Dictionary Learning and Sparse Coding on Grassmann Manifolds: An Extrinsic Solution (original) (raw)

Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds

International Journal of Computer Vision, 2015

Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in Euclidean spaces. With the aim of building a bridge between the two realms, we address the problem of sparse coding and dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping. This in turn enables us to extend two sparse coding schemes to Grassmann manifolds. Furthermore, we propose closed-form solutions for learning a Grassmann dictionary, atom by atom. Lastly, to handle non-linearity in data, we extend the proposed Grassmann sparse coding and dictionary learning algorithms through embedding into Hilbert spaces.

Riemannian coding and dictionary learning: Kernels to the rescue

2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015

While sparse coding on non-flat Riemannian manifolds has recently become increasingly popular, existing solutions either are dedicated to specific manifolds, or rely on optimization problems that are difficult to solve, especially when it comes to dictionary learning. In this paper, we propose to make use of kernels to perform coding and dictionary learning on Riemannian manifolds. To this end, we introduce a general Riemannian coding framework with its kernel-based counterpart. This lets us (i) generalize beyond the special case of sparse coding; (ii) introduce efficient solutions to two coding schemes; (iii) learn the kernel parameters; (iv) perform unsupervised and supervised dictionary learning in a much simpler manner than previous Riemannian coding methods. We demonstrate the effectiveness of our approach on three different types of non-flat manifolds, and illustrate its generality by applying it to Euclidean spaces, which also are Riemannian manifolds.

Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach

Lecture Notes in Computer Science, 2012

Recent advances suggest that a wide range of computer vision problems can be addressed more appropriately by considering non-Euclidean geometry. This paper tackles the problem of sparse coding and dictionary learning in the space of symmetric positive definite matrices, which form a Riemannian manifold. With the aid of the recently introduced Stein kernel (related to a symmetric version of Bregman matrix divergence), we propose to perform sparse coding by embedding Riemannian manifolds into reproducing kernel Hilbert spaces. This leads to a convex and kernel version of the Lasso problem, which can be solved efficiently. We furthermore propose an algorithm for learning a Riemannian dictionary (used for sparse coding), closely tied to the Stein kernel. Experiments on several classification tasks (face recognition, texture classification, person reidentification) show that the proposed sparse coding approach achieves notable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as tensor sparse coding, Riemannian locality preserving projection, and symmetry-driven accumulation of local features.

Sparse Coding on Symmetric Positive Definite Manifolds Using Bregman Divergences

IEEE Transactions on Neural Networks and Learning Systems, 2015

This paper introduces sparse coding and dictionary learning for Symmetric Positive Definite (SPD) matrices, which are often used in machine learning, computer vision and related areas. Unlike traditional sparse coding schemes that work in vector spaces, in this paper we discuss how SPD matrices can be described by sparse combination of dictionary atoms, where the atoms are also SPD matrices. We propose to seek sparse coding by embedding the space of SPD matrices into Hilbert spaces through two types of Bregman matrix divergences. This not only leads to an efficient way of performing sparse coding, but also an online and iterative scheme for dictionary learning. We apply the proposed methods to several computer vision tasks where images are represented by region covariance matrices. Our proposed algorithms outperform state-of-the-art methods on a wide range of classification tasks, including face recognition, action recognition, material classification and texture categorization.

Kernel analysis on Grassmann manifolds for action recognition

Pattern Recognition Letters, 2013

Modelling video sequences by subspaces has recently shown promise for recognising human actions. Subspaces are able to accommodate the effects of various image variations and can capture the dynamic properties of actions. Subspaces form a non-Euclidean and curved Riemannian manifold known as a Grassmann manifold. Inference on manifold spaces usually is achieved by embedding the manifolds in higher dimensional Euclidean spaces. In this paper, we instead propose to embed the Grassmann manifolds into reproducing kernel Hilbert spaces and then tackle the problem of discriminant analysis on such manifolds. To achieve efficient machinery, we propose graph-based local discriminant analysis that utilises withinclass and between-class similarity graphs to characterise intra-class compactness and inter-class separability, respectively. Experiments on KTH, UCF Sports, and Ballet datasets show that the proposed approach obtains marked improvements in discrimination accuracy in comparison to several state-of-the-art methods, such as the kernel version of affine hull image-set distance, tensor canonical correlation analysis, spatial-temporal words and hierarchy of discriminative space-time neighbourhood features. (M.T. Harandi). 1 The geodesic distance takes into account the curvature of manifolds; an example is the distance between two points on a sphere.

