Cross-covariance matrix: Time-shifted correlations for 3D action recognition

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data.For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.

<p><b>Metric Learning has proved valuable in information retrieval and classification problems, with many algorithms using statistical formulations to their advantage. Optimal Transport, a powerful framework for computing distances between probability distributions, has seen significant uptake in Machine Learning and some applications in Metric Learning. The Bures-Wasserstein distance, a particular formulation of Optimal Transport, has seen limited application in Metric Learning despite being well suited to the field. In this thesis Linear Metric Learning algorithms incorporating the Bures-Wasserstein distance are developed, utilising its novel properties. The first set of algorithms we contribute use the Bures-Wasserstein distance for learning linear metrics using Cyclic Projections, Riemannian barycenters and gradient-based algorithms. The convergence properties and classification performance of these algorithms are then compared to benchmark Linear Metric Learning algorithms. Our algorithms achieve competitive classification performance on feature and image datasets and converge reliability.</b></p> <p>The second set of contributions extend Linear Metric Learning with the Bures-Wasserstein distance to a low-rank context to address issues related to scalability. An extension of the previous Cyclic Projections algorithm and a Riemannian optimisation method are developed. Convergence and classification experiments analogous to the full-rank case are conducted to examine how the reduced rank affects both of these performance metrics. Both algorithms converge reliably under a variety of synthetic conditions. Reduction in rank generally decreased the training time for algorithms until effects caused by the size of the training data dominated. Reducing rank either left classification error unchanged or increased it. With respect to other low-rank Linear Metric Learning algorithms our algorithms again performed competitively with respect to classification error. </p> <p>The last set of contributions consider Linear Metric Learning on Symmetric Positive Definite (SPD) matrices. Some datasets are expressed as or are suited to a SPD representation. As such, Metric Learning algorithms that respect the structure of these matrices have found success when adapted to this kind of data. Another extension of the Cyclic Projections algorithm is developed to use the Generalised Bures-Wasserstein distance on SPD matrices. Two other methods based on a ground metric formulation of the Bures-Wasserstein distance for learning linear metrics on SPD data are also introduced. One ground metric formulation uses an approximation (inspired by the Sinkhorn distance) to increase its scalability. Convergence experiments on synthetic problems adapted to SPD matrices are conducted on the ground metric algorithms to establish their convergence properties. Classification experiments are also conducted with respect to several types of SPD data. While the Cyclic Projections method is competitive, the ground metric algorithms have poor performance. Despite their poor classification performance, the approximation method does evaluate faster than the Generalised Bures-Wasserstein distance, potentially offering some scalability improvements.</p>

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.

We propose a new kernel entropy component analysis approach to object recognition with image set by fusing the information entropy. Since the geometry of Symmetric Positive Definite (SPD) matrices, we model the image set with its covariance matrix (nonsingular). Thus the object recognition with image set can be model as classifying problem on the riemannian manifold space. Given the proper kernel function derived from efficient metric for SPD matrices, points lie on manifold space represented by covariance matrices can be cast into high dimensional euclidean space. In this euclidean space, traditional methods used for dimensional reduction can be applied directly. Accounting for the measurability to information of the Renyi entropy, we fuse the information entropy to the dimensional reduction progress of our method. Similar to Kernel Principal Component Analysis (KPCA), the method accomplish data dimensionality reduction by projection onto a subset of entropy preserving KPCA axes. But this subset does not need to correspond to the top eigenvalues of the kernel matrix. The positive results of experiment on datasets demonstrated effectiveness of our method.

A cross-session motor imagery classification method based on Riemannian geometry and deep domain adaptation

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics and we discuss the relation between this new principle of balanced metrics and the concept of dual connections in information geometry.

The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

We present a distributed memory algorithm for the approximate hierarchical of symmetric positive definite (SPD) matrices. Our method is based on the distributed memory GOFMM, an algorithm that appeared in SC18 (doi:10.1109/SC.2018.00018). GOFMM constructs a hierarchical matrix approximation of an arbitrary SPD matrix that compresses the matrix by creating low-rank approximations of the off-diagonal blocks. GOFMM method has no guarantees of success for arbitrary SPD matrices. (This is similar to the SVD; not every matrix admits a good low-rank approximation.) But for many SPD matrices, GOFMM does enable compression that results in fast matrix-vector multiplication that can reach N logN time—as opposed to N2 required for a dense matrix. GOFMM supports shared and distributed memory parallelism. In this paper, we build an approximate based on the Hierarchically Semi-Separable (HSS) compression of the GOFMM. This requires O(N) work (given the compressed matrix) and O(N=p) + O(log p) time on p MPI processes (assuming a hypercube topology). The previous state-of-the-art required O(N logN) work. We present the algorithm, discuss its complexity, and present weak and strong scaling results for the factorization and phases of our algorithm. We also discuss the performance of the inexact ULV as a preconditioner for a few exemplary large dense linear systems. In our largest run, we were able to factorize a 67M-by-67M matrix in less than one second; and solve a system with 64 right-hand sides in less than one-tenth of a second. This run was on 6,144 Intel Skylake cores on the SKX partition of the Stampede2 system at the Texas Advanced Computing Center.

The region covariance descriptors have recently become a popular method for detection and tracking of objects in an image. However, these descriptors are not suitable for classification of images with heterogeneous contents. In this paper, we present an image-level descriptor obtained using an affine-invariant geodesic mean of region covariance descriptors on the Riemannian manifold of symmetric positive definite (SPD) matrices. The resulting image descriptors are also SPD matrices, lending themselves to tractable geodesic distance based k-nearest neighbour classification using efficient kernels. We show that the proposed descriptor yields high classification accuracy on a challenging problem of nuclear pleomorphism scoring in breast cancer histology images.

