The Sixth International Workshop on Automation in Machine Learning aims to identify opportunities and challenges for automation in machine learning, to provide an opportunity for researchers to discuss best practices for automation in machine learning potentially leading to definition of standards, and to provide a forum for researchers to speak out and debate on different ideas in automation in machine learning. The workshop agenda includes four invited keynote speakers and four accepted paper presentations chosen from a peer review process. The workshop seeks to drive engaging and interactive exchange of thoughts and ideas on AutoML.

The Fifth International Workshop on Automation in Machine Learning aims to identify opportunities and challenges for automation in machine learning, to provide an opportunity for researchers to discuss best practices for automation in machine learning potentially leading to definition of standards, and to provide a forum for researchers to speak out and debate on different ideas in automation in machine learning. The workshop agenda includes four invited keynote speakers and four accepted paper presentations chosen from a peer review process. A panel discussion will close out the workshop to allow for an engaging and interactive exchange of thoughts and ideas on AutoML.

Decentralized optimization algorithms have attracted intensive interests recently, as it has a balanced communication pattern, especially when solving large-scale machine learning problems. Stochastic Path Integrated Differential Estimator Stochastic First-Order method (SPIDER-SFO) nearly achieves the algorithmic lower bound in certain regimes for nonconvex problems. However, whether we can find a decentralized algorithm which achieves a similar convergence rate to SPIDER-SFO is still unclear. To tackle this problem, we propose a decentralized variant of SPIDER-SFO, called decentralized SPIDER-SFO (D-SPIDER-SFO). We show that D-SPIDER-SFO achieves a similar gradient computation cost—that is, O(ε−3) for finding an ϵ-approximate first-order stationary point—to its centralized counterpart. To the best of our knowledge, D-SPIDER-SFO achieves the state-of-the-art performance for solving nonconvex optimization problems on decentralized networks in terms of the computational cost. Experiments on different network configurations demonstrate the efficiency of the proposed method.

In many real-world applications, the samples/features acquired are in spatial or temporal order. In such cases, the magnitudes of adjacent samples/features are typically close to each other. Meanwhile, in the high-dimensional scenario, identifying the most relevant samples/features is also desired. In this paper, we consider a regularized model which can simultaneously identify important features and group similar features together. The model is based on a penalty called Absolute Fused Lasso (AFL). The AFL penalty encourages sparsity in the coefficients as well as their successive differences of absolute values' i.e., local constancy of the coefficient components in absolute values. Due to the non-convexity of AFL, it is challenging to develop efficient algorithms to solve the optimization problem. To this end, we employ the Difference of Convex functions (DC) programming to optimize the proposed non-convex problem. At each DC iteration, we adopt the proximal algorithm to solve a convex regularized sub-problem. One of the major contributions of this paper is to develop a highly efficient algorithm to compute the proximal operator. Empirical studies on both synthetic and real-world data sets from Genome-Wide Association Studies demonstrate the efficiency and effectiveness of the proposed approach in simultaneous identifying important features and grouping similar features.

Sparse learning techniques have been routinely used for feature selection as the resulting model usually has a small number of non-zero entries. Safe screening, which eliminates the features that are guaranteed to have zero coefficients for a certain value of the regularization parameter, is a technique for improving the computational efficiency. Safe screening is gaining increasing attention since 1) solving sparse learning formulations usually has a high computational cost especially when the number of features is large and 2) one needs to try several regularization parameters to select a suitable model. In this paper, we propose an approach called “Sasvi" (Safe screening with variational inequalities). Sasvi makes use of the variational inequality that provides the sufficient and necessary optimality condition for the dual problem. Several existing approaches for Lasso screening can be casted as relaxed versions of the proposed Sasvi, thus Sasvi provides a stronger safe screening rule. We further study the monotone properties of Sasvi for Lasso, based on which a sure removal regularization parameter can be identified for each feature. Experimental results on both synthetic and real data sets are reported to demonstrate the effectiveness of the proposed Sasvi for Lasso screening.

