12 research outputs found
Joint Covariance Estimation with Mutual Linear Structure
We consider the problem of joint estimation of structured covariance
matrices. Assuming the structure is unknown, estimation is achieved using
heterogeneous training sets. Namely, given groups of measurements coming from
centered populations with different covariances, our aim is to determine the
mutual structure of these covariance matrices and estimate them. Supposing that
the covariances span a low dimensional affine subspace in the space of
symmetric matrices, we develop a new efficient algorithm discovering the
structure and using it to improve the estimation. Our technique is based on the
application of principal component analysis in the matrix space. We also derive
an upper performance bound of the proposed algorithm in the Gaussian scenario
and compare it with the Cramer-Rao lower bound. Numerical simulations are
presented to illustrate the performance benefits of the proposed method
Tyler's Covariance Matrix Estimator in Elliptical Models with Convex Structure
We address structured covariance estimation in elliptical distributions by
assuming that the covariance is a priori known to belong to a given convex set,
e.g., the set of Toeplitz or banded matrices. We consider the General Method of
Moments (GMM) optimization applied to robust Tyler's scatter M-estimator
subject to these convex constraints. Unfortunately, GMM turns out to be
non-convex due to the objective. Instead, we propose a new COCA estimator - a
convex relaxation which can be efficiently solved. We prove that the relaxation
is tight in the unconstrained case for a finite number of samples, and in the
constrained case asymptotically. We then illustrate the advantages of COCA in
synthetic simulations with structured compound Gaussian distributions. In these
examples, COCA outperforms competing methods such as Tyler's estimator and its
projection onto the structure set.Comment: arXiv admin note: text overlap with arXiv:1311.059
Regularized estimation of linear functionals of precision matrices for high-dimensional time series
This paper studies a Dantzig-selector type regularized estimator for linear
functionals of high-dimensional linear processes. Explicit rates of convergence
of the proposed estimator are obtained and they cover the broad regime from
i.i.d. samples to long-range dependent time series and from sub-Gaussian
innovations to those with mild polynomial moments. It is shown that the
convergence rates depend on the degree of temporal dependence and the moment
conditions of the underlying linear processes. The Dantzig-selector estimator
is applied to the sparse Markowitz portfolio allocation and the optimal linear
prediction for time series, in which the ratio consistency when compared with
an oracle estimator is established. The effect of dependence and innovation
moment conditions is further illustrated in the simulation study. Finally, the
regularized estimator is applied to classify the cognitive states on a real
fMRI dataset and to portfolio optimization on a financial dataset.Comment: 44 pages, 4 figure
Estimation of dynamic networks for high-dimensional nonstationary time series
This paper is concerned with the estimation of time-varying networks for
high-dimensional nonstationary time series. Two types of dynamic behaviors are
considered: structural breaks (i.e., abrupt change points) and smooth changes.
To simultaneously handle these two types of time-varying features, a two-step
approach is proposed: multiple change point locations are first identified
based on comparing the difference between the localized averages on sample
covariance matrices, and then graph supports are recovered based on a
kernelized time-varying constrained -minimization for inverse matrix
estimation (CLIME) estimator on each segment. We derive the rates of
convergence for estimating the change points and precision matrices under mild
moment and dependence conditions. In particular, we show that this two-step
approach is consistent in estimating the change points and the piecewise smooth
precision matrix function, under certain high-dimensional scaling limit. The
method is applied to the analysis of network structure of the S\&P 500 index
between 2003 and 2008
An overview of deep learning based methods for unsupervised and semi-supervised anomaly detection in videos
Videos represent the primary source of information for surveillance
applications and are available in large amounts but in most cases contain
little or no annotation for supervised learning. This article reviews the
state-of-the-art deep learning based methods for video anomaly detection and
categorizes them based on the type of model and criteria of detection. We also
perform simple studies to understand the different approaches and provide the
criteria of evaluation for spatio-temporal anomaly detection.Comment: 15 pages, double colum