The discriminative power of modern deep learning models for 3D human action recognition is growing ever so potent. In conjunction with the recent resurgence of 3D human action representation with 3D skeletons, the quality and the pace of recent progress have been significant. However, the inner workings of state-of-the-art learning based methods in 3D human action recognition still remain mostly black-box. In this work, we propose to use a new class of models known as Temporal Convolutional Neural Networks (TCN) for 3D human action recognition. TCN provides us a way to explicitly learn readily interpretable spatio-temporal representations for 3D human action recognition. Through this work, we wish to take a step towards a spatio-temporal model that is easier to understand, explain and interpret. The resulting model, Res-TCN, achieves state-of-the-art results on the largest 3D human action recognition dataset, NTU-RGBD.

The symmetric positive definite (SPD) matrices have been widely used in image and vision problems. Recently there are growing interests in studying sparse representation (SR) of SPD matrices, motivated by the great success of SR for vector data. Though the space of SPD matrices is well-known to form a Lie group that is a Riemannian manifold, existing work fails to take full advantage of its geometric structure. This paper attempts to tackle this problem by proposing a kernel based method for SR and dictionary learning (DL) of SPD matrices. We disclose that the space of SPD matrices, with the operations of logarithmic multiplication and scalar logarithmic multiplication defined in the Log-Euclidean framework, is a complete inner product space. We can thus develop a broad family of kernels that satisfies Mercer's condition. These kernels characterize the geodesic distance and can be computed efficiently. We also consider the geometric structure in the DL process by updating atom matrices in the Riemannian space instead of in the Euclidean space. The proposed method is evaluated with various vision problems and shows notable performance gains over state-of-the-arts.

The use of regional covariance descriptors to generate feature data represented by Symmetric Positive Definite (SPD) matrices from images or videos has become increasingly common in machine learning. However, SPD data itself does not constitute a vector space, and dictionary learning involves a large number of linear operations, so dictionary learning cannot be performed directly on SPD data. For this reason, a more common method is to map the SPD data to the Reproducing Kernel Hilbert Space (RKHS). The so-called kernel learning is to find the most suitable RKHS for specific tasks. RKHS can be uniquely generated by a kernel function. Therefore, RKHS learning can also be considered as kernel learning. In this article, there are two main contributions. The first contribution is to propose a framework which based on Kernel Learning and Riemannian Metric (KLRM). Usually the learnable kernel function framework is to learn some parameters in the kernel function. The second contribution is dictionary learning by applying KLRM to SPD data. The SPD data is transformed into the RKHS generated by KLRM, and RKHS after training provides the most suitable working space for dictionary learning. Under the proposed framework, we design a positive definite kernel function, which is defined by the Log-Euclidean metric. This function can be transformed into a corresponding Riemannian kernel. The experimental results provided in this paper is compared with other state-of-the-art algorithms for SPD data dictionary learning and show that the proposed algorithm achieves better results.

The existing end-to-end optimized 3D action recognition methods often suffer from high computational costs. Observing that different frames and different points in point cloud sequences often have different importance values for the 3D action recognition task, in this work, we propose a fully automatic model compression framework called 3D-Pruning (3DP) for efficient 3D action recognition. After performing model compression by using our 3DP framework, the compressed model can process different frames and different points in each frame by using different computational complexities based on their importance values, in which both the importance value and computational complexity for each frame/point can be automatically learned. Extensive experiments on five benchmark datasets demonstrate the effectiveness of our 3DP framework for model compression.

Symmetric positive-definite (SPD) matrices are ubiquitous in Computer Vision, Machine Learning and Medical Image Analysis. Finding the center/average of a population of such matrices is a common theme in many algorithms such as clustering, segmentation, principal geodesic analysis, etc. The center of a population of such matrices can be defined using a variety of distance/divergence measures as the minimizer of the sum of squared distances/divergences from the unknown center to the members of the population. It is well known that the computation of the Karcher mean for the space of SPD matrices which is a negatively-curved Riemannian manifold is computationally expensive. Recently, the LogDet divergence-based center was shown to be a computationally attractive alternative. However, the LogDet-based mean of more than two matrices can not be computed in closed form, which makes it computationally less attractive for large populations. In this paper we present a novel recursive estimator for center based on the Stein distance - which is the square root of the LogDet divergence - that is significantly faster than the batch mode computation of this center. The key theoretical contribution is a closed-form solution for the weighted Stein center of two SPD matrices, which is used in the recursive computation of the Stein center for a population of SPD matrices. Additionally, we show experimental evidence of the convergence of our recursive Stein center estimator to the batch mode Stein center. We present applications of our recursive estimator to K-means clustering and image indexing depicting significant time gains over corresponding algorithms that use the batch mode computations. For the latter application, we develop novel hashing functions using the Stein distance and apply it to publicly available data sets, and experimental results have shown favorable comparisons to other competing methods.

Personal authentication using electroencephalographic (EEG) signals, is one of the important applications in brain computer interface (BCI). In this work we investigate the use of EEG signals as a biometric trait. Multidimensional EEG signals were represented as symmetric positive-definite (SPD) matrices on a Riemannian manifold. Two experiments are performed in the first; we use minimum distance to Riemannian mean (MDRM) as a classifier. In the second; SPD matrices are vectorized, and the generated vectors are used to train various machine learning (ML) classifiers. MDRM classifier achieved a correct recognition rate (CRR) of 96.92% , while ML classifiers achieved CRR from 95.39% to 99.45%.