Xiaoming Huo

Theoretical Results on Sparse Representations of Multiple-Measurement Vectors

The sparse representation of a multiple-measurement vector (MMV) is a relatively new problem in sparse representation. Efficient methods have been proposed. Although many theoretical results that are available in a simple case-single-measurement vector (SMV)-the theoretical analysis regarding MMV is lacking. In this paper, some known results of SMV are generalized to MMV. Some of these new results take advantages of additional information in the formulation of MMV. We consider the uniqueness under both an lscr0-norm-like criterion and an lscr1-norm-like criterion. The consequent equivalence between the lscr0-norm approach and the lscr1-norm approach indicates a computationally efficient way of finding the sparsest representation in a redundant dictionary. For greedy algorithms, it is proven that under certain conditions, orthogonal matching pursuit (OMP) can find the sparsest representation of an MMV with computational efficiency, just like in SMV. Simulations show that the predictions made by the proved theorems tend to be very conservative; this is consistent with some recent advances in probabilistic analysis based on random matrix theory. The connections will be discussed

Beamlets and Multiscale Image Analysis

We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis.

Sparse representations for multiple measurement vectors (MMV) in an over-complete dictionary

The multiple measurement vector (MMV), a newly emerged problem in sparse representation in an over-complete dictionary motivated by a neuro-magnetic inverse problem that arises in magnetoencephalography (MEG) - a modality for imaging the possible activation regions in the brain, poses new challenges. Efficient methods have been designed to search for sparse representations; however, we have not seen substantial development in the theoretical analysis, considering what has been done in a simpler case - single measurement vector (SMV) - in which many theoretical results are known. This paper extends the known results of SMV to MMV. Our theoretical results show the fundamental limitation on when a sparse representation is unique. Moreover, the relation between the solutions of /spl lscr//sub 0/-norm minimization and the solutions of /spl lscr//sub 1/-norm minimization indicates a computationally efficient approach to find a sparse representation. Interestingly, simulations show that the predictions made by these theorems tend to be conservative.

Wavelet-Based Data Reduction Techniques for Process Fault Detection

This article presents new data reduction methods based on the discrete wavelet transform to handle potentially large and complicated nonstationary data curves. The methods minimize objective functions to balance the trade-off between data reduction and modeling accuracy. Theoretic investigations provide the optimality of the methods and the large-sample distribution of a closed-form estimate of the thresholding parameter. An upper bound of errors in signal approximation (or estimation) is derived. Based on evaluation studies with popular testing curves and real-life datasets, the proposed methods demonstrate their competitiveness with the existing engineering data compression and statistical data denoising methods for achieving the data reduction goals. Further experimentation with a tree-based classification procedure for identifying process fault classes illustrates the potential of the data reduction tools. Extension of the engineering scalogram to the reduced-size semiconductor fabrication data leads to a visualization tool for monitoring and understanding process problems.

Electricity Price Curve Modeling and Forecasting by Manifold Learning

This paper proposes a novel nonparametric approach for the modeling and analysis of electricity price curves by applying the manifold learning methodology-locally linear embedding (LLE). The prediction method based on manifold learning and reconstruction is employed to make short-term and medium-term price forecasts. Our method not only performs accurately in forecasting one-day-ahead prices, but also has a great advantage in predicting one-week-ahead and one-month-ahead prices over other methods. The forecast accuracy is demonstrated by numerical results using historical price data taken from the Eastern U.S. electric power markets.

When do stepwise algorithms meet subset selection criteria

Recent results in homotopy and solution paths demonstrate that certain well-designed greedy algorithms, with a range of values of the algorithmic parameter, can provide solution paths to a sequence of convex optimization problems. On the other hand, in regression many existing criteria in subset selection (including Cp, AIC, BIC, MDL, RIC, etc.) involve optimizing an objective function that contains a counting measure. The two optimization problems are formulated as (P1) and (P0) in the present paper. The latter is generally combinatoric and has been proven to be NP-hard. We study the conditions under which the two optimization problems have common solutions. Hence, in these situations a stepwise algorithm can be used to solve the seemingly unsolvable problem. Our main result is motivated by recent work in sparse representation, while two others emerge from different angles: a direct analysis of sufficiency and necessity and a condition on the mostly correlated covariates. An extreme example connected with least angle regression is of independent interest.

Incremental Multi-Scale Search Algorithm for Dynamic Path Planning With Low Worst-Case Complexity

Path-planning (equivalently, path-finding) problems are fundamental in many applications, such as transportation, VLSI design, robot navigation, and many more. In this paper, we consider dynamic shortest path-planning problems on a graph with a single endpoint pair and with potentially changing edge weights over time. Several algorithms exist in the literature that solve this problem, notably among them the Lifelong Planning algorithm. The algorithm is an incremental search algorithm that replans the path when there are changes in the environment. In numerical experiments, however, it was observed that the performance of is sensitive in the number of vertex expansions required to update the graph when an edge weight value changes or when a vertex is added or deleted. Although, in most cases, the classical requires a relatively small number of updates, in some other cases the amount of work required by the to find the optimal path can be overwhelming. To address this issue, in this paper, we propose an extension of the baseline algorithm, by making efficient use of a multiscale representation of the environment. This multiscale representation allows one to quickly localize the changed edges, and subsequently update the priority queue efficiently. This incremental multiscale ( for short) algorithm leads to an improvement both in terms of robustness and computational complexity-in the worst case-when compared to the classical . Numerical experiments validate the aforementioned claims.

Connect the dots: how many random points can a regular curve pass through?

Given a class Γ of curves in [0, 1]2, we ask: in a cloud of n uniform random points, how many points can lie on some curve γ ∈ Γ? Classes studied here include curves of length less than or equal to L, Lipschitz graphs, monotone graphs, twice-differentiable curves, and graphs of smooth functions with m-bounded derivatives. We find, for example, that there are twice-differentiable curves containing as many as O P (n 1/3) uniform random points, but not essentially more than this. More generally, we consider point clouds in higher-dimensional cubes [0, 1] d and regular hypersurfaces of specified codimension, finding, for example, that twice-differentiable k-dimensional hypersurfaces in R d may contain as many as O P (n k/(2d-k)) uniform random points. We also consider other notions of ‘incidence’, such as curves passing through given location/direction pairs, and find, for example, that twice-differentiable curves in R 2 may pass through at most O P (n 1/4) uniform random location/direction pairs. Idealized applications in image processing and perceptual psychophysics are described and several open mathematical questions are identified for the attention of the probability community.

Beamlet pyramids: a new form of multiresolution analysis suited for extracting lines, curves, and objects from very noisy image data

We describe a multiscale pyramid of line segments and develop algorithms which exploit that pyramid to recover image features - lines, curves, and blobs - from very noisy data.

A statistical analysis of Fukunaga-Koontz transform

The Fukunaga-Koontz transform (FKT) has been proposed as a feature selection methodology for nearly 32 years. There are a huge number of citations. We have not seen a direct analysis between FKT, and a well established statistical method: Fisher quadratic discriminant analysis (QDA). In this letter, under certain assumptions, we establish such a connection. We speculate that a link in a more general situation is hard to find.