ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 05 Oct 2022 19:00:04 GMT2022-10-05T19:00:04Z5041- Characterization of all solutions for undersampled uncorrelated linear discriminant analysis problemshttps://scholarbank.nus.edu.sg/handle/10635/52823Title: Characterization of all solutions for undersampled uncorrelated linear discriminant analysis problems
Authors: Chu, D.; Goh, S.T.; Hung, Y.S.
Abstract: In this paper the uncorrelated linear discriminant analysis (ULDA) for undersampled problems is studied. The main contributions of the present work include the following: (i) all solutions of the optimization problem used for establishing the ULDA are parameterized explicitly; (ii) the optimal solutions among all solutions of the corresponding optimization problem are characterized in terms of both the ratio of between-class distance to within-class distance and the maximum likelihood classification, and it is proved that these optimal solutions are exactly the solutions of the corresponding optimization problem with minimum Frobenius norm, also minimum nuclear norm; these properties provide a good mathematical justification for preferring the minimum-norm transformation over other possible solutions as the optimal transformation in ULDA; (iii) explicit necessary and sufficient conditions are provided to ensure that these minimal solutions lead to a larger ratio of between-class distance to within-class distance, thereby achieving larger discrimination in the reduced subspace than that in the original data space, and our numerical experiments show that these necessary and sufficient conditions hold true generally. Furthermore, a new and fast ULDA algorithm is developed, which is eigendecomposition-free and SVD-free, and its effectiveness is demonstrated by some real-world data sets. © 2011 Society for Industrial and Applied Mathematics.
Sat, 01 Jan 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/528232011-01-01T00:00:00Z
- A new and fast implementation for null space based linear discriminant analysishttps://scholarbank.nus.edu.sg/handle/10635/115561Title: A new and fast implementation for null space based linear discriminant analysis
Authors: Chu, D.; Thye, G.S.
Abstract: In this paper we present a new implementation for the null space based linear discriminant analysis. The main features of our implementation include: (i) the optimal transformation matrix is obtained easily by only orthogonal transformations without computing any eigendecomposition and singular value decomposition (SVD), consequently, our new implementation is eigendecomposition-free and SVD-free; (ii) its main computational complexity is from a economic QR factorization of the data matrix and a economic QR factorization of a n×n matrix with column pivoting, here n is the sample size, thus our new implementation is a fast one. The effectiveness of our new implementation is demonstrated by some real-world data sets. © 2009 Elsevier Ltd. All rights reserved.
Thu, 01 Apr 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1155612010-04-01T00:00:00Z
- A new and fast orthogonal linear discriminant analysis on undersampled problemshttps://scholarbank.nus.edu.sg/handle/10635/52755Title: A new and fast orthogonal linear discriminant analysis on undersampled problems
Authors: Chu, D.; Goh, S.T.
Abstract: Dimensionality reduction has become a ubiquit ous preprocessing step in many applications. Linear discriminant analysis (LDA) has been known to be one of the most optimal dimensionality reduction methods for classification. However, a main disadvantage of LDA is that the so-called total scatter matrix must be nonsingular. But, in many applications, the scatter matrices can be singular since the data points are from a very high-dimensional space, and thus usually the number of the data samples is smaller than the data dimension. This is known as the undersampled problem. Many generalized LDA methods have been proposed in the past to overcome this singularity problem. There is a commonality for these generalized LDA methods; that is, they compute the optimal linear transformations by computing some eigen-decompositions and involving some matrix inversions. However, the eigen-decomposition is computationally expensive, and the involvement of matrix inverses may lead to the methods not numerically stable if the associated matrices are ill-conditioned. Hence, many existing LDA methods have high computational cost and have potential numerical instability problems. In this paper we present a new orthogonal LDA method for the undersampled problem. The main features of our proposed LDA method include the following: (i) the optimal transformation matrix is obtained easily by only orthogonal transformations without computing any eigen-decomposition and matrix inverse, and, consequently, our LDA method is inverse-free and numerically stable; (ii) our LDA method is implemented by using several QR factorizations and is a fast one. The effectiveness of our new method is illustrated by some real-world data sets. © 2010 Society for Industrial and Applied Mathematics.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/527552010-01-01T00:00:00Z
- Several classes of even-variable balanced boolean functions with optimal algebraic immunityhttps://scholarbank.nus.edu.sg/handle/10635/115492Title: Several classes of even-variable balanced boolean functions with optimal algebraic immunity
Authors: Tan, C.-H.; Goh, S.-T.
Abstract: In this paper, we constructed six infinite classes of balanced Boolean functions. These six classes of Boolean functions achieved optimal algebraic degree, optimal algebraic immunity and high nonlinear ity. Furthermore, we gave the proof of the lower bound of the nonlinearities of these balanced Boolean functions and proved the better lower bound of nonlinearity for Carlet-Feng's Boolean function. Copyright © 2011 The Institute of Electronics, Information and Communication Engineers.
Sat, 01 Jan 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1154922011-01-01T00:00:00Z