Efficient algorithms for learning kernels from multiple similarity matrices with general convex loss functions

In this paper we consider the problem of learning an n × n kernel matrix from m(1) similarity matrices under general convex loss. Past research have extensively studied the m = 1 case and have derived several algorithms which require sophisticated techniques like ACCP, SOCP, etc. The existing algorithms do not apply if one uses arbitrary losses and often can not handle m > 1 case. We present several provably convergent iterative algorithms, where each iteration requires either an SVM or a Multiple Kernel Learning (MKL) solver for m > 1 case. One of the major contributions of the paper is to extend the well knownMirror Descent(MD) framework to handle Cartesian product of psd matrices. This novel extension leads to an algorithm, called EMKL, which solves the problem in O(m2 log n 2) iterations; in each iteration one solves an MKL involving m kernels and m eigen-decomposition of n × n matrices. By suitably defining a restriction on the objective function, a faster version of EMKL is proposed, called REKL,which avoids the eigen-decomposition. An alternative to both EMKL and REKL is also suggested which requires only an SVMsolver. Experimental results on real world protein data set involving several similarity matrices illustrate the efficacy of the proposed algorithms.

Superconductivity of embedded lead nano particles in metallic and amorphous matrices

Nanodispersed lead in metallic and amorphous matrices was synthesized by rapid solidification processing. The optimum microstructure was tailored to avoid percolation of the particles. With these embedded particles it is possible to study quantitatively the effect of size on the superconducting transition temperature by carrying out quantitative microstructural characterization and magnetic measurements. Our results suggest the role of the matrices in enhancement or depression of superconducting transition temperature of lead. The origin of this difference in behavior with respect to different matrices and sizes is discussed.

Closed form solutions for element equilibrium and flexibility matrices of eight node rectangular plate bending element using integrated force method

Closed form solutions for equilibrium and flexibility matrices of the Mindlin-Reissner theory based eight-node rectangular plate bending element (MRP8) using integrated Force Method (IFM) are presented in this paper. Though these closed form solutions of equilibrium and flexibility matrices are applicable to plate bending problems with square/rectangular boundaries, they reduce the computational time significantly and give more exact solutions. Presented closed form solutions are validated by solving large number of standard square/rectangular plate bending benchmark problems for deflections and moments and the results are compared with those of similar displacement-based eight-node quadrilateral plate bending elements available in the literature. The results are also compared with the exact solutions.

Efficient Methods for Robust Classification Under Uncertainty in Kernel Matrices

In this paper we study the problem of designing SVM classifiers when the kernel matrix, K, is affected by uncertainty. Specifically K is modeled as a positive affine combination of given positive semi definite kernels, with the coefficients ranging in a norm-bounded uncertainty set. We treat the problem using the Robust Optimization methodology. This reduces the uncertain SVM problem into a deterministic conic quadratic problem which can be solved in principle by a polynomial time Interior Point (IP) algorithm. However, for large-scale classification problems, IP methods become intractable and one has to resort to first-order gradient type methods. The strategy we use here is to reformulate the robust counterpart of the uncertain SVM problem as a saddle point problem and employ a special gradient scheme which works directly on the convex-concave saddle function. The algorithm is a simplified version of a general scheme due to Juditski and Nemirovski (2011). It achieves an O(1/T-2) reduction of the initial error after T iterations. A comprehensive empirical study on both synthetic data and real-world protein structure data sets show that the proposed formulations achieve the desired robustness, and the saddle point based algorithm outperforms the IP method significantly.

Fast Likelihood Computation in Speech Recognition using Matrices

Acoustic modeling using mixtures of multivariate Gaussians is the prevalent approach for many speech processing problems. Computing likelihoods against a large set of Gaussians is required as a part of many speech processing systems and it is the computationally dominant phase for Large Vocabulary Continuous Speech Recognition (LVCSR) systems. We express the likelihood computation as a multiplication of matrices representing augmented feature vectors and Gaussian parameters. The computational gain of this approach over traditional methods is by exploiting the structure of these matrices and efficient implementation of their multiplication. In particular, we explore direct low-rank approximation of the Gaussian parameter matrix and indirect derivation of low-rank factors of the Gaussian parameter matrix by optimum approximation of the likelihood matrix. We show that both the methods lead to similar speedups but the latter leads to far lesser impact on the recognition accuracy. Experiments on 1,138 work vocabulary RM1 task and 6,224 word vocabulary TIMIT task using Sphinx 3.7 system show that, for a typical case the matrix multiplication based approach leads to overall speedup of 46 % on RM1 task and 115 % for TIMIT task. Our low-rank approximation methods provide a way for trading off recognition accuracy for a further increase in computational performance extending overall speedups up to 61 % for RM1 and 119 % for TIMIT for an increase of word error rate (WER) from 3.2 to 3.5 % for RM1 and for no increase in WER for TIMIT. We also express pairwise Euclidean distance computation phase in Dynamic Time Warping (DTW) in terms of matrix multiplication leading to saving of approximately of computational operations. In our experiments using efficient implementation of matrix multiplication, this leads to a speedup of 5.6 in computing the pairwise Euclidean distances and overall speedup up to 3.25 for DTW.

