C. Wang and Y. M. Lu, “

Online Learning for Sparse PCA in High Dimensions: Exact Dynamics and Phase Transitions,” in

IEEE Information Theory Workshop (ITW), 2016.

AbstractWe study the dynamics of an online algorithm for learning a sparse leading eigenvector from samples generated from a spiked covariance model. This algorithm combines the classical Oja's method for online PCA with an element-wise nonlinearity at each iteration to promote sparsity. In the high-dimensional limit, the joint empirical measure of the underlying sparse eigenvector and its estimate provided by the algorithm is shown to converge weakly to a deterministic, measure-valued process. This scaling limit is characterized as the unique solution of a nonlinear PDE, and it provides exact information regarding the asymptotic performance of the algorithm. For example, performance metrics such as the cosine similarity and the misclassification rate in sparse support recovery can be obtained by examining the limiting dynamics. A steady-state analysis of the nonlinear PDE also reveals an interesting phase transition phenomenon. Although our analysis is asymptotic in nature, numerical simulations show that the theoretical predictions are accurate for moderate signal dimensions.

spca.pdf Y. Chi and Y. M. Lu, “

Kaczmarz Method for Solving Quadratic Equations,”

IEEE Signal Processing Letters, vol. 23, no. 9, pp. 1183-1187, 2016.

AbstractEstimating low-rank positive-semidefinite (PSD) matrices from symmetric rank-one measurements is of great importance in many applications, such as high-dimensional data processing, quantum state tomography, and phase retrieval. When the rank is known a priori, this problem can be regarded as solving a system of quadratic equations of a low-dimensional subspace. The authors develop a fast iterative algorithm based on an adaptation of the Kaczmarz method, which is traditionally used for solving overdetermined linear systems. In particular, the authors characterize the dynamics of the algorithm when the measurement vectors are composed of standard Gaussian entries in the online setting. Numerical simulations demonstrate the compelling performance of the proposed algorithm.

kaczmarz_quadratic_final.pdf F. Sroubek, J. Kamenicky, and Y. M. Lu, “

Decomposition space-variant blur in image deconvolution,”

IEEE Signal Processing Letters, vol. 23, no. 3, pp. 346-350, 2016.

Abstract
Standard convolution as a model of radiometric degradation is in majority of cases inaccurate as the blur varies in space and we are thus required to work with a computationally demanding space-variant model. Space-variant degradation can be approximately decomposed to a set of standard convolutions. We explain in detail the properties of the space-variant degrada- tion operator and show two possible decomposition models and two approximation approaches. Our target application is space- variant image deconvolution, on which we illustrate theoretical differences between these models. We propose a computationally efficient restoration algorithm that belongs to a category of alternating direction methods of multipliers, which consists of four update steps with closed-form solutions. Depending on the used decomposition, two variations of the algorithm exist with distinct properties. We test the effectiveness of the decomposition models under different levels of approximation on synthetic and real examples, and conclude the letter by drawing several practical observations.

main_rev.pdf D. M. Merfeld, T. K. Clark, Y. M. Lu, and F. Karmali, “

Dynamics of Individual Perceptual Decisions,”

Journal of Neurophysiology, vol. 115, no. 1, 2016.

Publisher's VersionAbstractPerceptual decision-making is fundamental to a broad range of fields including neurophysiology, economics, medicine, advertising, law, etc. While recent findings have yielded major advances in our understanding of perceptual decision-making, decision-making as a function of time and frequency (i.e., decision-making dynamics) is not well understood. To limit the review length, we focus most of this review on human findings. Animal findings, which are extensively reviewed elsewhere, are included when beneficial or necessary. We attempt to put these various findings and datasets - which can appear to be unrelated in the absence of a formal dynamic analysis - into context using published models. Specifically, by adding appropriate dynamic mechanisms (e.g., high-pass filters) to existing models, it appears that a number of otherwise seemingly disparate findings from the literature might be explained. One hypothesis that arises through this dynamic analysis is that decision-making includes phasic (high-pass) neural mechanisms, an evidence accumulator and/or some sort of mid-trial decision-making mechanism (e.g., peak detector and/or decision boundary).

C. Hu, J. Sepulcre, K. A. Johnson, G. E. Fakhri, Y. M. Lu, and Q. Li, “

Matched Signal Detection on Graphs: Theory and Application to Brain Imaging Data Classification,”

NeuroImage, vol. 125, pp. 587-600, 2016.

Abstract
Motivated by recent progresses in signal processing on graphs, we have developed a matched signal detection (MSD) theory for signals with intrinsic structures described by weighted graphs. First, we regard graph Laplacian eigenvalues as frequencies of graph-signals and assume that the signal is in a subspace spanned by the first few graph Laplacian eigenvectors associated with lower eigenvalues. The conventional matched subspace detector can be applied to this case. Furthermore, we study signals that may not merely live in a subspace. Namely, we consider signals with bounded variation on graphs and more general signals that are randomly drawn from a prior distribution. For bounded variation signals, the test is a weighted energy detector. For the random signals, the test statistic is the difference of signal variations on associated graphs, if a degenerate Gaussian distribution specified by the graph Laplacian is adopted. We evaluate the effectiveness of the MSD on graphs both on simulated and real data sets. Specifically, we apply MSD to the brain imaging data classification problem of Alzheimer’s disease (AD) based on two independent data sets: 1) positron emission tomography data with Pittsburgh compound-B tracer of 30 AD and 40 normal control (NC) subjects, 2) resting-state functional magnetic resonance imaging (R-fMRI) data of 30 early mild cognitive impairment and 20 NC subjects. Our results demonstrate that the MSD approach is able to outperform the traditional methods and help detect AD at an early stage, probably due to the success of exploiting the manifold structure of the data.

A. Minot, Y. M. Lu, and N. Li, “

A Distributed Gauss-Newton Method for Power System State Estimation,”

IEEE Transactions on Power Systems, vol. 31, no. 5, pp. 3804-3815, 2016.

AbstractWe propose a fully distributed Gauss-Newton algorithm for state estimation of electric power systems. At each Gauss-Newton iteration, matrix-splitting techniques are utilized to carry out the matrix inversion needed for calculating the Gauss-Newton step in a distributed fashion. In order to reduce the communication burden as well as increase robustness of state estimation, the proposed distributed scheme relies only on local information and a limited amount of information from neighboring areas. The matrix-splitting scheme is designed to calculate the Gauss-Newton step with exponential convergence speed. The effectiveness of the method is demonstrated in various numerical experiments.

I. Dokmanic and Y. M. Lu, “

Sampling Sparse Signals on the Sphere: Algorithms and Applications,”

IEEE Transactions on Signal Processing, vol. 64, no. 1, pp. 189-202, 2016.

arXiv:1502.07577AbstractWe propose a sampling scheme that can perfectly reconstruct a collection of

spikes on the sphere from samples of their lowpass-filtered observations.

Central to our algorithm is a generalization of the annihilating filter

method, a tool widely used in array signal processing and finite-rate-of-innovation

(FRI) sampling. The proposed algorithm can reconstruct $K$ spikes

from $(K+\sqrt{K})^2$ spatial samples---a sampling requirement that

improves over known sparse sampling schemes on the sphere by a factor of up

to four.

We showcase the versatility of the proposed algorithm by applying it to

three different problems: 1) sampling diffusion processes induced by

localized sources on the sphere, 2) shot-noise removal, and 3) sound source

localization (SSL) by a spherical microphone array. In particular, we show

how SSL can be reformulated as a spherical sparse sampling problem.