Non-Redundant Spectral Dimensionality Reduction

Spectral dimensionality reduction algorithms are widely used in numerous domains, including for recognition, segmentation, tracking and visualization. However, despite their popularity, these algorithms suffer from a major limitation known as the "repeated Eigen-directions" phenomenon. That is, many of the embedding coordinates they produce typically capture the same direction along the data manifold. This leads to redundant and inefficient representations that do not reveal the true intrinsic dimensionality of the data. In this paper, we propose a general method for avoiding redundancy in spectral algorithms. Our approach relies on replacing the orthogonality constraints underlying those methods by unpredictability constraints. Specifically, we require that each embedding coordinate be unpredictable (in the statistical sense) from all previous ones. We prove that these constraints necessarily prevent redundancy, and provide a simple technique to incorporate them into existing methods. As we illustrate on challenging high-dimensional scenarios, our approach produces significantly more informative and compact representations, which improve visualization and classification tasks

Differential geometric regularization for supervised learning of classifiers

We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to an estimator of the class probability P(y|\vec x). The regularization term measures the volume of this submanifold, based on the intuition that overfitting produces rapid local oscillations and hence large volume of the estimator. This technique can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. In experiments, we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification.http://proceedings.mlr.press/v48/baia16.pdfPublished versio

Active Semi-Supervised Learning Using Sampling Theory for Graph Signals

We consider the problem of offline, pool-based active semi-supervised learning on graphs. This problem is important when the labeled data is scarce and expensive whereas unlabeled data is easily available. The data points are represented by the vertices of an undirected graph with the similarity between them captured by the edge weights. Given a target number of nodes to label, the goal is to choose those nodes that are most informative and then predict the unknown labels. We propose a novel framework for this problem based on our recent results on sampling theory for graph signals. A graph signal is a real-valued function defined on each node of the graph. A notion of frequency for such signals can be defined using the spectrum of the graph Laplacian matrix. The sampling theory for graph signals aims to extend the traditional Nyquist-Shannon sampling theory by allowing us to identify the class of graph signals that can be reconstructed from their values on a subset of vertices. This approach allows us to define a criterion for active learning based on sampling set selection which aims at maximizing the frequency of the signals that can be reconstructed from their samples on the set. Experiments show the effectiveness of our method.Comment: 10 pages, 6 figures, To appear in KDD'1

Contribution to supervised representation learning: algorithms and applications.

278 p.In this thesis, we focus on supervised learning methods for pattern categorization. In this context, itremains a major challenge to establish efficient relationships between the discriminant properties of theextracted features and the inter-class sparsity structure.Our first attempt to address this problem was to develop a method called "Robust Discriminant Analysiswith Feature Selection and Inter-class Sparsity" (RDA_FSIS). This method performs feature selectionand extraction simultaneously. The targeted projection transformation focuses on the most discriminativeoriginal features while guaranteeing that the extracted (or transformed) features belonging to the sameclass share a common sparse structure, which contributes to small intra-class distances.In a further study on this approach, some improvements have been introduced in terms of theoptimization criterion and the applied optimization process. In fact, we proposed an improved version ofthe original RDA_FSIS called "Enhanced Discriminant Analysis with Class Sparsity using GradientMethod" (EDA_CS). The basic improvement is twofold: on the first hand, in the alternatingoptimization, we update the linear transformation and tune it with the gradient descent method, resultingin a more efficient and less complex solution than the closed form adopted in RDA_FSIS.On the other hand, the method could be used as a fine-tuning technique for many feature extractionmethods. The main feature of this approach lies in the fact that it is a gradient descent based refinementapplied to a closed form solution. This makes it suitable for combining several extraction methods andcan thus improve the performance of the classification process.In accordance with the above methods, we proposed a hybrid linear feature extraction scheme called"feature extraction using gradient descent with hybrid initialization" (FE_GD_HI). This method, basedon a unified criterion, was able to take advantage of several powerful linear discriminant methods. Thelinear transformation is computed using a descent gradient method. The strength of this approach is thatit is generic in the sense that it allows fine tuning of the hybrid solution provided by different methods.Finally, we proposed a new efficient ensemble learning approach that aims to estimate an improved datarepresentation. The proposed method is called "ICS Based Ensemble Learning for Image Classification"(EM_ICS). Instead of using multiple classifiers on the transformed features, we aim to estimate multipleextracted feature subsets. These were obtained by multiple learned linear embeddings. Multiple featuresubsets were used to estimate the transformations, which were ranked using multiple feature selectiontechniques. The derived extracted feature subsets were concatenated into a single data representationvector with strong discriminative properties.Experiments conducted on various benchmark datasets ranging from face images, handwritten digitimages, object images to text datasets showed promising results that outperformed the existing state-ofthe-art and competing methods