Optimal Riemannian quantization with an application to air traffic analysis

The goal of optimal quantization is to find the best approximation of a probability distribution by a discrete measure with finite support. When dealing with empirical distributions, this boils down to finding the best summary of the data by a smaller number of points, and automatically yields a K-means-type clustering. In this paper, we introduce Competitive Learning Riemannian Quantization (CLRQ), an online algorithm that computes the optimal summary when the data does not belong to a vector space, but rather a Riemannian manifold. We prove its convergence and show simulated examples on the sphere and the hyperbolic plane. We also provide an application to real data by using CLRQ to create summaries of images of covariance matrices estimated from air traffic images. These summaries are representative of the air traffic complexity and yield clusterings of the airspaces into zones that are homogeneous with respect to that criterion. They can then be compared using discrete optimal transport and be further used as inputs of a machine learning algorithm or as indexes in a traffic database

Clustering in Hilbert simplex geometry

Clustering categorical distributions in the probability simplex is a fundamental task met in many applications dealing with normalized histograms. Traditionally, the differential-geometric structures of the probability simplex have been used either by (i) setting the Riemannian metric tensor to the Fisher information matrix of the categorical distributions, or (ii) defining the dualistic information-geometric structure induced by a smooth dissimilarity measure, the Kullback-Leibler divergence. In this work, we introduce for this clustering task a novel computationally-friendly framework for modeling the probability simplex termed {\em Hilbert simplex geometry}. In the Hilbert simplex geometry, the distance function is described by a polytope. We discuss the pros and cons of those different statistical modelings, and benchmark experimentally these geometries for center-based $k$-means and $k$-center clusterings. We show that Hilbert metric in the probability simplex satisfies the property of information monotonicity. Furthermore, since a canonical Hilbert metric distance can be defined on any bounded convex subset of the Euclidean space, we also consider Hilbert's projective geometry of the elliptope of correlation matrices and study its clustering performances.Comment: 42 page

Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices

Data encoded as symmetric positive definite (SPD) matrices frequently arise in many areas of computer vision and machine learning. While these matrices form an open subset of the Euclidean space of symmetric matrices, viewing them through the lens of non-Euclidean Riemannian geometry often turns out to be better suited in capturing several desirable data properties. However, formulating classical machine learning algorithms within such a geometry is often non-trivial and computationally expensive. Inspired by the great success of dictionary learning and sparse coding for vector-valued data, our goal in this paper is to represent data in the form of SPD matrices as sparse conic combinations of SPD atoms from a learned dictionary via a Riemannian geometric approach. To that end, we formulate a novel Riemannian optimization objective for dictionary learning and sparse coding in which the representation loss is characterized via the affine invariant Riemannian metric. We also present a computationally simple algorithm for optimizing our model. Experiments on several computer vision datasets demonstrate superior classification and retrieval performance using our approach when compared to sparse coding via alternative non-Riemannian formulations

Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.Comment: arXiv admin note: text overlap with arXiv:1407.112

Learning Mid-level Words on Riemannian Manifold for Action Recognition

Human action recognition remains a challenging task due to the various sources of video data and large intra-class variations. It thus becomes one of the key issues in recent research to explore effective and robust representation to handle such challenges. In this paper, we propose a novel representation approach by constructing mid-level words in videos and encoding them on Riemannian manifold. Specifically, we first conduct a global alignment on the densely extracted low-level features to build a bank of corresponding feature groups, each of which can be statistically modeled as a mid-level word lying on some specific Riemannian manifold. Based on these mid-level words, we construct intrinsic Riemannian codebooks by employing K-Karcher-means clustering and Riemannian Gaussian Mixture Model, and consequently extend the Riemannian manifold version of three well studied encoding methods in Euclidean space, i.e. Bag of Visual Words (BoVW), Vector of Locally Aggregated Descriptors (VLAD), and Fisher Vector (FV), to obtain the final action video representations. Our method is evaluated in two tasks on four popular realistic datasets: action recognition on YouTube, UCF50, HMDB51 databases, and action similarity labeling on ASLAN database. In all cases, the reported results achieve very competitive performance with those most recent state-of-the-art works.Comment: 10 page

