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Distance Metric Learning for Large Margin Nearest Neighbor Classification

JOURNAL OF MACHINE LEARNING RESEARCH 2009年第2期10卷 207-244页

作者： Weinberger, Kilian Q. Saul, Lawrence K. Yahoo Res Santa Clara CA USA Univ Calif San Diego Dept Comp Sci & Engn La Jolla CA 92093 USA

The accuracy of k-nearest neighbor (kNN) classification depends significantly on the metric used to compute distances between different examples. In this paper, we show how to learn a Mahalanobis distance metric for kNN classification from labeled examples. The Mahalanobis metric can equivalently be viewed as a global linear transformation of the input space that precedes kNN classification using Euclidean distances. In our approach, the metric is trained with the goal that the k-nearest neighbors always belong to the same class while examples from different classes are separated by a large margin. As in support vector machines (SVMs), the margin criterion leads to a convex optimization based on the hinge loss. Unlike learning in SVMs, however, our approach requires no modification or extension for problems in multiway (as opposed to binary) classification. In our framework, the Mahalanobis distance metric is obtained as the solution to a semidefinite program. On several data sets of varying size and difficulty, we find that metrics trained in this way lead to significant improvements in kNN classification. Sometimes these results can be further improved by clustering the training examples and learning an individual metric within each cluster. We show how to learn and combine these local metrics in a globally integrated manner.

关键词： convex optimization semi-definite programming Mahalanobis distance metric learning multi-class classification support vector machines

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On the Reconstruction of Block-Sparse Signals With an Optimal Number of Measurements

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2009年第8期57卷 3075-3085页

作者： Stojnic, Mihailo Parvaresh, Farzad Hassibi, Babak Purdue Univ Sch Ind Engn W Lafayette IN 47907 USA CALTECH Ctr Math Informat Pasadena CA 91125 USA CALTECH Dept Elect Engn Pasadena CA 91125 USA

Let A be an M by N matrix (M infinity, (*) finds the sparsest solution to A x = y, with overwhelming probability in A, for any x whose sparsity is k/n (1/2) - O(epsilon), provided m/n > 1 - 1/d, and d = Omega(log(... 详细信息

Let A be an M by N matrix (M < N) which is an instance of a real random Gaussian ensemble. In compressed sensing we are interested in finding the sparsest solution to the system of equations Ax = y for a given y. In general, whenever the sparsity of x: is smaller than half the dimension of y then with overwhelming probability over A the sparsest solution is unique and can be found by an exhaustive search over x with an exponential time complexity for any y. The recent work of Candes, Donoho, and Tho shows that minimization of the l(1) norm of x subject to A x = y results in the sparsest solution provided the sparsity of x, say K, is smaller than a certain threshold for a given number of measurements. Specifically, if the dimension of y approaches the dimension of x, the sparsity of x should be K < 0.239 N. Here, we consider the case where x is block sparse, i.e., x consists of n = N/d blocks where each block is of length d and is either a zero vector or a nonzero vector (under nonzero vector we consider a vector that can have both, zero and nonzero components). Instead of l(1)-norm relaxation, we consider the following relaxation: min parallel to X-1 parallel to(2) + parallel to X-2 parallel to(2) +...+ parallel to X-n parallel to(2), subject to Ax = y where X-i = (X(i-1)d+1, X(i-1)d+2,...,X-id)(T) for i = 1, 2,..., N. Our main result is that as n -> infinity, (*) finds the sparsest solution to A x = y, with overwhelming probability in A, for any x whose sparsity is k/n (1/2) - O(epsilon), provided m/n > 1 - 1/d, and d = Omega(log(1/epsilon)/epsilon(3)). The relaxation given in (*) can be solved in polynomial time using semi-definite programming.

