Robust Formulations for Handling Uncertainty in Kernel Matrices

We study the problem of uncertainty in the entries of the Kernel matrix, arising in SVM formulation. Using Chance Constraint Programming and a novel large deviation inequality we derive a formulation which is robust to such noise. The resulting formulation applies when the noise is Gaussian, or has finite support. The formulation in general is non-convex, but in several cases of interest it reduces to a convex program. The problem of uncertainty in kernel matrix is motivated from the real world problem of classifying proteins when the structures are provided with some uncertainty. The formulation derived here naturally incorporates such uncertainty in a principled manner leading to significant improvements over the state of the art. 1.

Snakes with Ellipse-Reproducing Property

We present a new class of continuously defined parametric snakes using a special kind of exponential splines as basis functions. We have enforced our bases to have the shortestpossible support subject to some design constraints to maximize efficiency. While the resulting snakes are versatile enough to provide a good approximation of any closed curve in the plane, their most important feature is the fact that they admit ellipses within their span. Thus, they can perfectly generate circular and elliptical shapes. These features are appropriate to delineate cross sections of cylindrical-like conduits and to outline blob-like objects. We address the implementation details and illustrate the capabilities of our snake with synthetic and real data.

Time Series Classification for Online Tamil Handwritten Character Recognition – A Kernel Based Approach

In this paper, we consider the problem of time series classification. Using piecewise linear interpolation various novel kernels are obtained which can be used with Support vector machines for designing classifiers capable of deciding the class of a given time series. The approach is general and is applicable in many scenarios. We apply the method to the task of Online Tamil handwritten character recognition with promising results.

Snakes With an Ellipse-Reproducing Property

We present a new class of continuously defined parametric snakes using a special kind of exponential splines as basis functions. We have enforced our bases to have the shortest possible support subject to some design constraints to maximize efficiency. While the resulting snakes are versatile enough to provide a good approximation of any closed curve in the plane, their most important feature is the fact that they admit ellipses within their span. Thus, they can perfectly generate circular and elliptical shapes. These features are appropriate to delineate cross sections of cylindrical-like conduits and to outline bloblike objects. We address the implementation details and illustrate the capabilities of our snake with synthetic and real data.

Spacelike pion form factor from analytic continuation and the onset of perturbative QCD

The factorization theorem for exclusive processes in perturbative QCD predicts the behavior of the pion electromagnetic form factor F(t) at asymptotic spacelike momenta t(= -Q(2)) < 0. We address the question of the onset energy using a suitable mathematical framework of analytic continuation, which uses as input the phase of the form factor below the first inelastic threshold, known with great precision through the Fermi-Watson theorem from pi pi elastic scattering, and the modulus measured from threshold up to 3 GeV by the BABAR Collaboration. The method leads to almost model-independent upper and lower bounds on the spacelike form factor. Further inclusion of the value of the charge radius and the experimental value at -2.45 GeV2 measured at JLab considerably increases the strength of the bounds in the region Q(2) less than or similar to 10 GeV2, excluding the onset of the asymptotic perturbative QCD regime for Q(2) < 7 GeV2. We also compare the bounds with available experimental data and with several theoretical models proposed for the low and intermediate spacelike region.

On the structure of analytic vectors for the Schrodinger representation

This article deals with the structure of analytic and entire vectors for the Schrodinger representations of the Heisenberg group. Using refined versions of Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.

A Technique to Compute Smooth Amplitude, Phase, and Frequency Modulations From the Analytic Signal

Gabor's analytic signal (AS) is a unique complex signal corresponding to a real signal, but in general, it admits infinitely-many combinations of amplitude and frequency modulations (AM and FM, respectively). The standard approach is to enforce a non-negativity constraint on the AM, but this results in discontinuities in the corresponding phase modulation (PM), and hence, an FM with discontinuities particularly when the underlying AM-FM signal is over-modulated. In this letter, we analyze the phase discontinuities and propose a technique to compute smooth AM and FM from the AS, by relaxing the non-negativity constraint on the AM. The proposed technique is effective at handling over-modulated signals. We present simulation results to support the theoretical calculations.

Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane

We examine the large-order behavior of a recently proposed renormalization-group-improved expansion of the Adler function in perturbative QCD, which sums in an analytically closed form the leading logarithms accessible from renormalization-group invariance. The expansion is first written as an effective series in powers of the one-loop coupling, and its leading singularities in the Borel plane are shown to be identical to those of the standard ``contour-improved'' expansion. Applying the technique of conformal mappings for the analytic continuation in the Borel plane, we define a class of improved expansions, which implement both the renormalization-group invariance and the knowledge about the large-order behavior of the series. Detailed numerical studies of specific models for the Adler function indicate that the new expansions have remarkable convergence properties up to high orders. Using these expansions for the determination of the strong coupling from the hadronic width of the tau lepton we obtain, with a conservative estimate of the uncertainty due to the nonperturbative corrections, alpha(s)(M-tau(2)) = 0.3189(-0.0151)(+0.0173), which translates to alpha(s)(M-Z(2)) = 0.1184(-0.0018)(+0.0021). DOI: 10.1103/PhysRevD.87.014008

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.

On Computing Amplitude, Phase, and Frequency Modulations Using a Vector Interpretation of the Analytic Signal

The amplitude-modulation (AM) and phase-modulation (PM) of an amplitude-modulated frequency-modulated (AM-FM) signal are defined as the modulus and phase angle, respectively, of the analytic signal (AS). The FM is defined as the derivative of the PM. However, this standard definition results in a PM with jump discontinuities in cases when the AM index exceeds unity, resulting in an FM that contains impulses. We propose a new approach to define smooth AM, PM, and FM for the AS, where the PM is computed as the solution to an optimization problem based on a vector interpretation of the AS. Our approach is directly linked to the fractional Hilbert transform (FrHT) and leads to an eigenvalue problem. The resulting PM and AM are shown to be smooth, and in particular, the AM turns out to be bipolar. We show an equivalence of the eigenvalue formulation to the square of the AS, and arrive at a simple method to compute the smooth PM. Some examples on synthesized and real signals are provided to validate the theoretical calculations.

Fractional Hilbert transform extensions and associated analytic signal construction

The analytic signal (AS) was proposed by Gabor as a complex signal corresponding to a given real signal. The AS has a one-sided spectrum and gives rise to meaningful spectral averages. The Hilbert transform (HT) is a key component in Gabor's AS construction. We generalize the construction methodology by employing the fractional Hilbert transform (FrHT), without going through the standard fractional Fourier transform (FrFT) route. We discuss some properties of the fractional Hilbert operator and show how decomposition of the operator in terms of the identity and the standard Hilbert operators enables the construction of a family of analytic signals. We show that these analytic signals also satisfy Bedrosian-type properties and that their time-frequency localization properties are unaltered. We also propose a generalized-phase AS (GPAS) using a generalized-phase Hilbert transform (GPHT). We show that the GPHT shares many properties of the FrHT, in particular, selective highlighting of singularities, and a connection with Lie groups. We also investigate the duality between analyticity and causality concepts to arrive at a representation of causal signals in terms of the FrHT and GPHT. On the application front, we develop a secure multi-key single-sideband (SSB) modulation scheme and analyze its performance in noise and sensitivity to security key perturbations. (C) 2013 Elsevier B.V. All rights reserved.

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order H-m (H-n), m is an element of N-n, under the heat kernel transform on H-n, using direct sum and direct integral of Bergmann spaces and certain unitary representations of H-n which can be realized on the Hilbert space of Hilbert-Schmidt operators on L-2 (R-n). We also show that the image of Sobolev space of negative order H-s (H-n), s(> 0) is an element of R is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on H-n under the heat kernel transform. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim