+ → M+ with µ → G(µ) = ν. We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution ν and the distribution of the Lyapunov exponents for the sequence {T k } ∞ k=1 introduced in [13].

In this work we study the asymptotic traffic flow in Gromov's hyperbolic graphs. We prove that under certain mild hypotheses the traffic flow in a hyperbolic graph tends to pass through a finite set of highly congested nodes. These nodes are called the "core" of the graph. We provide a formal definition of the core in a very general context and we study the properties of this set for several graphs.

In this work we prove that the giant component of the Erdös-Renyi random graph G(n, c/n) for c a constant greater than 1 (sparse regime), is not Gromov δ-hyperbolic for any δ with probability tending to one as n → ∞. As a corollary we provide an alternative proof that the giant component of G(n, c/n) when c > 1 has zero spectral gap almost surely as n → ∞.

Abstract. We show that if A is a Hilbert-space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra vN (A) that is generated by A, is independent of the representation of vN (A), thought of as an abstract W * -algebra.We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras.We introduce the B-circular operators as a special case of Speicher's B-Gaussian operators in free probability theory, and we prove several results about a B-circular operator z, including formulas for the B-valued Cauchy-and R-transforms of z * z. We show that a large class of L ∞ ([0, 1])-circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F 3 ). These results generalize some of what is known about the quasinilpotent DT-operator.

This work explains how to analyze the aggregate electricity consumption of many consumers, and extract key components such as heating, ventilation and air conditioning (HVAC), residential lighting, and street lighting consumption from the total consumption. To avoid explicit modeling of dependencies on time of day and on working versus non-working days, leastsquares fitting for outside temperature and natural illumination dependency proceeds independently for each hour of the day. Cubic polynomials model dependencies on Steadman apparent temperature and on log-scale illumination, but spline surfaces are best when considering these variables jointly. The primary focus is on residential consumption, but the same techniques can be used for studying street lighting, commercial and industrial consumption.

This paper examines various statistical distributions in connection with random N ×N Vandermonde matrices and their generalization to d-dimensional phase distributions. Upper and lower bound asymptotics for the maximum eigenvalue are found to be O(log N d ) and O(log N d / log log N d ) respectively. The behavior of the minimum eigenvalue is considered by studying the behavior of the maximum eigenvalue of the inverse matrix. In particular, we prove that the minimum eigenvalue λ1 is shown to be at most O(exp(− √ N W * N )) where W * N is a positive random variable converging weakly to a random variable constructed from a realization of the Brownian Bridge on [0, 2π). Additional results for (V * V) −1 , a trace log formula for V * V, as well as a some numerical examinations of the size of the atom at 0 for the random Vandermonde eigenvalue distribution are also presented.