Nodal length fluctuations for arithmetic random waves

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (''arithmetic random waves''). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

Field extension code based dispersion matrices for coherently detected space-time shift keying

Motivated by the recent Coherent Space-Time Shift Keying (CSTSK) philosophy, we construct new dispersion matrices for rotationally invariant PSK signaling sets. Given a specific PSK signal constellation, the dispersion matrices of the existing CSTSK scheme were chosen by maximizing the mutual information over randomly generated sets of dispersion matrices. In this contribution we propose a general method for constructing a set of structured dispersion matrices for arbitrary PSK signaling sets using Field Extension (FE) codes and then study the attainable Symbol Error Rate (SER) performance of some example constructions. We demonstrate that the proposed dispersion scheme is capable of outperforming the existing dispersion arrangement at medium to high SNRs.

Structured Dispersion Matrices From Division Algebra Codes for Space-Time Shift Keying

We propose a novel method of constructing Dispersion Matrices (DM) for Coherent Space-Time Shift Keying (CSTSK) relying on arbitrary PSK signal sets by exploiting codes from division algebras. We show that classic codes from Cyclic Division Algebras (CDA) may be interpreted as DMs conceived for PSK signal sets. Hence various benefits of CDA codes such as their ability to achieve full diversity are inherited by CSTSK. We demonstrate that the proposed CDA based DMs are capable of achieving a lower symbol error ratio than the existing DMs generated using the capacity as their optimization objective function for both perfect and imperfect channel estimation.

Burst error correction using partial fourier matrices and block sparse representation

There is a strong relation between sparse signal recovery and error control coding. It is known that burst errors are block sparse in nature. So, here we attempt to solve burst error correction problem using block sparse signal recovery methods. We construct partial Fourier based encoding and decoding matrices using results on difference sets. These constructions offer guaranteed and efficient error correction when used in conjunction with reconstruction algorithms which exploit block sparsity.

Lipschitz Correspondence between Metric Measure Spaces and Random Distance Matrices

Given a metric space with a Borel probability measure, for each integer N, we obtain a probability distribution on N x N distance matrices by considering the distances between pairs of points in a sample consisting of N points chosen independently from the metric space with respect to the given measure. We show that this gives an asymptotically bi-Lipschitz relation between metric measure spaces and the corresponding distance matrices. This is an effective version of a result of Vershik that metric measure spaces are determined by associated distributions on infinite random matrices.

Nodal domains of the equilateral triangle billiard

We characterize the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems.

Automatic finite element formulation and assembly of hyperelastic higher order structural models

The formulation of higher order structural models and their discretization using the finite element method is difficult owing to their complexity, especially in the presence of non-linearities. In this work a new algorithm for automating the formulation and assembly of hyperelastic higher-order structural finite elements is developed. A hierarchic series of kinematic models is proposed for modeling structures with special geometries and the algorithm is formulated to automate the study of this class of higher order structural models. The algorithm developed in this work sidesteps the need for an explicit derivation of the governing equations for the individual kinematic modes. Using a novel procedure involving a nodal degree-of-freedom based automatic assembly algorithm, automatic differentiation and higher dimensional quadrature, the relevant finite element matrices are directly computed from the variational statement of elasticity and the higher order kinematic model. Another significant feature of the proposed algorithm is that natural boundary conditions are implicitly handled for arbitrary higher order kinematic models. The validity algorithm is illustrated with examples involving linear elasticity and hyperelasticity. (C) 2013 Elsevier Inc. All rights reserved.

A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, nu of the eigenfunctions with Dirichlet boundary conditions are considered. The billiards for which the time-independent Schrodinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and nonseparable integrable billiards, nu satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of m mod kn, given a particular k, for a set of quantum numbers, m, n. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations. (C) 2014 Elsevier Inc. All rights reserved.

LDPC Code Designs Based on root I Matrices

LDPC codes can be constructed by tiling permutation matrices that belong to the square root of identity type and similar algebraic structures. We investigate into the properties of such codes. We also present code structures that are amenable for efficient encoding.