82 research outputs found

    Grassmann Learning for Recognition and Classification

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    Computational performance associated with high-dimensional data is a common challenge for real-world classification and recognition systems. Subspace learning has received considerable attention as a means of finding an efficient low-dimensional representation that leads to better classification and efficient processing. A Grassmann manifold is a space that promotes smooth surfaces, where points represent subspaces and the relationship between points is defined by a mapping of an orthogonal matrix. Grassmann learning involves embedding high dimensional subspaces and kernelizing the embedding onto a projection space where distance computations can be effectively performed. In this dissertation, Grassmann learning and its benefits towards action classification and face recognition in terms of accuracy and performance are investigated and evaluated. Grassmannian Sparse Representation (GSR) and Grassmannian Spectral Regression (GRASP) are proposed as Grassmann inspired subspace learning algorithms. GSR is a novel subspace learning algorithm that combines the benefits of Grassmann manifolds with sparse representations using least squares loss §¤1-norm minimization for improved classification. GRASP is a novel subspace learning algorithm that leverages the benefits of Grassmann manifolds and Spectral Regression in a framework that supports high discrimination between classes and achieves computational benefits by using manifold modeling and avoiding eigen-decomposition. The effectiveness of GSR and GRASP is demonstrated for computationally intensive classification problems: (a) multi-view action classification using the IXMAS Multi-View dataset, the i3DPost Multi-View dataset, and the WVU Multi-View dataset, (b) 3D action classification using the MSRAction3D dataset and MSRGesture3D dataset, and (c) face recognition using the ATT Face Database, Labeled Faces in the Wild (LFW), and the Extended Yale Face Database B (YALE). Additional contributions include the definition of Motion History Surfaces (MHS) and Motion Depth Surfaces (MDS) as descriptors suitable for activity representations in video sequences and 3D depth sequences. An in-depth analysis of Grassmann metrics is applied on high dimensional data with different levels of noise and data distributions which reveals that standardized Grassmann kernels are favorable over geodesic metrics on a Grassmann manifold. Finally, an extensive performance analysis is made that supports Grassmann subspace learning as an effective approach for classification and recognition

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

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    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

    The Structure Transfer Machine Theory and Applications

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    Representation learning is a fundamental but challenging problem, especially when the distribution of data is unknown. We propose a new representation learning method, termed Structure Transfer Machine (STM), which enables feature learning process to converge at the representation expectation in a probabilistic way. We theoretically show that such an expected value of the representation (mean) is achievable if the manifold structure can be transferred from the data space to the feature space. The resulting structure regularization term, named manifold loss, is incorporated into the loss function of the typical deep learning pipeline. The STM architecture is constructed to enforce the learned deep representation to satisfy the intrinsic manifold structure from the data, which results in robust features that suit various application scenarios, such as digit recognition, image classification and object tracking. Compared to state-of-the-art CNN architectures, we achieve the better results on several commonly used benchmarks\footnote{The source code is available. https://github.com/stmstmstm/stm }

    Biview learning for human posture segmentation from 3D points cloud

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    Posture segmentation plays an essential role in human motion analysis. The state-of-the-art method extracts sufficiently high-dimensional features from 3D depth images for each 3D point and learns an efficient body part classifier. However, high-dimensional features are memory-consuming and difficult to handle on large-scale training dataset. In this paper, we propose an efficient two-stage dimension reduction scheme, termed biview learning, to encode two independent views which are depth-difference features (DDF) and relative position features (RPF). Biview learning explores the complementary property of DDF and RPF, and uses two stages to learn a compact yet comprehensive low-dimensional feature space for posture segmentation. In the first stage, discriminative locality alignment (DLA) is applied to the high-dimensional DDF to learn a discriminative low-dimensional representation. In the second stage, canonical correlation analysis (CCA) is used to explore the complementary property of RPF and the dimensionality reduced DDF. Finally, we train a support vector machine (SVM) over the output of CCA. We carefully validate the effectiveness of DLA and CCA utilized in the two-stage scheme on our 3D human points cloud dataset. Experimental results show that the proposed biview learning scheme significantly outperforms the state-of-the-art method for human posture segmentation. © 2014 Qiao et al

    Comparing sets of data sets on the Grassmann and flag manifolds with applications to data analysis in high and low dimensions

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    Includes bibliographical references.2020 Summer.This dissertation develops numerical algorithms for comparing sets of data sets utilizing shape and orientation of data clouds. Two key components for "comparing" are the distance measure between data sets and correspondingly the geodesic path in between. Both components will play a core role which connects two parts of this dissertation, namely data analysis on the Grassmann manifold and flag manifold. For the first part, we build on the well known geometric framework for analyzing and optimizing over data on the Grassmann manifold. To be specific, we extend the classical self-organizing mappings to the Grassamann manifold to visualize sets of high dimensional data sets in 2D space. We also propose an optimization problem on the Grassmannian to recover missing data. In the second part, we extend the geometric framework to the flag manifold to encode the variability of nested subspaces. There we propose a numerical algorithm for computing a geodesic path and distance between nested subspaces. We also prove theorems to show how to reduce the dimension of the algorithm for practical computations. The approach is shown to have advantages for analyzing data when the number of data points is larger than the number of features

    Regularization Methods for High-Dimensional Inference

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    High dimensionality is a common problem in statistical inference, and is becoming more prevalent in modern data analysis settings. While often data of interest may have a large -- often unmanageable -- dimension, modifications to various well-known techniques can be made to improve performance and aid interpretation. We typically assume that although predictors lie in a high-dimensional ambient space, they have a lower-dimensional structure that can be exploited through either prior knowledge or estimation. In performing regression, the structure in the predictors can be taken into account implicitly through regularization. In the case where the underlying structure in the predictors is known, using knowledge of this structure can yield improvements in prediction. We approach this problem through regularization using a known projection based on knowledge of the structure of the Grassmannian. Using this projection, we can obtain improvements over many classical and recent techniques in both regression and classification problems with only minor modification to a typical least squares problem. The structure of the predictors can also be taken into account explicitly through methods of dimension reduction. We often wish to have a lower-dimensional representation of our data in order to build potentially more interpretable models or to explore possible connections between predictors. In many problems, we are faced with data that does not have a similar distribution between estimating the model parameters and performing prediction. This results in problems when estimating a lower-dimensional structure of the predictors, as it may change. We pose methods for estimating a linear dimension reduction that will take into account these discrepancies between data distributions, while also incorporating as much of the information as possible in the data into construction of the predictor structure. These methods are built on regularized maximum likelihood and yield improvements in many cases of regression and classification, including those cases in which predictor dimension changes between training and testing
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