Sparse support vector machine (SVM) is a robust predictive model that can effectively remove noise and preserve signals. Like Lasso, it can efficiently learn a solution path based on a set of predefined parameters and therefore provides strong support for model selection. Sparse SVM has been successfully applied in a variety of data mining applications including text mining, bioinformatics, and image processing. The emergence of big-data analysis poses new challenges for model selection with large-scale data that consist of tens of millions samples and features. In this paper, a novel screening technique is proposed to accelerate model selection for l1-regularized l2-SVM and effectively improve its scalability. This technique can precisely identify inactive features in the optimal solution of a l1-regularized l2-SVM model and remove them before training. The technique makes use of the variational inequality and provides a closed-form solution for screening inactive features in different situations. Every feature that is removed by the screening technique is guaranteed to be inactive in the optimal solution. Therefore, when l1-regularized l2-SVM uses the features selected by the technique, it achieves exactly the same result as when it uses the full feature set. Because the technique can remove a large number of inactive features, it can greatly increase the efficiency of model selection for l1-regularized l2-SVM. Experimental results on five high-dimensional benchmark data sets demonstrate the power of the proposed technique.

In this paper, we consider the problem of learning from multiple related tasks for improved generalization performance by extracting their shared structures. The alternating structure optimization (ASO) algorithm, which couples all tasks using a shared feature representation, has been successfully applied in various multitask learning problems. However, ASO is nonconvex and the alternating algorithm only finds a local solution. We first present an improved ASO formulation (iASO) for multitask learning based on a new regularizer. We then convert iASO, a nonconvex formulation, into a relaxed convex one (rASO). Interestingly, our theoretical analysis reveals that rASO finds a globally optimal solution to its nonconvex counterpart iASO under certain conditions. rASO can be equivalently reformulated as a semidefinite program (SDP), which is, however, not scalable to large datasets. We propose to employ the block coordinate descent (BCD) method and the accelerated projected gradient (APG) algorithm separately to find the globally optimal solution to rASO; we also develop efficient algorithms for solving the key subproblems involved in BCD and APG. The experiments on the Yahoo webpages datasets and the Drosophila gene expression pattern images datasets demonstrate the effectiveness and efficiency of the proposed algorithms and confirm our theoretical analysis.

The group Lasso is an extension of the Lasso for feature selection on (predefined) nonoverlapping groups of features. The nonoverlapping group structure limits its applicability in practice. There have been several recent attempts to study a more general formulation where groups of features are given, potentially with overlaps between the groups. The resulting optimization is, however, much more challenging to solve due to the group overlaps. In this paper, we consider the efficient optimization of the overlapping group Lasso penalized problem. We reveal several key properties of the proximal operator associated with the overlapping group Lasso, and compute the proximal operator by solving the smooth and convex dual problem, which allows the use of the gradient descent type of algorithms for the optimization. Our methods and theoretical results are then generalized to tackle the general overlapping group Lasso formulation based on the l(q) norm. We further extend our algorithm to solve a nonconvex overlapping group Lasso formulation based on the capped norm regularization, which reduces the estimation bias introduced by the convex penalty. We have performed empirical evaluations using both a synthetic and the breast cancer gene expression dataset, which consists of 8,141 genes organized into (overlapping) gene sets. Experimental results show that the proposed algorithm is more efficient than existing state-of-the-art algorithms. Results also demonstrate the effectiveness of the nonconvex formulation for overlapping group Lasso.

