Dennis Wei

Ensemble Estimators for Multivariate Entropy Estimation

The problem of estimation of density functionals like entropy and mutual information has received much attention in the statistics and information theory communities. A large class of estimators of functionals of the probability density suffer from the curse of dimensionality, wherein the mean squared error decays increasingly slowly as a function of the sample size T as the dimension d of the samples increases. In particular, the rate is often glacially slow of order O(T-γ/d), where γ > 0 is a rate parameter. Examples of such estimators include kernel density estimators, k -nearest neighbor (k-NN) density estimators, k-NN entropy estimators, intrinsic dimension estimators, and other examples. In this paper, we propose a weighted affine combination of an ensemble of such estimators, where optimal weights can be chosen such that the weighted estimator converges at a much faster dimension invariant rate of O(T1). Furthermore, we show that these optimal weights can be determined by solving a convex optimization problem which can be performed offline and does not require training data. We illustrate the superior performance of our weighted estimator for two important applications: 1) estimating the Panter-Dite distortion-rate factor; and 2) estimating the Shannon entropy for testing the probability distribution of a random sample.

Multistage adaptive estimation of sparse signals

This paper considers sequential adaptive estimation of sparse signals under a constraint on the total sensing effort. The advantage of adaptivity in this context is the ability to focus more resources on regions of space where signal components exist, thereby improving performance. A dynamic programming formulation is derived for the allocation of sensing effort to minimize the expected estimation loss. Based on the method of open-loop feedback control, allocation policies are then developed for a variety of loss functions. The policies are optimal in the two-stage case, generalizing an optimal two-stage policy proposed by Bashan , and improve monotonically thereafter with the number of stages. Numerical simulations show gains up to several dB as compared to recently proposed adaptive methods, and dramatic gains compared to non-adaptive estimation. An application to radar imaging is also presented.

A Dictionary Approach to Electron Backscatter Diffraction Indexing.

We propose a framework for indexing of grain and subgrain structures in electron backscatter diffraction patterns of polycrystalline materials. We discretize the domain of a dynamical forward model onto a dense grid of orientations, producing a dictionary of patterns. For each measured pattern, we identify the most similar patterns in the dictionary, and identify boundaries, detect anomalies, and index crystal orientations. The statistical distribution of these closest matches is used in an unsupervised binary decision tree (DT) classifier to identify grain boundaries and anomalous regions. The DT classifies a pattern as an anomaly if it has an abnormally low similarity to any pattern in the dictionary. It classifies a pixel as being near a grain boundary if the highly ranked patterns in the dictionary differ significantly over the pixels neighborhood. Indexing is accomplished by computing the mean orientation of the closest matches to each pattern. The mean orientation is estimated using a maximum likelihood approach that models the orientation distribution as a mixture of Von Mises-Fisher distributions over the quaternionic three sphere. The proposed dictionary matching approach permits segmentation, anomaly detection, and indexing to be performed in a unified manner with the additional benefit of uncertainty quantification.

Sparse Filter Design Under a Quadratic Constraint: Low-Complexity Algorithms

This paper considers three problems in sparse filter design, the first involving a weighted least-squares constraint on the frequency response, the second a constraint on mean squared error in estimation, and the third a constraint on signal-to-noise ratio in detection. The three problems are unified under a single framework based on sparsity maximization under a quadratic performance constraint. Efficient and exact solutions are developed for specific cases in which the matrix in the quadratic constraint is diagonal, block-diagonal, banded, or has low condition number. For the more difficult general case, a low-complexity algorithm based on backward greedy selection is described with emphasis on its efficient implementation. Examples in wireless channel equalization and minimum-variance distortionless-response beamforming show that the backward selection algorithm yields optimally sparse designs in many instances while also highlighting the benefits of sparse design.

A Branch-and-Bound Algorithm for Quadratically-Constrained Sparse Filter Design

This paper presents an exact algorithm for sparse filter design under a quadratic constraint on filter performance. The algorithm is based on branch-and-bound, a combinatorial optimization procedure that can either guarantee an optimal solution or produce a sparse solution with a bound on its deviation from optimality. To reduce the complexity of branch-and-bound, several methods are developed for bounding the optimal filter cost. Bounds based on infeasibility yield incrementally accumulating improvements with minimal computation, while two convex relaxations, referred to as linear and diagonal relaxations, are derived to provide stronger bounds. The approximation properties of the two relaxations are characterized analytically as well as numerically. Design examples involving wireless channel equalization and minimum-variance distortionless-response beamforming show that the complexity of obtaining certifiably optimal solutions can often be significantly reduced by incorporating diagonal relaxations, especially in more difficult instances. In the case of early termination due to computational constraints, diagonal relaxations strengthen the bound on the proximity of the final solution to the optimum.

