Latent Semantic Learning with Structured Sparse Representation for Human Action Recognition

This paper proposes a novel latent semantic learning method for extracting high-level features (i.e. latent semantics) from a large vocabulary of abundant mid-level features (i.e. visual keywords) with structured sparse representation, which can help to bridge the semantic gap in the challenging task of human action recognition. To discover the manifold structure of midlevel features, we develop a spectral embedding approach to latent semantic learning based on L1-graph, without the need to tune any parameter for graph construction as a key step of manifold learning. More importantly, we construct the L1-graph with structured sparse representation, which can be obtained by structured sparse coding with its structured sparsity ensured by novel L1-norm hypergraph regularization over mid-level features. In the new embedding space, we learn latent semantics automatically from abundant mid-level features through spectral clustering. The learnt latent semantics can be readily used for human action recognition with SVM by defining a histogram intersection kernel. Different from the traditional latent semantic analysis based on topic models, our latent semantic learning method can explore the manifold structure of mid-level features in both L1-graph construction and spectral embedding, which results in compact but discriminative high-level features. The experimental results on the commonly used KTH action dataset and unconstrained YouTube action dataset show the superior performance of our method.Comment: The short version of this paper appears in ICCV 201

Graph Construction for Manifold Discovery

Manifold learning is a class of machine learning methods that exploits the observation that high-dimensional data tend to lie on a smooth lower-dimensional manifold. Manifold discovery is the essential first component of manifold learning methods, in which the manifold structure is inferred from available data. This task is typically posed as a graph construction problem: selecting a set of vertices and edges that most closely approximates the true underlying manifold. The quality of this learned graph is critical to the overall accuracy of the manifold learning method. Thus, it is essential to develop accurate, efficient, and reliable algorithms for constructing manifold approximation graphs. To aid in this investigation of graph construction methods, we propose new methods for evaluating graph quality. These quality measures act as a proxy for ground-truth manifold approximation error and are applicable even when prior information about the dataset is limited. We then develop an incremental update scheme for some quality measures, demonstrating their usefulness for efficient parameter tuning. We then propose two novel methods for graph construction, the Manifold Spanning Graph and the Mutual Neighbors Graph algorithms. Each method leverages assumptions about the structure of both the input data and the subsequent manifold learning task. The algorithms are experimentally validated against state of the art graph construction techniques on a multi-disciplinary set of application domains, including image classification, directional audio prediction, and spectroscopic analysis. The final contribution of the thesis is a method for aligning sequential datasets while still respecting each set’s internal manifold structure. The use of high quality manifold approximation graphs enables accurate alignments with few ground-truth correspondences

New forms of non-relativistic and relativistic hydrodynamic equations as derived by the renormalization-group method - possible functional ansatz in the moment method consistent with Chapman-Enskog theory -

After a brief account of the derivation of the first-order relativistic hydrodynamic equation as a construction of the invariant manifold of relativistic Boltzmann equation, we give a sketch of derivation of the second-order hydrodynamic equation (extended thermodynamics) both in the nonrelativistic and relativistic cases. We show that the resultant equation suggests a novel ansatz for the functional form to be used in Grad moment method, which turns out to give the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for the relaxation times and lenghts allowing natural physical interpretaion.Comment: Typos are corrected. Accepted version. 10 pages. To be published in Suppl. Prog. Theor. Phy

Constructing Generalized Synchronization Manifolds by Manifold Equation

Full understanding of synchronous behavior in coupled dynamical systems beyond the identical case requires an explicit construction of the generalized synchronization manifold, whether we wish to compare the systems, or to understand their stability. Nonetheless, while synchronization has become an extremely popular topic, the bulk of the research in this area has been focused on the identical case, specifically because its invariant manifold is simply the identity function, and there have yet to be any generally workable methods to compute the generalized synchronization manifolds for non-identical systems. Here, we derive time dependent PDEs whose stationary solution mirrors exactly the generalized synchronization manifold, respecting its stability. We introduce a novel method for dealing with subtle issues with boundary conditions in the numerical scheme to solve the PDE, and we develop first order expansions close to the identical case. We give several examples of increasing sophistication, including coupled non-identical Van der Pol oscillators. By using the manifold equation, we also discuss the design of coupling to achieve desired synchronization

Out-of-sample generalizations for supervised manifold learning for classification

Supervised manifold learning methods for data classification map data samples residing in a high-dimensional ambient space to a lower-dimensional domain in a structure-preserving way, while enhancing the separation between different classes in the learned embedding. Most nonlinear supervised manifold learning methods compute the embedding of the manifolds only at the initially available training points, while the generalization of the embedding to novel points, known as the out-of-sample extension problem in manifold learning, becomes especially important in classification applications. In this work, we propose a semi-supervised method for building an interpolation function that provides an out-of-sample extension for general supervised manifold learning algorithms studied in the context of classification. The proposed algorithm computes a radial basis function (RBF) interpolator that minimizes an objective function consisting of the total embedding error of unlabeled test samples, defined as their distance to the embeddings of the manifolds of their own class, as well as a regularization term that controls the smoothness of the interpolation function in a direction-dependent way. The class labels of test data and the interpolation function parameters are estimated jointly with a progressive procedure. Experimental results on face and object images demonstrate the potential of the proposed out-of-sample extension algorithm for the classification of manifold-modeled data sets

Partially integrable systems in multidimensions by a variant of the dressing method. 1

In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially integrable''. Such a construction is achieved using a suitable modification of the classical dressing scheme, consisting in assuming that the kernel of the basic integral operator of the dressing formalism be nontrivial. This new hypothesis leads to the construction of: 1) a linear system of compatible spectral problems for the solution $U(\lambda;x)$ of the integral equation in 3 independent variables each (while the usual dressing method generates spectral problems in 1 or 2 dimensions); 2) a system of nonlinear partial differential equations in $n$ dimensions ($n>3$), possessing a manifold of analytic solutions of dimension ($n-2$), which includes one largely arbitrary relation among the fields. These nonlinear equations can also contain an arbitrary forcing.Comment: 21 page

Mirror Symmetry and Other Miracles in Superstring Theory

The dominance of string theory in the research landscape of quantum gravity physics (despite any direct experimental evidence) can, I think, be justified in a variety of ways. Here I focus on an argument from mathematical fertility, broadly similar to Hilary Putnam's 'no miracles argument' that, I argue, many string theorists in fact espouse. String theory leads to many surprising, useful, and well-confirmed mathematical 'predictions' - here I focus on mirror symmetry. These predictions are made on the basis of general physical principles entering into string theory. The success of the mathematical predictions are then seen as evidence for framework that generated them. I attempt to defend this argument, but there are nonetheless some serious objections to be faced. These objections can only be evaded at a high (philosophical) price.Comment: For submission to a Foundations of Physics special issue on "Forty Years Of String Theory: Reflecting On the Foundations" (edited by G. `t Hooft, E. Verlinde, D. Dieks and S. de Haro)