Learning an optimal kernel plays a pivotal role in kernel-based methods. Recently, an approach called optimal neighborhood kernel learning (ONKL) has been proposed, showing promising classification performance. It assumes that the optimal kernel will reside in the neighborhood of a "pre-specified" kernel. Nevertheless, how to specify such a kernel in a principled way remains unclear. To solve this issue, this paper treats the pre-specified kernel as an extra variable and jointly learns it with the optimal neighborhood kernel and the structure parameters of support vector machines. To avoid trivial solutions, we constrain the pre-specified kernel with a parameterized model. We first discuss the characteristics of our approach and in particular highlight its adaptivity. After that, two instantiations are demonstrated by modeling the pre-specified kernel as a common Gaussian radial basis function kernel and a linear combination of a set of base kernels in the way of multiple kernel learning (MKL), respectively. We show that the optimization in our approach is a min-max problem and can be efficiently solved by employing the extended level method and Nesterov's method. Also, we give the probabilistic interpretation for our approach and apply it to explain the existing kernel learning methods, providing another perspective for their commonness and differences. Comprehensive experimental results on 13 UCI data sets and another two real-world data sets show that via the joint learning process, our approach not only adaptively identifies the pre-specified kernel, but also achieves superior classification performance to the original ONKL and the related MKL algorithms.

Image matting refers to the problem of accurately extracting foreground objects in images and video. The most recent works in natural image matting relies on the local and manifold smoothness assumptions on foreground and background colors on which a cost function is established. In this paper, we present a framework of formulating new regularization for robust solutions and illustrate new algorithms using the standard benchmark images.

Recovering a large matrix from a small subset of its entries is a challenging problem arising in many real world applications, such as recommender system and image in-painting. These problems can be formulated as a general matrix completion problem. The Singular Value Thresholding (SVT) algorithm is a simple and efficient first-order matrix completion method to recover the missing values when the original data matrix is of low rank. SVT has been applied successfully in many applications. However, SVT is computationally expensive when the size of the data matrix is large, which significantly limits its applicability. In this paper, we propose an Accelerated Singular Value Thresholding (ASVT) algorithm which improves the convergence rate from O(1/N) for SVT to O(1/N2), where N is the number of iterations during optimization. Specifically, the dual problem of the nuclear norm minimization problem is derived and an adaptive line search scheme is introduced to solve this dual problem. Consequently, the optimal solution of the primary problem can be readily obtained from that of the dual problem. We have conducted a series of experiments on a synthetic dataset, a distance matrix dataset and a large movie rating dataset. The experimental results have demonstrated the efficiency and effectiveness of the proposed algorithm.

Traditional co-clustering methods identify block structures from static data matrices. However, the data matrices in many applications are dynamic; that is, they evolve smoothly over time. Consequently, the hidden block structures embedded into the matrices are also expected to vary smoothly along the temporal dimension. It is therefore desirable to encourage smoothness between the block structures identified from temporally adjacent data matrices. In this paper, we propose an evolutionary co-clustering formulation for identifying co-cluster structures from time-varying data. The proposed formulation encourages smoothness between temporally adjacent blocks by employing the fused Lasso type of regularization. Our formulation is very flexible and allows for imposing smoothness constraints over only one dimension of the data matrices, thereby enabling its applicability to a large variety of settings. The optimization problem for the proposed formulation is non-convex, non-smooth, and non-separable. We develop an iterative procedure to compute the solution. Each step of the iterative procedure involves a convex, but non-smooth and non-separable problem. We propose to solve this problem in its dual form, which is convex and smooth. This leads to a simple gradient descent algorithm for computing the dual optimal solution. We evaluate the proposed formulation using the Allen Developing Mouse Brain Atlas data. Results show that our formulation consistently outperforms methods without the temporal smoothness constraints.

Alzheimer's Disease (AD) is the most common neurodegenerative disorder associated with aging. Understanding how the disease progresses and identifying related pathological biomarkers for the progression is of primary importance in the clinical diagnosis and prognosis of Alzheimer's disease. In this paper, we develop novel multi-task learning techniques to predict the disease progression measured by cognitive scores and select biomarkers predictive of the progression. In multi-task learning, the prediction of cognitive scores at each time point is considered as a task, and multiple prediction tasks at different time points are performed simultaneously to capture the temporal smoothness of the prediction models across different time points. Specifically, we propose a novel convex fused sparse group Lasso (cFSGL) formulation that allows the simultaneous selection of a common set of biomarkers for multiple time points and specific sets of biomarkers for different time points using the sparse group Lasso penalty and in the meantime incorporates the temporal smoothness using the fused Lasso penalty. The proposed formulation is challenging to solve due to the use of several non-smooth penalties. One of the main technical contributions of this paper is to show that the proximal operator associated with the proposed formulation exhibits a certain decomposition property and can be computed efficiently; thus cFSGL can be solved efficiently using the accelerated gradient method. To further improve the model, we propose two non-convex formulations to reduce the shrinkage bias inherent in the convex formulation. We employ the difference of convex (DC) programming technique to solve the non-convex formulations. We have performed extensive experiments using data from the Alzheimer's Disease Neuroimaging Initiative (ADNI). Results demonstrate the effectiveness of the proposed progression models in comparison with existing methods for disease progression. We also perform longitudinal stability selection to identify and analyze the temporal patterns of biomarkers in disease progression.