Dictionary Learning and Sparse Coding on Statistical Manifolds

In this paper, we propose a novel information theoretic framework for dictionary learning (DL) and sparse coding (SC) on a statistical manifold (the manifold of probability distributions). Unlike the traditional DL and SC framework, our new formulation does not explicitly incorporate any sparsity inducing norm in the cost function being optimized but yet yields sparse codes. Our algorithm approximates the data points on the statistical manifold (which are probability distributions) by the weighted Kullback-Leibeler center/mean (KL-center) of the dictionary atoms. The KL-center is defined as the minimizer of the maximum KL-divergence between itself and members of the set whose center is being sought. Further, we prove that the weighted KL-center is a sparse combination of the dictionary atoms. This result also holds for the case when the KL-divergence is replaced by the well known Hellinger distance. From an applications perspective, we present an extension of the aforementioned framework to the manifold of symmetric positive definite matrices (which can be identified with the manifold of zero mean gaussian distributions), $\mathcal{P}_n$. We present experiments involving a variety of dictionary-based reconstruction and classification problems in Computer Vision. Performance of the proposed algorithm is demonstrated by comparing it to several state-of-the-art methods in terms of reconstruction and classification accuracy as well as sparsity of the chosen representation.Comment: arXiv admin note: substantial text overlap with arXiv:1604.0693

Image segmentation with superpixel-based covariance descriptors in low-rank representation

This paper investigates the problem of image segmentation using superpixels. We propose two approaches to enhance the discriminative ability of the superpixel's covariance descriptors. In the first one, we employ the Log-Euclidean distance as the metric on the covariance manifolds, and then use the RBF kernel to measure the similarities between covariance descriptors. The second method is focused on extracting the subspace structure of the set of covariance descriptors by extending a low rank representation algorithm on to the covariance manifolds. Experiments are carried out with the Berkly Segmentation Dataset, and compared with the state-of-the-art segmentation algorithms, both methods are competitive.Comment: 7 pages, 2 figures, 1 tabl

A Novel Space-Time Representation on the Positive Semidefinite Con for Facial Expression Recognition

In this paper, we study the problem of facial expression recognition using a novel space-time geometric representation. We describe the temporal evolution of facial landmarks as parametrized trajectories on the Riemannian manifold of positive semidefinite matrices of fixed-rank. Our representation has the advantage to bring naturally a second desirable quantity when comparing shapes -- the spatial covariance -- in addition to the conventional affine-shape representation. We derive then geometric and computational tools for rate-invariant analysis and adaptive re-sampling of trajectories, grounding on the Riemannian geometry of the manifold. Specifically, our approach involves three steps: 1) facial landmarks are first mapped into the Riemannian manifold of positive semidefinite matrices of rank 2, to build time-parameterized trajectories; 2) a temporal alignment is performed on the trajectories, providing a geometry-aware (dis-)similarity measure between them; 3) finally, pairwise proximity function SVM (ppfSVM) is used to classify them, incorporating the latter (dis-)similarity measure into the kernel function. We show the effectiveness of the proposed approach on four publicly available benchmarks (CK+, MMI, Oulu-CASIA, and AFEW). The results of the proposed approach are comparable to or better than the state-of-the-art methods when involving only facial landmarks.Comment: To be appeared at ICCV 201

Graph Quantization

Vector quantization(VQ) is a lossy data compression technique from signal processing, which is restricted to feature vectors and therefore inapplicable for combinatorial structures. This contribution presents a theoretical foundation of graph quantization (GQ) that extends VQ to the domain of attributed graphs. We present the necessary Lloyd-Max conditions for optimality of a graph quantizer and consistency results for optimal GQ design based on empirical distortion measures and stochastic optimization. These results statistically justify existing clustering algorithms in the domain of graphs. The proposed approach provides a template of how to link structural pattern recognition methods other than GQ to statistical pattern recognition.Comment: 24 pages; submitted to CVI

Deep manifold-to-manifold transforming network for action recognition

Symmetric positive definite (SPD) matrices (e.g., covariances, graph Laplacians, etc.) are widely used to model the relationship of spatial or temporal domain. Nevertheless, SPD matrices are theoretically embedded on Riemannian manifolds. In this paper, we propose an end-to-end deep manifold-to-manifold transforming network (DMT-Net) which can make SPD matrices flow from one Riemannian manifold to another more discriminative one. To learn discriminative SPD features characterizing both spatial and temporal dependencies, we specifically develop three novel layers on manifolds: (i) the local SPD convolutional layer, (ii) the non-linear SPD activation layer, and (iii) the Riemannian-preserved recursive layer. The SPD property is preserved through all layers without any requirement of singular value decomposition (SVD), which is often used in the existing methods with expensive computation cost. Furthermore, a diagonalizing SPD layer is designed to efficiently calculate the final metric for the classification task. To evaluate our proposed method, we conduct extensive experiments on the task of action recognition, where input signals are popularly modeled as SPD matrices. The experimental results demonstrate that our DMT-Net is much more competitive over state-of-the-art