Graph Regularized Tensor Sparse Coding for Image Representation

Sparse coding (SC) is an unsupervised learning scheme that has received an increasing amount of interests in recent years. However, conventional SC vectorizes the input images, which destructs the intrinsic spatial structures of the images. In this paper, we propose a novel graph regularized tensor sparse coding (GTSC) for image representation. GTSC preserves the local proximity of elementary structures in the image by adopting the newly proposed tubal-tensor representation. Simultaneously, it considers the intrinsic geometric properties by imposing graph regularization that has been successfully applied to uncover the geometric distribution for the image data. Moreover, the returned sparse representations by GTSC have better physical explanations as the key operation (i.e., circular convolution) in the tubal-tensor model preserves the shifting invariance property. Experimental results on image clustering demonstrate the effectiveness of the proposed scheme

Structure-Aware Classification using Supervised Dictionary Learning

In this paper, we propose a supervised dictionary learning algorithm that aims to preserve the local geometry in both dimensions of the data. A graph-based regularization explicitly takes into account the local manifold structure of the data points. A second graph regularization gives similar treatment to the feature domain and helps in learning a more robust dictionary. Both graphs can be constructed from the training data or learned and adapted along the dictionary learning process. The combination of these two terms promotes the discriminative power of the learned sparse representations and leads to improved classification accuracy. The proposed method was evaluated on several different datasets, representing both single-label and multi-label classification problems, and demonstrated better performance compared with other dictionary based approaches

Optimization with Sparsity-Inducing Penalties

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view

Graph Signal Representation with Wasserstein Barycenters

In many applications signals reside on the vertices of weighted graphs. Thus, there is the need to learn low dimensional representations for graph signals that will allow for data analysis and interpretation. Existing unsupervised dimensionality reduction methods for graph signals have focused on dictionary learning. In these works the graph is taken into consideration by imposing a structure or a parametrization on the dictionary and the signals are represented as linear combinations of the atoms in the dictionary. However, the assumption that graph signals can be represented using linear combinations of atoms is not always appropriate. In this paper we propose a novel representation framework based on non-linear and geometry-aware combinations of graph signals by leveraging the mathematical theory of Optimal Transport. We represent graph signals as Wasserstein barycenters and demonstrate through our experiments the potential of our proposed framework for low-dimensional graph signal representation

Convex and Network Flow Optimization for Structured Sparsity

We consider a class of learning problems regularized by a structured sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over groups of variables. Whereas much effort has been put in developing fast optimization techniques when the groups are disjoint or embedded in a hierarchy, we address here the case of general overlapping groups. To this end, we present two different strategies: On the one hand, we show that the proximal operator associated with a sum of l_infinity-norms can be computed exactly in polynomial time by solving a quadratic min-cost flow problem, allowing the use of accelerated proximal gradient methods. On the other hand, we use proximal splitting techniques, and address an equivalent formulation with non-overlapping groups, but in higher dimension and with additional constraints. We propose efficient and scalable algorithms exploiting these two strategies, which are significantly faster than alternative approaches. We illustrate these methods with several problems such as CUR matrix factorization, multi-task learning of tree-structured dictionaries, background subtraction in video sequences, image denoising with wavelets, and topographic dictionary learning of natural image patches.Comment: to appear in the Journal of Machine Learning Research (JMLR

Distributed Machine Learning via Sufficient Factor Broadcasting

Matrix-parametrized models, including multiclass logistic regression and sparse coding, are used in machine learning (ML) applications ranging from computer vision to computational biology. When these models are applied to large-scale ML problems starting at millions of samples and tens of thousands of classes, their parameter matrix can grow at an unexpected rate, resulting in high parameter synchronization costs that greatly slow down distributed learning. To address this issue, we propose a Sufficient Factor Broadcasting (SFB) computation model for efficient distributed learning of a large family of matrix-parameterized models, which share the following property: the parameter update computed on each data sample is a rank-1 matrix, i.e., the outer product of two "sufficient factors" (SFs). By broadcasting the SFs among worker machines and reconstructing the update matrices locally at each worker, SFB improves communication efficiency --- communication costs are linear in the parameter matrix's dimensions, rather than quadratic --- without affecting computational correctness. We present a theoretical convergence analysis of SFB, and empirically corroborate its efficiency on four different matrix-parametrized ML models

Graph Regularized Non-negative Matrix Factorization By Maximizing Correntropy

Non-negative matrix factorization (NMF) has proved effective in many clustering and classification tasks. The classic ways to measure the errors between the original and the reconstructed matrix are $l_2$ distance or Kullback-Leibler (KL) divergence. However, nonlinear cases are not properly handled when we use these error measures. As a consequence, alternative measures based on nonlinear kernels, such as correntropy, are proposed. However, the current correntropy-based NMF only targets on the low-level features without considering the intrinsic geometrical distribution of data. In this paper, we propose a new NMF algorithm that preserves local invariance by adding graph regularization into the process of max-correntropy-based matrix factorization. Meanwhile, each feature can learn corresponding kernel from the data. The experiment results of Caltech101 and Caltech256 show the benefits of such combination against other NMF algorithms for the unsupervised image clustering