Lag selection and estimation of stable parameters for multiple autoregressive processes through convex programming

Motivated by a variety of applications, high-dimensional time series have become an active topic of research. In particular, several methods and finite-sample theories for individual stable autoregressive processes with known lag have become available very recently. We, instead, consider multiple stable autoregressive processes that share an unknown lag. We use information across the different processes to simultaneously select the lag and estimate the parameters. We prove that the estimated process is stable, and we establish rates for the forecasting error that can outmatch the known rate in our setting. Our insights on the lag selection and the stability are also of interest for the case of individual autoregressive processes.

Learning Mixtures of Spherical Gaussians via Fourier Analysis

Suppose that we are given independent, identically distributed samples $x_l$ from a mixture $\mu$ of no more than $k$ of $d$-dimensional spherical gaussian distributions $\mu_i$ with variance $1$, such that the minimum $\ell_2$ distance between two distinct centers $y_l$ and $y_j$ is greater than $\sqrt{d} \Delta$ for some $c \leq \Delta $, where $c\in (0,1)$ is a small positive universal constant. We develop a randomized algorithm that learns the centers $y_l$ of the gaussians, to within an $\ell_2$ distance of $\delta < \frac{\Delta\sqrt{d}}{2}$ and the weights $w_l$ to within $cw_{min}$ with probability greater than $1 - \exp(-k/c)$. The number of samples and the computational time is bounded above by $poly(k, d, \frac{1}{\delta})$. Such a bound on the sample and computational complexity was previously unknown when $\omega(1) \leq d \leq O(\log k)$. When $d = O(1)$, this follows from work of Regev and Vijayaraghavan. These authors also show that the sample complexity of learning a random mixture of gaussians in a ball of radius $\Theta(\sqrt{d})$ in $d$ dimensions, when $d$ is $\Theta( \log k)$ is at least $poly(k, \frac{1}{\delta})$, showing that our result is tight in this case.