Parameter Estimation in Spherical Symmetry Groups

This letter considers statistical estimation problems where the probability distribution of the observed random variable is invariant with respect to actions of a finite topological group. It is shown that any such distribution must satisfy a restricted finite mixture representation. When specialized to the case of distributions over the sphere that are invariant to the actions of a finite spherical symmetry group G, a group-invariant extension of the Von Mises Fisher (VMF) distribution is obtained. The G-invariant VMF is parameterized by location and scale parameters that specify the distributions mean orientation and its concentration about the mean, respectively. Using the restricted finite mixture representation these parameters can be estimated using an Expectation Maximization (EM) maximum likelihood (ML) estimation algorithm. This is illustrated for the problem of mean crystal orientation estimation under the spherically symmetric group associated with the crystal form, e.g., cubic or octahedral or hexahedral. Simulations and experiments establish the advantages of the extended VMF EM-ML estimator for data acquired by Electron Backscatter Diffraction (EBSD) microscopy of a polycrystalline Nickel alloy sample.

Marginal Likelihoods for Distributed Parameter Estimation of Gaussian Graphical Models

We consider distributed estimation of the inverse covariance matrix, also called the concentration or precision matrix, in Gaussian graphical models. Traditional centralized estimation often requires global inference of the covariance matrix, which can be computationally intensive in large dimensions. Approximate inference based on message-passing algorithms, on the other hand, can lead to unstable and biased estimation in loopy graphical models. Here, we propose a general framework for distributed estimation based on a maximum marginal likelihood (MML) approach. This approach computes local parameter estimates by maximizing marginal likelihoods defined with respect to data collected from local neighborhoods. Due to the non-convexity of the MML problem, we introduce and solve a convex relaxation. The local estimates are then combined into a global estimate without the need for iterative message-passing between neighborhoods. The proposed algorithm is naturally parallelizable and computationally efficient, thereby making it suitable for high-dimensional problems. In the classical regime where the number of variables p is fixed and the number of samples T increases to infinity, the proposed estimator is shown to be asymptotically consistent and to improve monotonically as the local neighborhood size increases. In the high-dimensional scaling regime where both p and T increase to infinity, the convergence rate to the true parameters is derived and is seen to be comparable to centralized maximum-likelihood estimation. Extensive numerical experiments demonstrate the improved performance of the two-hop version of the proposed estimator, which suffices to almost close the gap to the centralized maximum likelihood estimator at a reduced computational cost.

A performance guarantee for adaptive estimation of sparse signals

This paper studies adaptive sensing for estimating the nonzero amplitudes of a sparse signal. We consider a previously proposed optimal two-stage policy for allocating sensing resources. We derive an upper bound on the mean squared error resulting from the optimal two-stage policy and a corresponding lower bound on the improvement over non-adaptive sensing. It is shown that the adaptation gain is related to the detectability of nonzero signal components as characterized by a Bhattacharyya coefficient, thus quantifying analytically the dependence on the sparsity level of the signal, the signal-to-noise ratio, and the sensing resource budget. The bound is shown to be a good approximation to the optimal two-stage gain through numerical simulations.

Learning sparse two-level boolean rules

This paper develops a novel optimization framework for learning accurate and sparse two-level Boolean rules for classification, both in Conjunctive Normal Form (CNF, i.e. AND-of-ORs) and in Disjunctive Normal Form (DNF, i.e. OR-of-ANDs). In contrast to opaque models (e.g. neural networks), sparse two-level Boolean rules gain the crucial benefit of interpretability, which is necessary in a wide range of applications such as law and medicine and is attracting considerable attention in machine learning. This paper introduces two principled objective functions to trade off classification accuracy and sparsity, where 0-1 error and Hamming loss are used to characterize accuracy. We propose efficient procedures to optimize these objectives based on linear programming (LP) relaxation, block coordinate descent, and alternating minimization. We also describe a new approach to rounding any fractional values in the optimal solutions of LP relaxations. Experiments show that our new algorithms based on the Hamming loss objective provide excellent tradeoffs between accuracy and sparsity with improvements over state-of-the-art methods.

Optigrow: People Analytics for Job Transfers

The information technology (IT) services industry is undergoing a rapid change with the growth of market interest in cloud, analytics, mobile, social, and security technologies. For service providers to match this pace, they must rapidly transform their workforce in terms of job roles, and do so without incurring excessive cost while continuing to deliver core services. In this paper, we describe a big data approach to enable such a transformation through internal job transfers of suitable employees from legacy areas to growth areas. Toward this end, we use data on employee expertise to mathematically profile skill sets required for growth area jobs and develop a statistical scoring algorithm to prioritize internal candidates to be transferred to those growth area jobs. We describe how we have enacted this analytics procedure within the IT services division of the IBM Corporation and provide empirical results. We also discuss the lessons learned during the deployment, focusing mostly on organizational reasons preventing wide uptake.