In Compressive Sensing, the Restricted Isometry Property (RIP) ensures that robust recovery of sparsevectors is possible from noisy, undersampled measurements via computationally tractable algorithms. Itis by now well-known that Gaussian (or, more generally, sub-Gaussian) random matrices satisfy the RIPunder certain conditions on the number of measurements. Their use can be limi...

We present a computationally efficient method to generate random variables from a univariate conditional probability density function (PDF) derived from a multivariate α-sub-Gaussian (αSG) distribution. The approach may be used to sequentially generate variates for sliding-window models that constrain immediately adjacent samples to be αSG random vectors. We initially derive and establish vario...

Vector autoregressive models characterize a variety of time series in which linear combinations of current and past observations can be used to accurately predict future observations. For instance, each element of an observation vector could correspond to a different node in a network, and the parameters of an autoregressive model would correspond to the impact of the network structure on the t...

Abstract. It is well known that the ratio of two independent standard Gaussian random variables follows a Cauchy distribution. Any convex combination of independent standard Cauchy random variables also follows a Cauchy distribution. In a recent joint work [PM16], the author proved a surprising multivariate generalization of the above facts. Fix m > 1 and let Σ be a m × m positive semi-definite...

Let K be a subset of the Euclidean sphere S d−1. As seen in Lecture #1, in analyzing how well a given random projection matrix S ∈ R m×d preserves vectors in K, a central object is the random variable Z(K) = sup u∈K Su 2 2 m − 1. (2.1) Suppose that our goal is to establish that, for some δ ∈ (0, 1), we have Z(K) ≤ δ with high probability. How large must the projection dimension m be, as a funct...

We present a Bayesian model for area-level count data that uses Gaussian random effects with a novel type of G-Wishart prior on the inverse variance- covariance matrix. Specifically, we introduce a new distribution called the truncated G-Wishart distribution that has support over precision matrices that lead to positive associations between the random effects of neighboring regions while preser...

A methodology was developed for estimating the parameters involved in a first-order autoregressive process; these parameters comprise a variance component associated with the random effect, a correlation coefficient, p, and a residual variance. These parameters were estimated using REML with an expectationmaximization algorithm. For two singletrait analyses (milk and fat production being the de...