Subspace learning based face recognition methods have attracted considerable interests in recent years, including Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Locality Preserving Projection (LPP), Neighborhood Preserving Embedding (NPE) and Marginal Fisher Analysis (MFA). However, a disadvantage of all these approaches is that their computations involve eigendecomposition of dense matrices which is expensive in both time and memory. In this paper, we propose a novel dimensionality reduction framework, called Spectral Regression (SR), for efficient regularized subspace learning. SR casts the problem of learning the projective functions into a regression framework, which avoids eigen-decomposition of dense matrices. Also, with the regression based framework, different kinds of regularizers can be naturally incorporated into our algorithm which makes it more flexible. Computational analysis shows that SR has only linear-time complexity which is a huge speed up comparing to the cubic-time complexity of the ordinary approaches. Experimental results on face recognition demonstrate the effectiveness and efficiency of our method. 1.

Linear Discriminant Analysis (LDA) has been a popular method for extracting features which preserve class separability. The projection vectors are commonly obtained by maximizing the between class covariance and simultaneously minimizing the within class covariance. LDA can be performed either in the original input space or in the reproducing kernel Hilbert space (RKHS) into which data points are mapped, which leads to Kernel Discriminant Analysis (KDA). When the data are highly nonlinear distributed, KDA can achieve better performance than LDA. However, computing the projective functions in KDA involves eigen-decomposition of kernel matrix, which is very expensive when a large number of training samples exist. In this paper, we present a new algorithm for kernel discriminant analysis, called Spectral Regression Kernel Discriminant Analysis (SRKDA). By using spectral graph analysis, SRKDA casts discriminant analysis into a regression framework which facilitates both efficient computation and the use of regularization techniques. Specifically, SRKDA only needs to solve a set of regularized regression problems and there is no eigenvector computation involved, which is a huge save of computational cost. Moreover, the new formulation makes it very easy to develop incremental version of the algorithm which can fully utilize the computational results of the existing training samples. Extensive experiments on spoken letter, handwritten digit image and face image data demonstrate the effectiveness and efficiency of the proposed algorithm.

published in Numerical Algorithms and the paper's text is reprinted here by kind permission

Recently the problem of dimensionality reduction (or, subspace learning) has received a lot of interests in many fields of information processing, including data mining, information retrieval, and pattern recognition. Some popular methods include Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA) and Locality Preserving Projection (LPP). However, a disadvantage of all these approaches is that the learned projective functions are linear combinations of all the original features, thus it is often difficult to interpret the results. In this paper, we propose a novel dimensionality reduction framework, called Unified Sparse Subspace Learning (USSL), for learning sparse projections. USSL casts the problem of learning the projective functions into a regression framework, which facilitates the use of different kinds of regularizers. By using a L1-norm regularizer (lasso), the sparse projections can be efficiently computed. Experimental results on real world classification and clustering problems demonstrate the effectiveness of our method.

Relevance feedback is a well established and effective framework for narrowing down the gap between low-level visual features and high-level semantic concepts in content-based image retrieval. In most of traditional implementations of relevance feedback, a distance metric or a classifier is usually learned from user’s provided negative and positive examples. However, due to the limitation of the user’s feedbacks and the high dimensionality of the feature space, one is often confront with the issue of the curse of the dimensionality. Recently, several researchers have considered manifold ways to address this issue, such as Locality Preserving Projections, Augmented Relation Embedding, and Semantic Subspace Projection. In this paper, by using techniques from spectral graph embedding and regression, we propose a unified framework, called spectral regression, for learning an image subspace. This framework facilitates the analysis of the differences and connections between the algorithms mentioned above. And more crucially, it provides much faster computation and therefore makes the retrieval system capable of responding to the user’s query more efficiently.

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We consider the problem of document indexing and representation. Recently, Locality Preserving Indexing (LPI) was proposed for learning a compact document subspace. Different from Latent Semantic Indexing (LSI) which is optimal in the sense of global Euclidean structure, LPI is optimal in the sense of local manifold structure. However, LPI is not efficient in time and memory which makes it difficult to be applied to very large data set. Specifically, the computation of LPI involves eigen-decompositions of two dense matrices which is expensive. In this paper, we propose a new algorithm called Regularized Locality Preserving Indexing (RLPI). Benefit from recent progresses on spectral graph analysis, we cast the original LPI algorithm into a regression framework which enable us to avoid eigen-decomposition of dense matrices. Also, with the regression based framework, different kinds of regularizers can be naturally incorporated into our algorithm which makes it more flexible. Extensive experimental results show that RLPI obtains similar or better results comparing to LPI and it is significantly faster, which makes it an efficient and effective data preprocessing method for large scale text clustering, classification and retrieval.

We present an iterative algorithm (BIN) for scaling all the rows and columns of a real symmetric matrix to unit 2-norm. We study the theoretical convergence properties and its relation to optimal conditioning. Numerical

We show how to compute an LU factorization of a matrix when the factors of a leading principle submatrix are already known. The approach incorporates pivoting akin to partial pivoting, a strategy we call incremental pivoting. An implementation using the Formal Linear Algebra Methods Environment (FLAME) Application Programming Interface (API) is described. Experimental results demonstrate practical numerical stability and high performance on an Intel Itanium2 processor based server.

We suggest a local hybrid approximation scheme based on polynomials and radial basis functions, and use it to improve the scattered data fitting algorithm of [7]. Similar to that algorithm, the new method has linear computational complexity and is therefore suitable for large real world data. Numerical examples suggest that it can produce high quality artifact–free approximations that are more accurate than those given by the original method where pure polynomial local approximations are used.