Alzheimer's Disease (AD), the most common type of dementia, is a severe neurodegenerative disorder. Identifying markers that can track the progress of the disease has recently received increasing attentions in AD research. A definitive diagnosis of AD requires autopsy confirmation, thus many clinical/cognitive measures including Mini Mental State Examination (MMSE) and Alzheimer's Disease Assessment Scale cognitive subscale (ADAS-Cog) have been designed to evaluate the cognitive status of the patients and used as important criteria for clinical diagnosis of probable AD. In this paper, we propose a multi-task learning formulation for predicting the disease progression measured by the cognitive scores and selecting markers predictive of the progression. Specifically, we formulate the prediction problem as a multi-task regression problem by considering the prediction at each time point as a task. We capture the intrinsic relatedness among different tasks by a temporal group Lasso regularizer. The regularizer consists of two components including an L2,1-norm penalty on the regression weight vectors, which ensures that a small subset of features will be selected for the regression models at all time points, and a temporal smoothness term which ensures a small deviation between two regression models at successive time points. We have performed extensive evaluations using various types of data at the baseline from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database for predicting the future MMSE and ADAS-Cog scores. Our experimental studies demonstrate the effectiveness of the proposed algorithm for capturing the progression trend and the cross-sectional group differences of AD severity. Results also show that most markers selected by the proposed algorithm are consistent with findings from existing cross-sectional studies.

We consider the problem of computing the Euclidean projection of a vector of length $p$ onto a non-negative max-heap---an ordered tree where the values of the nodes are all nonnegative and the value of any parent node is no less than the value(s) of its child node(s). This Euclidean projection plays a building block role in the optimization problem with a non-negative max-heap constraint. Such a constraint is desirable when the features follow an ordered tree structure, that is, a given feature is selected for the given regression/classification task only if its parent node is selected. In this paper, we show that such Euclidean projection problem admits an analytical solution and we develop a top-down algorithm where the key operation is to find the so-called \emph{maximal root-tree} of the subtree rooted at each node. A naive approach for finding the maximal root-tree is to enumerate all the possible root-trees, which, however, does not scale well. We reveal several important properties of the maximal root-tree, based on which we design a bottom-up algorithm with merge for efficiently finding the maximal root-tree. The proposed algorithm has a (worst-case) linear time complexity for a sequential list, and $O(p^2)$ for a general tree. We report simulation results showing the effectiveness of the max-heap for regression with an ordered tree structure. Empirical results show that the proposed algorithm has an expected linear time complexity for many special cases including a sequential list, a full binary tree, and a tree with depth 1.

The group Lasso is an extension of the Lasso for feature selection on (predefined) non-overlapping groups of features. The non-overlapping group structure limits its applicability in practice. There have been several recent attempts to study a more general formulation, where groups of features are given, potentially with overlaps between the groups. The resulting optimization is, however, much more challenging to solve due to the group overlaps. In this paper, we consider the efficient optimization of the overlapping group Lasso penalized problem. We reveal several key properties of the proximal operator associated with the overlapping group Lasso, and compute the proximal operator by solving the smooth and convex dual problem, which allows the use of the gradient descent type of algorithms for the optimization. We have performed empirical evaluations using both synthetic and the breast cancer gene expression data set, which consists of 8,141 genes organized into (overlapping) gene sets. Experimental results show that the proposed algorithm is more efficient than existing state-of-the-art algorithms.