关键词： Compressed sensing block-sparse signals semi-definite programming

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Accurate time delay estimation based passive localization

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SIGNAL PROCESSING 2009年第9期89卷 1835-1838页

作者： Lui, Kenneth W. K. Chan, Frankie K. W. So, H. C. City Univ Hong Kong Dept Elect Engn Kowloon Hong Kong Peoples R China

A conventional approach for source localization is to utilize time delay measurements of the emitted signal received at an array of sensors. The time delay information is then employed to construct a set of hyperbolic equations from which the target position can be determined. In this paper, we utilize semi-definite programming (SDP) technique to derive a passive source localization algorithm which can integrate the available a priori knowledge such as admissible target range and other cues. It is shown that the SDP method is superior to the well-known two-step weighted least squares method at lower signal-to-noise ratio conditions. Crown Copyright (C) 2009 Published by Elsevier B.V. All rights reserved.

关键词： Time delay estimation Source localization semi-definite programming

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ORTHOGONAL HERMITE FUNCTIONS BASED ORTHOGONAL PULSE SHAPING METHOD

ORTHOGONAL HERMITE FUNCTIONS BASED ORTHOGONAL PULSE SHAPING ...

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22nd IEEE Canadian Conference on Electrical and Computer Engineering

作者： Wu Xuan-li Qiu Xin Ning Xiao-yan Lin Di Harbin Inst Technol Commun Res Ctr Harbin 150006 Peoples R China

ISBN: (纸本)9781424435098

Ultra-Wideband (UWB) communication has attracted more attentions due to its advantages in short range applications. As one of the key techniques in UWB systems, many pulse shaping methods have been proposed. The semi-definite programming (SDP) based pulse shaping method can obtain the pulse with highest power efficiency by far. However, an accurate timing of 40ps is required, and such requirement cannot be implemented precisely using modem hardware. Furthermore, the power efficiency of the second orthogonal pulse will be lower than the first pulse. To solve such problems, the linear combination of orthogonal Hermite functions is used in the SDP method and a modification method is then proposed to eliminate the direct current (DC) component. Simulation results show that orthogonal pulses without DC component can be obtained and the power efficiency of the obtained pulses are more than 80%. Moreover, the obtained orthogonal pulses will have shorter pulse duration due to the energy concentration property of orthogonal spaces.

关键词： ultra wideband pulse shaping semi-definite programming orthogonal Hermite functions

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New Unsupervised Support Vector Machines

New Unsupervised Support Vector Machines

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20th International Conference on Multiple Criteria Decision Making

作者： Zhao, Kun Tian, Ying-jie Deng, Nai-yang China Agr Univ Coll Sci Beijing Peoples R China Beijing Wuzi Univ Logist Sch Beijing Peoples R China Chinese Acad Sci Res Ctr Fictitious Econ & Data Sci Beijing Peoples R China

ISBN: (纸本)9783642022975

Support Vector Machines (SVMs) have been dominant learning techniques for more than ten years, and mostly applied to supervised learning problems. Recently nice results are obtained by two-class unsupervised classification algorithms where the optimization problems based on Bounded C-SVMs, Bounded v-SVMs and Lagrangian SVMs respectively are relaxed to semi-definite programming. In this paper we propose another approach to solve unsupervised classification problem, which directly relaxes a modified version of primal problem of SVMs with label variables to a semi-definite programming. The preliminary numerical results show that Our new algorithm often obtains more accurate results than other unsupervised classification methods, although the relaxation has no tight bound, as shown by an example where its approximate ratio of optimal values can be arbitrarily large.

关键词： Support Vector Machines semi-definite programming unsupervised learning Kernel

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Efficient Design of 2-D Nonseparable Filters of Low Complexity

Efficient Design of 2-D Nonseparable Filters of Low Complexi...

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16th IEEE International Conference on Image Processing

作者： Fanian, K. Tuan, H. D. Nguyen, T. Q. Univ New South Wales Sch Elect Engn & Telecommun Unsw Sydney NSW 2052 Australia Univ Calif San Diego Dept Elect & Comp Engn La Jolla CA 92093 USA

ISBN: (纸本)9781424456536

By using semi-definite programming (SDP) as a tool, a new deign for Two-Dimensional (2-D) Diamond-Shaped (DS) filters is developed. Surprisingly, the diamond shape of the filter is exactly expressed by using simple 2-D trigonometric polynomial curves of second order. In contrast to the high-order polynomial transformation based methods, the order of the designed filters is kept moderate with no performance sacrifices. Unlike conventional non-separable 2-D filters, the designed filters allow fast digital implementation despite being nonseparable and hence they are of low complexity. Numerical Simulations with application to quincunx image sampling are also performed to illustrate the viability of our method.