Discriminative sparse coding on multi-manifolds

Knowledge-Based Systems, 2013

Sparse coding has been popularly used as an effective data representation method in various applications, such as computer vision, medical imaging and bioinformatics. However, the conventional sparse coding algorithms and their manifold-regularized variants (graph sparse coding and Laplacian sparse coding), learn codebooks and codes in an unsupervised manner and neglect class information that is available in the training set. To address this problem, we propose a novel discriminative sparse coding method based on multi-manifolds, that learns discriminative class-conditioned codebooks and sparse codes from both data feature spaces and class labels. First, the entire training set is partitioned into multiple manifolds according to the class labels. Then, we formulate the sparse coding as a manifold-manifold matching problem and learn class-conditioned codebooks and codes to maximize the manifold margins of different classes. Lastly, we present a data sample-manifold matching-based strategy to classify the unlabeled data samples. Experimental results on somatic mutations identification and breast tumor classification based on ultrasonic images demonstrate the efficacy of the proposed data representation and classification approach.

Graph embedding discriminant analysis on Grassmannian manifolds for improved image set matching

2011

A convenient way of dealing with image sets is to represent them as points on Grassmannian manifolds. While several recent studies explored the applicability of discriminant analysis on such manifolds, the conventional formalism of discriminant analysis suffers from not considering the local structure of the data. We propose a discriminant analysis approach on Grassmannian manifolds, based on a graph-embedding framework. We show that by introducing within-class and between-class similarity graphs to characterise intra-class compactness and inter-class separability, the geometrical structure of data can be exploited. Experiments on several image datasets (PIE, BANCA, MoBo, ETH-80) show that the proposed algorithm obtains considerable improvements in discrimination accuracy, in comparison to three recent methods: Grassmann Discriminant Analysis (GDA), Kernel GDA, and the kernel version of Affine Hull Image Set Distance. We further propose a Grassmannian kernel, based on canonical correlation between subspaces, which can increase discrimination accuracy when used in combination with previous Grassmannian kernels.

Online Dictionary Learning on Symmetric Positive Definite Manifolds with Vision Applications

Symmetric Positive Definite (SPD) matrices in the form of region covariances are considered rich descriptors for images and videos. Recent studies suggest that exploiting the Riemannian geometry of the SPD manifolds could lead to improved performances for vision applications. For tasks involving processing large-scale and dynamic data in computer vision, the underlying model is required to progressively and efficiently adapt itself to the new and unseen observations. Motivated by these requirements, this paper studies the problem of online dictionary learning on the SPD manifolds. We make use of the Stein divergence to recast the problem of online dictionary learning on the manifolds to a problem in Reproducing Kernel Hilbert Spaces, for which, we develop efficient algorithms by taking into account the geometric structure of the SPD manifolds. To our best knowledge, our work is the first study that provides a solution for online dictionary learning on the SPD manifolds. Empirical results on both large-scale image classification task and dynamic video processing tasks validate the superior performance of our approach as compared to several state-of-the-art algorithms.

Expanding the Family of Grassmannian Kernels: An Embedding Perspective

Lecture Notes in Computer Science, 2014

Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques developed for Euclidean spaces (e.g., support vector machines) with subspaces, several recent studies have proposed to embed the Grassmannian into a Hilbert space by making use of a positive definite kernel. Unfortunately, only two Grassmannian kernels are known, none of which -as we will show-is universal, which limits their ability to approximate a target function arbitrarily well. Here, we introduce several positive definite Grassmannian kernels, including universal ones, and demonstrate their superiority over previously-known kernels in various tasks, such as classification, clustering, sparse coding and hashing.

Grassmann Registration Manifolds for Face Recognition

Lecture Notes in Computer Science, 2008

Motivated by image perturbation and the geometry of manifolds, we present a novel method combining these two elements. First, we form a tangent space from a set of perturbed images and observe that the tangent space admits a vector space structure. Second, we embed the approximated tangent spaces on a Grassmann manifold and employ a chordal distance as the means for comparing subspaces. The matching process is accelerated using a coarse to fine strategy. Experiments on the FERET database suggest that the proposed method yields excellent results using both holistic and local features. Specifically, on the FERET Dup2 data set, our proposed method achieves 83.8% rank 1 recognition: to our knowledge the currently the best result among all non-trained methods. Evidence is also presented that peak recognition performance is achieved using roughly 100 distinct perturbed images.