The fused Lasso penalty enforces sparsity in both the coefficients and their successive differences, which is desirable for applications with features ordered in some meaningful way. The resulting problem is, however, challenging to solve, as the fused Lasso penalty is both non-smooth and non-separable. Existing algorithms have high computational complexity and do not scale to large-size problems. In this paper, we propose an Efficient Fused Lasso Algorithm (EFLA) for optimizing this class of problems. One key building block in the proposed EFLA is the Fused Lasso Signal Approximator (FLSA). To efficiently solve FLSA, we propose to reformulate it as the problem of finding an "appropriate" subgradient of the fused penalty at the minimizer, and develop a Subgradient Finding Algorithm (SFA). We further design a restart technique to accelerate the convergence of SFA, by exploiting the special "structures" of both the original and the reformulated FLSA problems. Our empirical evaluations show that, both SFA and EFLA significantly outperform existing solvers. We also demonstrate several applications of the fused Lasso.

We consider the tree structured group Lasso where the structure over the features can be represented as a tree with leaf nodes as features and internal nodes as clusters of the features. The structured regularization with a pre-defined tree structure is based on a group-Lasso penalty, where one group is defined for each node in the tree. Such a regularization can help uncover the structured sparsity, which is desirable for applications with some meaningful tree structures on the features. However, the tree structured group Lasso is challenging to solve due to the complex regularization. In this paper, we develop an efficient algorithm for the tree structured group Lasso. One of the key steps in the proposed algorithm is to solve the Moreau-Yosida regularization associated with the grouped tree structure. The main technical contributions of this paper include (1) we show that the associated Moreau-Yosida regularization admits an analytical solution, and (2) we develop an efficient algorithm for determining the effective interval for the regularization parameter. Our experimental results on the AR and JAFFE face data sets demonstrate the efficiency and effectiveness of the proposed algorithm.

Compared to singular value decomposition (SVD), generalized low-rank approximations of matrices (GLRAM) can consume less computation time, obtain higher compression ratio, and yield competitive classification performance. GLRAM has been successfully applied to applications such as image compression and retrieval, and quite a few extensions have been successively proposed. However, in literature, some basic properties and crucial problems with regard to GLRAM have not been explored or solved yet. For this sake, we revisit GLRAM in this paper. First, we reveal such a close relationship between GLRAM and SVD that GLRAM's objective function is identical to SVD's objective function except the imposed constraints. Second, we derive a lower bound of GLRAM's objective function, and discuss when the lower bound can be touched. Moreover, from the viewpoint of minimizing the lower bound, we answer one open problem raised by Ye (Machine Learning, 2005), i.e., a theoretical justification of the experimental phenomenon that, under given number of reduced dimension, the lowest reconstruction error is obtained when the left and right transformations have equal number of columns. Third, we explore when and why GLRAM can perform well in terms of compression, which is a fundamental problem concerning the usability of GLRAM.

Multi-task learning (MTL) aims to improve generalization performance by learning multiple related tasks simultaneously. In this paper, we consider the problem of learning shared structures from multiple related tasks. We present an improved formulation (iASO) for multi-task learning based on the non-convex alternating structure optimization (ASO) algorithm, in which all tasks are related by a shared feature representation. We convert iASO, a non-convex formulation, into a relaxed convex one, which is, however, not scalable to large data sets due to its complex constraints. We propose an alternating optimization (cASO) algorithm which solves the convex relaxation efficiently, and further show that cASO converges to a global optimum. In addition, we present a theoretical condition, under which cASO can find a globally optimal solution to iASO. Experiments on several benchmark data sets confirm our theoretical analysis. 1.

