Disturbance Grassmann Kernels for Subspace-Based Learning

In this paper, we focus on subspace-based learning problems, where data elements are linear subspaces instead of vectors. To handle this kind of data, Grassmann kernels were proposed to measure the space structure and used with classifiers, e.g., Support Vector Machines (SVMs). However, the existing discriminative algorithms mostly ignore the instability of subspaces, which would cause the classifiers misled by disturbed instances. Thus we propose considering all potential disturbance of subspaces in learning processes to obtain more robust classifiers. Firstly, we derive the dual optimization of linear classifiers with disturbance subject to a known distribution, resulting in a new kernel, Disturbance Grassmann (DG) kernel. Secondly, we research into two kinds of disturbance, relevant to the subspace matrix and singular values of bases, with which we extend the Projection kernel on Grassmann manifolds to two new kernels. Experiments on action data indicate that the proposed kernels perform better compared to state-of-the-art subspace-based methods, even in a worse environment.Comment: This paper include 3 figures, 10 pages, and has been accpeted to SIGKDD'1

Human action recognition using local spatiotemporal discriminant embedding

Human action video sequences can be considered as nonlinear dynamic shape manifolds in the space of image frames. In this paper, we address learning and classifying human actions on embedded low-dimensional manifolds. We propose a novel manifold embedding method, called Local Spatio-Temporal Discriminant Embedding (LSTDE). The discriminating capabilities of the proposed method are two-fold: (1) for local spatial discrimination, LSTDE projects data points (silhouette-based image frames of human action sequences) in a local neighborhood into the embedding space where data points of the same action class are close while those of different classes are far apart; (2) in such a local neighborhood, each data point has an associated short video segment, which forms a local temporal subspace on the embedded manifold. LSTDE finds an optimal embedding which maximizes the principal angles between those temporal subspaces associated with data points of different classes. Benefiting from the joint spatio-temporal discriminant embedding, our method is potentially more powerful for classifying human actions with similar space-time shapes, and is able to perform recognition on a frame-byframe or short video segment basis. Experimental results demonstrate that our method can accurately recognize human actions, and can improve the recognition performance over some representative manifold embedding methods, especially on highly confusing human action types. 1

Positive Definite Kernels in Machine Learning

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as reproducing kernel Hibert spaces, the natural extension of the set of functions $\{k(x,\cdot),x\in\mathcal{X}\}$ associated with a kernel $k$ defined on a space $\mathcal{X}$. We discuss at length the construction of kernel functions that take advantage of well-known statistical models. We provide an overview of numerous data-analysis methods which take advantage of reproducing kernel Hilbert spaces and discuss the idea of combining several kernels to improve the performance on certain tasks. We also provide a short cookbook of different kernels which are particularly useful for certain data-types such as images, graphs or speech segments.Comment: draft. corrected a typo in figure

Chapter Machine Learning in Volcanology: A Review

A volcano is a complex system, and the characterization of its state at any given time is not an easy task. Monitoring data can be used to estimate the probability of an unrest and/or an eruption episode. These can include seismic, magnetic, electromagnetic, deformation, infrasonic, thermal, geochemical data or, in an ideal situation, a combination of them. Merging data of different origins is a non-trivial task, and often even extracting few relevant and information-rich parameters from a homogeneous time series is already challenging. The key to the characterization of volcanic regimes is in fact a process of data reduction that should produce a relatively small vector of features. The next step is the interpretation of the resulting features, through the recognition of similar vectors and for example, their association to a given state of the volcano. This can lead in turn to highlight possible precursors of unrests and eruptions. This final step can benefit from the application of machine learning techniques, that are able to process big data in an efficient way. Other applications of machine learning in volcanology include the analysis and classification of geological, geochemical and petrological “static” data to infer for example, the possible source and mechanism of observed deposits, the analysis of satellite imagery to quickly classify vast regions difficult to investigate on the ground or, again, to detect changes that could indicate an unrest. Moreover, the use of machine learning is gaining importance in other areas of volcanology, not only for monitoring purposes but for differentiating particular geochemical patterns, stratigraphic issues, differentiating morphological patterns of volcanic edifices, or to assess spatial distribution of volcanoes. Machine learning is helpful in the discrimination of magmatic complexes, in distinguishing tectonic settings of volcanic rocks, in the evaluation of correlations of volcanic units, being particularly helpful in tephrochronology, etc. In this chapter we will review the relevant methods and results published in the last decades using machine learning in volcanology, both with respect to the choice of the optimal feature vectors and to their subsequent classification, taking into account both the unsupervised and the supervised approaches

Towards On-line Domain-Independent Big Data Learning: Novel Theories and Applications

