Pedestrian Detection via Classification on Riemannian Manifolds

We present a new algorithm to detect pedestrian in still images utilizing covariance matrices as object descriptors. Since the descriptors do not form a vector space, well known machine learning techniques are not well suited to learn the classifiers. The space of d-dimensional nonsingular covariance matrices can be represented as a connected Riemannian manifold. The main contribution of the paper is a novel approach for classifying points lying on a connected Riemannian manifold using the geometry of the space. The algorithm is tested on INRIA and DaimlerChrysler pedestrian datasets where superior detection rates are observed over the previous approaches

Locality Preserving Projections for Grassmann manifold

Learning on Grassmann manifold has become popular in many computer vision tasks, with the strong capability to extract discriminative information for imagesets and videos. However, such learning algorithms particularly on high-dimensional Grassmann manifold always involve with significantly high computational cost, which seriously limits the applicability of learning on Grassmann manifold in more wide areas. In this research, we propose an unsupervised dimensionality reduction algorithm on Grassmann manifold based on the Locality Preserving Projections (LPP) criterion. LPP is a commonly used dimensionality reduction algorithm for vector-valued data, aiming to preserve local structure of data in the dimension-reduced space. The strategy is to construct a mapping from higher dimensional Grassmann manifold into the one in a relative low-dimensional with more discriminative capability. The proposed method can be optimized as a basic eigenvalue problem. The performance of our proposed method is assessed on several classification and clustering tasks and the experimental results show its clear advantages over other Grassmann based algorithms.Comment: Accepted by IJCAI 201

Spatio-temporal covariance descriptors for action and gesture recognition

We propose a new action and gesture recognition method based on spatio-temporal covariance descriptors and a weighted Riemannian locality preserving projection approach that takes into account the curved space formed by the descriptors. The weighted projection is then exploited during boosting to create a final multiclass classification algorithm that employs the most useful spatio-temporal regions. We also show how the descriptors can be computed quickly through the use of integral video representations. Experiments on the UCF sport, CK+ facial expression and Cambridge hand gesture datasets indicate superior performance of the proposed method compared to several recent state-of-the-art techniques. The proposed method is robust and does not require additional processing of the videos, such as foreground detection, interest-point detection or tracking

Isometry and convexity in dimensionality reduction

The size of data generated every year follows an exponential growth. The number of data points as well as the dimensions have increased dramatically the past 15 years. The gap between the demand from the industry in data processing and the solutions provided by the machine learning community is increasing. Despite the growth in memory and computational power, advanced statistical processing on the order of gigabytes is beyond any possibility. Most sophisticated Machine Learning algorithms require at least quadratic complexity. With the current computer model architecture, algorithms with higher complexity than linear O(N) or O(N logN) are not considered practical. Dimensionality reduction is a challenging problem in machine learning. Often data represented as multidimensional points happen to have high dimensionality. It turns out that the information they carry can be expressed with much less dimensions. Moreover the reduced dimensions of the data can have better interpretability than the original ones. There is a great variety of dimensionality reduction algorithms under the theory of Manifold Learning. Most of the methods such as Isomap, Local Linear Embedding, Local Tangent Space Alignment, Diffusion Maps etc. have been extensively studied under the framework of Kernel Principal Component Analysis (KPCA). In this dissertation we study two current state of the art dimensionality reduction methods, Maximum Variance Unfolding (MVU) and Non-Negative Matrix Factorization (NMF). These two dimensionality reduction methods do not fit under the umbrella of Kernel PCA. MVU is cast as a Semidefinite Program, a modern convex nonlinear optimization algorithm, that offers more flexibility and power compared to iv KPCA. Although MVU and NMF seem to be two disconnected problems, we show that there is a connection between them. Both are special cases of a general nonlinear factorization algorithm that we developed. Two aspects of the algorithms are of particular interest: computational complexity and interpretability. In other words computational complexity answers the question of how fast we can find the best solution of MVU/NMF for large data volumes. Since we are dealing with optimization programs, we need to find the global optimum. Global optimum is strongly connected with the convexity of the problem. Interpretability is strongly connected with local isometry1 that gives meaning in relationships between data points. Another aspect of interpretability is association of data with labeled information. The contributions of this thesis are the following: 1. MVU is modified so that it can scale more efficient. Results are shown on 1 million speech datasets. Limitations of the method are highlighted. 2. An algorithm for fast computations for the furthest neighbors is presented for the first time in the literature. 3. Construction of optimal kernels for Kernel Density Estimation with modern convex programming is presented. For the first time we show that the Leave One Cross Validation (LOOCV) function is quasi-concave. 4. For the first time NMF is formulated as a convex optimization problem 5. An algorithm for the problem of Completely Positive Matrix Factorization is presented. 6. A hybrid algorithm of MVU and NMF the isoNMF is presented combining advantages of both methods. 7. The Isometric Separation Maps (ISM) a variation of MVU that contains classification information is presented. 8. Large scale nonlinear dimensional analysis on the TIMIT speech database is performed. 9. A general nonlinear factorization algorithm is presented based on sequential convex programming. Despite the efforts to scale the proposed methods up to 1 million data points in reasonable time, the gap between the industrial demand and the current state of the art is still orders of magnitude wide.Ph.D.Committee Chair: David Anderson; Committee Co-Chair: Alexander Gray; Committee Member: Anthony Yezzi; Committee Member: Hongyuan Zha; Committee Member: Justin Romberg; Committee Member: Ronald Schafe