We consider the problem of computing the Euclidean projection of a vector of length n onto a closed convex set including the l1 ball and the specialized polyhedra employed in (Shalev-Shwartz & Singer, 2006). These problems have played building block roles in solving several l1-norm based sparse learning problems. Existing methods have a worst-case time complexity of O(n log n). In this paper, we propose to cast both Euclidean projections as root finding problems associated with specific auxiliary functions, which can be solved in linear time via bisection. We further make use of the special structure of the auxiliary functions, and propose an improved bisection algorithm. Empirical studies demonstrate that the proposed algorithms are much more efficient than the competing ones for computing the projections.

Logistic Regression is a well-known classification method that has been used widely in many applications of data mining, machine learning, computer vision, and bioinformatics. Sparse logistic regression embeds feature selection in the classification framework using the l1-norm regularization, and is attractive in many applications involving high-dimensional data. In this paper, we propose Lassplore for solving large-scale sparse logistic regression. Specifically, we formulate the problem as the l1-ball constrained smooth convex optimization, and propose to solve the problem using the Nesterov's method, an optimal first-order black-box method for smooth convex optimization. One of the critical issues in the use of the Nesterov's method is the estimation of the step size at each of the optimization iterations. Previous approaches either applies the constant step size which assumes that the Lipschitz gradient is known in advance, or requires a sequence of decreasing step size which leads to slow convergence in practice. In this paper, we propose an adaptive line search scheme which allows to tune the step size adaptively and meanwhile guarantees the optimal convergence rate. Empirical comparisons with several state-of-the-art algorithms demonstrate the efficiency of the proposed Lassplore algorithm for large-scale problems.

Effective diagnosis of Alzheimer's disease (AD), the most common type of dementia in elderly patients, is of primary importance in biomedical research. Recent studies have demonstrated that AD is closely related to the structure change of the brain network, i.e., the connectivity among different brain regions. The connectivity patterns will provide useful imaging-based biomarkers to distinguish Normal Controls (NC), patients with Mild Cognitive Impairment (MCI), and patients with AD. In this paper, we investigate the sparse inverse covariance estimation technique for identifying the connectivity among different brain regions. In particular, a novel algorithm based on the block coordinate descent approach is proposed for the direct estimation of the inverse covariance matrix. One appealing feature of the proposed algorithm is that it allows the user feedback (e.g., prior domain knowledge) to be incorporated into the estimation process, while the connectivity patterns can be discovered automatically. We apply the proposed algorithm to a collection of FDG-PET images from 232 NC, MCI, and AD subjects. Our experimental results demonstrate that the proposed algorithm is promising in revealing the brain region connectivity differences among these groups.

Recent advances in neuroimaging techniques provide great potentials for effective diagnosis of Alzheimer’s disease (AD), the most common form of dementia. Previous studies have shown that AD is closely related to alternation in the functional brain network, i.e., the functional connectivity among different brain regions. In this paper, we consider the problem of learning functional brain connectivity from neuroimaging, which holds great promise for identifying image-based markers used to distinguish Normal Controls (NC), patients with Mild Cognitive Impairment (MCI), and patients with AD. More specifically, we study sparse inverse covariance estimation (SICE), also known as exploratory Gaussian graphical models, for brain connectivity modeling. In particular, we apply SICE to learn and analyze functional brain connectivity patterns from different subject groups, based on a key property of SICE, called the “monotone property” we established in this paper. Our experimental results on neuroimaging PET data of 42 AD, 116 MCI, and 67 NC subjects reveal several interesting connectivity patterns consistent with literature findings, and also some new patterns that can help the knowledge discovery of AD.

We consider the reconstruction of sparse signals in the multiple measurement vector (MMV) model,in which the signal, represented as a matrix, consists of a set of jointly sparse vectors. MMV is an extension of the single measurement vector (SMV) model employed in standard compressive sensing (CS). Recent theoretical studies focus on the convex relaxation of the MMV problem based on the $(2,1)$-norm minimization, which is an extension of the well-known $1$-norm minimization employed in SMV. However, the resulting convex optimization problem in MMV is significantly much more difficult to solve than the one in SMV. Existing algorithms reformulate it as a second-order cone programming (SOCP) or semidefinite programming (SDP), which is computationally expensive to solve for problems of moderate size. In this paper, we propose a new (dual) reformulation of the convex optimization problem in MMV and develop an efficient algorithm based on the prox-method. Interestingly, our theoretical analysis reveals the close connection between the proposed reformulation and multiple kernel learning. Our simulation studies demonstrate the scalability of the proposed algorithm.

