The sparse matrix transform for covariance estimation and analysis of high dimensional signals.
Covariance estimation for high dimensional signals is a classically difficult problem in statistical signal analysis and machine learning. In this paper, we propose a maximum likelihood (ML) approach to covariance estimation, which employs a novel non-linear sparsity constraint. More specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy optimization of the log-likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The resulting estimator is generally positive definite and well-conditioned, even when the sample size is limited. Experiments on a combination of simulated data, standard hyperspectral data, and face image sets show that the SMT-based covariance estimates are consistently more accurate than both traditional shrinkage estimates and recently proposed graphical lasso estimates for a variety of different classes and sample sizes. An important property of the new covariance estimate is that it naturally yields a fast implementation of the estimated eigen-transformation using the SMT representation. In fact, the SMT can be viewed as a generalization of the classical fast Fourier transform (FFT) in that it uses "butterflies" to represent an orthonormal transform. However, unlike the FFT, the SMT can be used for fast eigen-signal analysis of general non-stationary signals.
Cao
Guangzhi
G
School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA. lbachega@purdue.edu
Bachega
Leonardo R
LR
Bouman
Charles A
CA
D000465:Algorithms; D056667:Biometric Identification; D016208:Databases, Factual; D005145:Face / Q000033:anatomy & histology; D005583:Fourier Analysis; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D017711:Nonlinear Dynamics; D015203:Reproducibility of Results; D012815:Signal Processing, Computer-Assisted