Feature extraction is an extremely important pre-processing step to pattern recognition, and machine learning problems. This thesis highlights how one can best extract features from the data in an exhaustively online and purely adaptive manner. The solution to this problem is given for both labeled and unlabeled datasets, by presenting a number of novel on-line learning approaches. Specifically, the differential equation method for solving the generalized eigenvalue problem is used to derive a number of novel machine learning and feature extraction algorithms. The incremental eigen-solution method is used to derive a novel incremental extension of linear discriminant analysis (LDA). Further the proposed incremental version is combined with extreme learning machine (ELM) in which the ELM is used as a preprocessor before learning. In this first key contribution, the dynamic random expansion characteristic of ELM is combined with the proposed incremental LDA technique, and shown to offer a significant improvement in maximizing the discrimination between points in two different classes, while minimizing the distance within each class, in comparison with other standard state-of-the-art incremental and batch techniques. In the second contribution, the differential equation method for solving the generalized eigenvalue problem is used to derive a novel state-of-the-art purely incremental version of slow feature analysis (SLA) algorithm, termed the generalized eigenvalue based slow feature analysis (GENEIGSFA) technique. Further the time series expansion of echo state network (ESN) and radial basis functions (EBF) are used as a pre-processor before learning. In addition, the higher order derivatives are used as a smoothing constraint in the output signal. Finally, an online extension of the generalized eigenvalue problem, derived from James Stone’s criterion, is tested, evaluated and compared with the standard batch version of the slow feature analysis technique, to demonstrate its comparative effectiveness. In the third contribution, light-weight extensions of the statistical technique known as canonical correlation analysis (CCA) for both twinned and multiple data streams, are derived by using the same existing method of solving the generalized eigenvalue problem. Further the proposed method is enhanced by maximizing the covariance between data streams while simultaneously maximizing the rate of change of variances within each data stream. A recurrent set of connections used by ESN are used as a pre-processor between the inputs and the canonical projections in order to capture shared temporal information in two or more data streams. A solution to the problem of identifying a low dimensional manifold on a high dimensional dataspace is then presented in an incremental and adaptive manner. Finally, an online locally optimized extension of Laplacian Eigenmaps is derived termed the generalized incremental laplacian eigenmaps technique (GENILE). Apart from exploiting the benefit of the incremental nature of the proposed manifold based dimensionality reduction technique, most of the time the projections produced by this method are shown to produce a better classification accuracy in comparison with standard batch versions of these techniques - on both artificial and real datasets

A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold

Although Deep Learning (DL) has achieved success in complex Artificial Intelligence (AI) tasks, it suffers from various notorious problems (e.g., feature redundancy, and vanishing or exploding gradients), since updating parameters in Euclidean space cannot fully exploit the geometric structure of the solution space. As a promising alternative solution, Riemannian-based DL uses geometric optimization to update parameters on Riemannian manifolds and can leverage the underlying geometric information. Accordingly, this article presents a comprehensive survey of applying geometric optimization in DL. At first, this article introduces the basic procedure of the geometric optimization, including various geometric optimizers and some concepts of Riemannian manifold. Subsequently, this article investigates the application of geometric optimization in different DL networks in various AI tasks, e.g., convolution neural network, recurrent neural network, transfer learning, and optimal transport. Additionally, typical public toolboxes that implement optimization on manifold are also discussed. Finally, this article makes a performance comparison between different deep geometric optimization methods under image recognition scenarios.Comment: 41 page

On Linear Separation Capacity of Self-Supervised Representation Learning

Recent advances in self-supervised learning have highlighted the efficacy of data augmentation in learning data representation from unlabeled data. Training a linear model atop these enhanced representations can yield an adept classifier. Despite the remarkable empirical performance, the underlying mechanisms that enable data augmentation to unravel nonlinear data structures into linearly separable representations remain elusive. This paper seeks to bridge this gap by investigating under what conditions learned representations can linearly separate manifolds when data is drawn from a multi-manifold model. Our investigation reveals that data augmentation offers additional information beyond observed data and can thus improve the information-theoretic optimal rate of linear separation capacity. In particular, we show that self-supervised learning can linearly separate manifolds with a smaller distance than unsupervised learning, underscoring the additional benefits of data augmentation. Our theoretical analysis further underscores that the performance of downstream linear classifiers primarily hinges on the linear separability of data representations rather than the size of the labeled data set, reaffirming the viability of constructing efficient classifiers with limited labeled data amid an expansive unlabeled data set

Data-Driven Shape Analysis and Processing

Data-driven methods play an increasingly important role in discovering geometric, structural, and semantic relationships between 3D shapes in collections, and applying this analysis to support intelligent modeling, editing, and visualization of geometric data. In contrast to traditional approaches, a key feature of data-driven approaches is that they aggregate information from a collection of shapes to improve the analysis and processing of individual shapes. In addition, they are able to learn models that reason about properties and relationships of shapes without relying on hard-coded rules or explicitly programmed instructions. We provide an overview of the main concepts and components of these techniques, and discuss their application to shape classification, segmentation, matching, reconstruction, modeling and exploration, as well as scene analysis and synthesis, through reviewing the literature and relating the existing works with both qualitative and numerical comparisons. We conclude our report with ideas that can inspire future research in data-driven shape analysis and processing.Comment: 10 pages, 19 figure