关键词： 2-D filter Diamond-Shaped filters semi-definite programming

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Exact calculation of robustness of entanglement via convex semi-definite programming

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INTERNATIONAL JOURNAL OF QUANTUM INFORMATION 2005年第3期3卷 511-533页

作者： Jafarizadeh, MA Mirzaee, M Rezaee, M Tabriz Univ Dept Theoret Phys & Asdtrophys Tabriz 51664 Iran Inst Studies Theoret Phys & Math Tehran 19395 Iran Res Inst Fundamental Sci Tabriz 51664 Iran

In general, the calculation of robustness of entanglement for the mixed entangled quantum states is rather difficult to handle analytically. Using the convex semi-definite programming method, we claculate exactly the robustness of entanglement of some mixed entangled quantum states such as the generic two-qubit state in the Wootters basis, 2 circle times 2 Bell decomposable (BD) states, iso-concurrence decomposable states, 2 circle times 3 Bell decomposable states, d circle times d Werner and isotropic states, a one parameter 3 circle times 3 state and a multi-partite isotropic state. The results are in agreement with those of 2 circle times 2 density matrices and have already been calculated in Refs. 1 and 2. Also, an analytic expression is given for separable states that wipe out all the entanglements. It is further shown that they are on the boundary of separable states, as pointed out in Ref. 3.

关键词： robustness of entanglement semi-definite programming Bell decomposable states Werner and isotropic states

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An augmented Lagrangian SQP method for solving some special class of nonlinear semi-definite programming problems

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COMPUTATIONAL & APPLIED MATHEMATICS 2005年第3期24卷 461-486页

作者： Mostafa, El-Sayed M. E. Univ Alexandria Fac Sci Dept Math Alexandria Egypt

In this paper, we consider a special class of nonlinear semi-definite programming problems that represents the fixed order H(2)/H(infinity) synthesis problem. An augmented Lagrangian sequential quadratic programming method combined with a trust region globalization strategy is described, taking advantage of the problem structure and using inexact computations. Some numerical examples that illustrate the performance of the method are given.

关键词： semi-definite programming linear quadratic control nonlinear programming trust region methods

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A semi-definite programming-based Underestimation method for global optimization in molecular docking*

A Semi-Definite programming-based Underestimation method for...

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44th IEEE Conference on Decision Control/European Control Conference (CCD-ECC)

作者： Paschalidis, Ioannis Ch. Shen, Yang Vajda, Sandor Vakili, Pirooz Boston Univ Ctr Informat & Syst Engn Brookline MA 02446 USA

ISBN: (纸本)0780395670

The paper introduces a new global optimization method that is targeted to solve molecular docking problems, an important class of problems in computational biology. The search method is based on finding general convex quadratic underestimators to the binding energy function that is funnel-like. Finding the optimum underestimator requires solving a semi-definite programming problem, hence the name semi-definite programming based Underestimation (SDU). The optimal underestimator is used to bias sampling in the search region. A detailed comparison of SDU with a related method of Convex Global Underestimator (CGU), a discussion of the convergence properties of SDU, and computational results of the application of SDU to a number of rigid protein-protein docking problems are provided.

关键词： computational biology global optimization semi-definite programming molecular docking.

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Complete characterizations of stable Farkas' lemma and cone-convex programming duality

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MATHEMATICAL programming 2008年第2期114卷 335-347页

作者： Jeyakumar, V. Lee, G. M. Univ New S Wales Dept Appl Math Sydney NSW 2052 Australia Pukyong Natl Univ Dept Appl Math Pusan 608737 South Korea

We establish necessary and sufficient conditions for a stable Farkas' lemma. We then derive necessary and sufficient conditions for a stable duality of a cone-convex optimization problem, where strong duality holds for each linear perturbation of a given convex objective function. As an application, we obtain stable duality results for convex semi-definite programs and convex second-order cone programs.

关键词： stable Farkas lemma stable duality convex optimization semi-definite programming second-order cone programming

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