Recognition in uncontrolled situations is one of the most important bottlenecks for practical face recognition systems. In particular, few researchers have addressed the challenge to recognize noncooperative or even uncooperative subjects who try to cheat the recognition system by deliberately changing their facial appearance through such tricks as variant expressions or disguise (e.g., by partial occlusions). This paper addresses these problems within the framework of similarity matching. A novel perception-inspired nonmetric partial similarity measure is introduced, which is potentially useful in dealing with the concerned problems because it can help capture the prominent partial similarities that are dominant in human perception. Two methods, based on the general golden section rule and the maximum margin criterion, respectively, are proposed to automatically set the similarity threshold. The effectiveness of the proposed method in handling large expressions, partial occlusions, and other distortions is demonstrated on several well-known face databases.

The matrix, as an extended pattern representation to the vector, has proven to be effective in feature extraction. However, the subsequent classifier following the matrix-pattern- oriented feature extraction is generally still based on the vector pattern representation (namely, MatFE + VecCD), where it has been demonstrated that the effectiveness in classification just attributes to the matrix representation in feature extraction. This paper looks at the possibility of applying the matrix pattern representation to both feature extraction and classifier design. To this end, we propose a so-called fully matrixized approach, i.e., the matrix-pattern-oriented feature extraction followed by the matrix-pattern-oriented classifier design (MatFE + MatCD). To more comprehensively validate MatFE + MatCD, we further consider all the possible combinations of feature extraction (FE) and classifier design (CD) on the basis of patterns represented by matrix and vector respectively, i.e., MatFE + MatCD, MatFE + VecCD, just the matrix-pattern-oriented classifier design (MatCD), the vector-pattern-oriented feature extraction followed by the matrix-pattern-oriented classifier design (VecFE + MatCD), the vector-pattern-oriented feature extraction followed by the vector-pattern-oriented classifier design (VecFE + VecCD) and just the vector-pattern-oriented classifier design (VecCD). The experiments on the combinations have shown the following: 1) the designed fully matrixized approach (MatFE + MatCD) has an effective and efficient performance on those patterns with the prior structural knowledge such as images; and 2) the matrix gives us an alternative feasible pattern representation in feature extraction and classifier designs, and meanwhile provides a necessary validation for "ugly duckling" and "no free lunch" theorems.

Pseudoinverse Linear Discriminant Analysis (PLDA) is a classical and pioneer method that deals with the Small Sample Size (SSS) problem in LDA when applied to such applications as face recognition. However, it is expensive in computation and storage due to direct manipulation on extremely large d × d matrices, where d is the dimension of the sample image. As a result, although frequently cited in literature, PLDA is hardly compared in terms of classification performance with the newly proposed methods. In this paper, we propose a new feature extraction method named RSw + LDA, which is (1) much more efficient than PLDA in both computation and storage; and (2) theoretically equivalent to PLDA, meaning that it produces the same projection matrix as PLDA. Further, to make PLDA deal better with data of nonlinear distribution, we propose a Kernel PLDA (KPLDA) method with the well-known kernel trick. Finally, our experimental results on AR face dataset, a challenging dataset with variations in expression, lighting and occlusion, show that PLDA (or RSw + LDA) can achieve significantly higher classification accuracy than the recently proposed Linear Discriminant Analysis via QR decomposition and Discriminant Common Vectors, and KPLDA can yield better classification performance compared to PLDA and Kernel PCA.

The goal of this comment is to first point out two loopholes in the paper by Li (2006): (1) so-designed efficient maximal margin criterion (MMC) algorithm for small sample size (SSS) problem is problematic and (2) the discussion on the equivalence with the null-space-based methods in SSS problem does not hold. Then, we will present a really efficient MMC algorithm for SSS problem.