Covariance matrix estimates are an essential part of many signal processing
algorithms, and are often used to determine a low-dimensional principal
subspace via their spectral decomposition. However, exact eigenanalysis is
computationally intractable for sufficiently high-dimensional matrices, and in
the case of small sample sizes, sample eigenvalues and eigenvectors are known
to be poor estimators of their true counterparts. To address these issues, we
propose a covariance estimator that is computationally efficient while also
performing shrinkage on the sample eigenvalues.

Vocal tract resonance characteristics in acoustic speech signals are
classically tracked using frame-by-frame point estimates of formant frequencies
followed by candidate selection and smoothing using dynamic programming methods
that minimize ad hoc cost functions. The goal of the current work is to provide
both point estimates and associated uncertainties of center frequencies and
bandwidths in a statistically principled state-space framework.

In conventional supervised pattern recognition tasks, model selection is
typically accomplished by minimizing the classification error rate on a set of
so-called development data, subject to ground-truth labeling by human experts
or some other means. In the context of speech processing systems and other
large-scale practical applications, however, such labeled development data are
typically costly and difficult to obtain.

Rank estimation is a classical model order selection problem that arises in a
variety of important statistical signal and array processing systems, yet is
addressed relatively infrequently in the extant literature. Here we present
sample covariance asymptotics stemming from random matrix theory, and bring
them to bear on the problem of optimal rank estimation in the context of the
standard array observation model with additive white Gaussian noise.

Material indentation studies, in which a probe is brought into controlled
physical contact with an experimental sample, have long been a primary means by
which scientists characterize the mechanical properties of materials. More
recently, the advent of atomic force microscopy, which operates on the same
fundamental principle, has in turn revolutionized the nanoscale analysis of
soft biomaterials such as cells and tissues.