Transferable Linear Discriminant Analysis

On the Equivalence of Linear Discriminant Analysis and Least Squares Regression

Studying the relationship between linear discriminant analysis (LDA) and least squares regression (LSR) is of great theoretical and practical significance. It is well-known that the two-class LDA is equivalent to an LSR problem, and directly casting multiclass LDA as an LSR problem, however, becomes more challenging. Recent study reveals that the equivalence between multiclass LDA and LSR can be established based on a special class indicator matrix, but under a mild condition which may not hold under the scenarios with low-dimensional or oversampled data. In this article, we show that the equivalence between multiclass LDA and LSR can be established based on arbitrary linearly independent class indicator vectors and without any condition. In addition, we show that LDA is also equivalent to a constrained LSR based on the data-dependent indicator vectors. It can be concluded that under exactly the same mild condition, such two regressions are both equivalent to the null space LDA method. Illuminated by the equivalence of LDA and LSR, we propose a direct LDA classifier to replace the conventional framework of LDA plus extra classifier. Extensive experiments well validate the above theoretic analysis.

Understanding How Pretraining Regularizes Deep Learning Algorithms

Deep learning algorithms have led to a series of breakthroughs in computer vision, acoustical signal processing, and others. However, they have only been popularized recently due to the groundbreaking techniques developed for training deep architectures. Understanding the training techniques is important if we want to further improve them. Through extensive experimentation, Erhan et al. (2010) empirically illustrated that unsupervised pretraining has an effect of regularization for deep learning algorithms. However, theoretical justifications for the observation remain elusive. In this article, we provide theoretical supports by analyzing how unsupervised pretraining regularizes deep learning algorithms. Specifically, we interpret deep learning algorithms as the traditional Tikhonov-regularized batch learning algorithms that simultaneously learn predictors in the input feature spaces and the parameters of the neural networks to produce the Tikhonov matrices. We prove that unsupervised pretraining helps in learning meaningful Tikhonov matrices, which will make the deep learning algorithms uniformly stable and the learned predictor will generalize fast w.r.t. the sample size. Unsupervised pretraining, therefore, can be interpreted as to have the function of regularization.

TRCLA: A Transfer Learning Approach to Reduce Negative Transfer for Cellular Learning Automata

In most traditional machine learning algorithms, the training and testing datasets have identical distributions and feature spaces. However, these assumptions have not held in many real applications. Although transfer learning methods have been invented to fill this gap, they introduce new challenges as negative transfers (NTs). Most previous research considered NT a significant problem, but they pay less attention to solving it. This study will propose a transductive learning algorithm based on cellular learning automata (CLA) to alleviate the NT issue. Two famous learning automata (LA) entitled estimators are applied as estimator CLA in the proposed algorithms. A couple of new decision criteria called merit and and attitude parameters are introduced to CLA to limit NT. The proposed algorithms are applied to standard LA environments. The experiments show that the proposed algorithm leads to higher accuracy and less NT results.

Ratio Sum Versus Sum Ratio for Linear Discriminant Analysis

Dimension reduction is a critical technology for high-dimensional data processing, where Linear Discriminant Analysis (LDA) and its variants are effective supervised methods. However, LDA prefers to feature with smaller variance, which causes feature with weak discriminative ability retained. In this paper, we propose a novel Ratio Sum for Linear Discriminant Analysis (RSLDA), which aims at maximizing discriminative ability of each feature in subspace. To be specific, it maximizes the sum of ratio of the between-class distance to the within-class distance in each dimension of subspace. Since the original RSLDA problem is difficult to obtain the closed solution, an equivalent problem is developed which can be solved by an alternative optimization algorithm. For solving the equivalent problem, it is transformed into two sub-problems, one of which can be solved directly, the other is changed into a convex optimization problem, where singular value decomposition is employed instead of matrix inversion. Consequently, performance of algorithm cannot be affected by the non-singularity of covariance matrix. Furthermore, Kernel RSLDA (KRSLDA) is presented to improve the robustness of RSLDA. Additionally, time complexity of RSLDA and KRSLDA are analyzed. Extensive experiments show that RSLDA and KRSLDA outperforms other comparison methods on toy datasets and multiple public datasets.

Robust Latent Common Subspace Learning for Transferable Feature Representation

This paper proposes a novel robust latent common subspace learning (RLCSL) method by integrating low-rank and sparse constraints into a joint learning framework. Specifically, we transform the data from source and target domains into a latent common subspace to perform the data reconstruction, i.e., the transformed source data is used to reconstruct the transformed target data. We impose joint low-rank and sparse constraints on the reconstruction coefficient matrix which can achieve following objectives: (1) the data from different domains can be interlaced by using the low-rank constraint; (2) the data from different domains but with the same label can be aligned together by using the sparse constraint. In this way, the new feature representation in the latent common subspace is discriminative and transferable. To learn a suitable classifier, we also integrate the classifier learning and feature representation learning into a unified objective and thus the high-level semantics label (data label) is fully used to guide the learning process of these two tasks. Experiments are conducted on diverse data sets for image, object, and document classifications, and encouraging experimental results show that the proposed method outperforms some state-of-the-arts methods.