The Role of Riemannian Manifolds in Computer Vision: From Coding to Deep Metric Learning

A diverse number of tasks in computer vision and machine learning enjoy from representations of data that are compact yet discriminative, informative and robust to critical measurements. Two notable representations are offered by Region Covariance Descriptors (RCovD) and linear subspaces which are naturally analyzed through the manifold of Symmetric Positive Definite (SPD) matrices and the Grassmann manifold, respectively, two widely used types of Riemannian manifolds in computer vision. As our first objective, we examine image and video-based recognition applications where the local descriptors have the aforementioned Riemannian structures, namely the SPD or linear subspace structure. Initially, we provide a solution to compute Riemannian version of the conventional Vector of Locally aggregated Descriptors (VLAD), using geodesic distance of the underlying manifold as the nearness measure. Next, by having a closer look at the resulting codes, we formulate a new concept which we name Local Difference Vectors (LDV). LDVs enable us to elegantly expand our Riemannian coding techniques to any arbitrary metric as well as provide intrinsic solutions to Riemannian sparse coding and its variants when local structured descriptors are considered. We then turn our attention to two special types of covariance descriptors namely infinite-dimensional RCovDs and rank-deficient covariance matrices for which the underlying Riemannian structure, i.e. the manifold of SPD matrices is out of reach to great extent. %Generally speaking, infinite-dimensional RCovDs offer better discriminatory power over their low-dimensional counterparts. To overcome this difficulty, we propose to approximate the infinite-dimensional RCovDs by making use of two feature mappings, namely random Fourier features and the Nystrom method. As for the rank-deficient covariance matrices, unlike most existing approaches that employ inference tools by predefined regularizers, we derive positive definite kernels that can be decomposed into the kernels on the cone of SPD matrices and kernels on the Grassmann manifolds and show their effectiveness for image set classification task. Furthermore, inspired by attractive properties of Riemannian optimization techniques, we extend the recently introduced Keep It Simple and Straightforward MEtric learning (KISSME) method to the scenarios where input data is non-linearly distributed. To this end, we make use of the infinite dimensional covariance matrices and propose techniques towards projecting on the positive cone in a Reproducing Kernel Hilbert Space (RKHS). We also address the sensitivity issue of the KISSME to the input dimensionality. The KISSME algorithm is greatly dependent on Principal Component Analysis (PCA) as a preprocessing step which can lead to difficulties, especially when the dimensionality is not meticulously set. To address this issue, based on the KISSME algorithm, we develop a Riemannian framework to jointly learn a mapping performing dimensionality reduction and a metric in the induced space. Lastly, in line with the recent trend in metric learning, we devise end-to-end learning of a generic deep network for metric learning using our derivation