Product Kernel Research Articles

SURFACE SUBGROUPS OF GRAPH PRODUCTS OF GROUPS

Let G be a graph product of a collection of groups and H be the direct product of the same collection of groups, so that there is a natural surjection p : G → H. The kernel of this map p is called a graph product kernel. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of this yields conditions for a graph over which the graph product of arbitrary nontrivial groups (or some cyclic groups, or some finite groups) contains a hyperbolic surface group. In particular, the graph product of arbitrary nontrivial groups over a cycle of length at least five, or over its opposite graph, contains a hyperbolic surface group. For the case when the defining graphs have at most seven vertices, we completely characterize right-angled Coxeter groups with hyperbolic surface subgroups.

Supersingular mass distributions in gelling systems

This paper considers the time evolution of disperse systems in which binary coagulation is so rapid that it leads to formation of gels during a finite but nonzero interval of time. Right before the transition point an algebraic particle mass spectrum forms whose behavior at small particle masses does not permit one to define the total particle mass concentration as ∫(0) (∞)gc(g,t)dg, because the integral diverges at low limit. This divergency prevents the formulation of an asymptotic self-similarity spectrum in its traditional form. This difficulty is avoided by introducing the second moment of the particle mass spectrum as a basic scale defining its behavior at the pregelation stage. The equations for the universality mass spectra are formulated. It is shown that these equations correctly reproduce the asymptotic form of the particle mass spectrum for the system with the product kernel. The postcritical behavior in this case is investigated. The time dependencies of the gel mass, the second moment, and particle number concentrations in the postcritical period are found.

GPU-accelerated preconditioned iterative linear solvers

This work is an overview of our preliminary experience in developing a high-performance iterative linear solver accelerated by GPU coprocessors. Our goal is to illustrate the advantages and difficulties encountered when deploying GPU technology to perform sparse linear algebra computations. Techniques for speeding up sparse matrix-vector product (SpMV) kernels and finding suitable preconditioning methods are discussed. Our experiments with an NVIDIA TESLA M2070 show that for unstructured matrices SpMV kernels can be up to 8 times faster on the GPU than the Intel MKL on the host Intel Xeon X5675 Processor. Overall performance of the GPU-accelerated Incomplete Cholesky (IC) factorization preconditioned CG method can outperform its CPU counterpart by a smaller factor, up to 3, and GPU-accelerated The incomplete LU (ILU) factorization preconditioned GMRES method can achieve a speed-up nearing 4. However, with better suited preconditioning techniques for GPUs, this performance can be further improved.

Ergodicity and stability of the conditional distributions of nondegenerate Markov chains

We consider a bivariate stationary Markov chain $(X_{n},Y_{n})_{n\ge0}$ in a Polish state space, where only the process $(Y_{n})_{n\ge0}$ is presumed to be observable. The goal of this paper is to investigate the ergodic theory and stability properties of the measure-valued process $(\Pi_{n})_{n\ge0}$, where $\Pi_{n}$ is the conditional distribution of $X_{n}$ given $Y_{0},\ldots,Y_{n}$. We show that the ergodic and stability properties of $(\Pi_{n})_{n\ge0}$ are inherited from the ergodicity of the unobserved process $(X_{n})_{n\ge0}$ provided that the Markov chain $(X_{n},Y_{n})_{n\ge0}$ is nondegenerate, that is, its transition kernel is equivalent to the product of independent transition kernels. Our main results generalize, subsume and in some cases correct previous results on the ergodic theory of nonlinear filters.

Trajectory clustering in CCTV traffic videos using probability product kernels with hidden Markov models

This article presents a method for clustering the trajectories obtained by tracking vehicles in traffic videos, recorded from CCTV cameras in public spaces. The proposed method employs a model-based approach in which (1) each trajectory (position and velocity) is modelled using a hidden Markov model (HMM), and (2) the distance between two trajectories is computed as the probabilistic similarity between HMMs, by means of the probability product kernel. Experiments on a set of real traffic video sequences reveal very good results of the proposed approach, outperforming two non-trivial baselines. To the best of the authors' knowledge, this approach is novel for trajectory grouping in traffic videos.

Multi-Objects Detection in Remote Sensing Images Using Multiple Kernel Learning

The objective of this work is multiple objects detection in remote sensing images. Many classifiers have been proposed to detect military objects. In this paper, we demonstrate that linear combination of kernels can get a better classification precision than product of kernels. Starting with base kernels, we obtain different weights for each class through learning. Experiment on Caltech-101 dataset shows the learnt kernels yields superior classification results compared with single-kernel SVM. While such a powerful classifier act as a sliding-window detector to search planes in images collected from Google Earth, results shows the effectiveness of using MKL detector to locate military objects in remote sensing images.

Average case tractability of non-homogeneous tensor product problems

We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure νd. Our interest is focused on measures having the structure of a non-homogeneous tensor product, where the covariance kernel of νd is a product of univariate kernels: Kd(s,t)=∏k=1dRk(sk,tk)for s,t∈[0,1]d. We consider the normalized average error of algorithms that use finitely many evaluations of arbitrary linear functionals. The information complexity is defined as the minimal number navg(ε,d) of such evaluations for error in the d-variate case to be at most ε. The growth of navg(ε,d) as a function of ε−1 and d depends on the eigenvalues of the covariance operator of νd and determines whether a problem is tractable or not. Four types of tractability are studied and for each of them we find the necessary and sufficient conditions in terms of the eigenvalues of the integral operator with kernel Rk.We illustrate our results by considering approximation problems related to the product of Korobov kernels Rk. Each Rk is characterized by a weight gk and a smoothness rk. We assume that weights are non-increasing and smoothness parameters are non-decreasing. Furthermore they may be related; for instance gk=g(rk) for some non-increasing function g. In particular, we show that the approximation problem is strongly polynomially tractable, i.e., navg(ε,d)≤Cε−p for all d∈N,ε∈(0,1], where C and p are independent of ε and d, iff lim infk→∞ln1gklnk>1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences gk and rk.

Automatic speaker age and gender recognition using acoustic and prosodic level information fusion

The paper presents a novel automatic speaker age and gender identification approach which combines seven different methods at both acoustic and prosodic levels to improve the baseline performance. The three baseline subsystems are (1) Gaussian mixture model (GMM) based on mel-frequency cepstral coefficient (MFCC) features, (2) Support vector machine (SVM) based on GMM mean supervectors and (3) SVM based on 450-dimensional utterance level features including acoustic, prosodic and voice quality information. In addition, we propose four subsystems: (1) SVM based on UBM weight posterior probability supervectors using the Bhattacharyya probability product kernel, (2) Sparse representation based on UBM weight posterior probability supervectors, (3) SVM based on GMM maximum likelihood linear regression (MLLR) matrix supervectors and (4) SVM based on the polynomial expansion coefficients of the syllable level prosodic feature contours in voiced speech segments. Contours of pitch, time domain energy, frequency domain harmonic structure energy and formant for each syllable (segmented using energy information in the voiced speech segment) are considered for analysis in subsystem (4). The proposed four subsystems have been demonstrated to be effective and able to achieve competitive results in classifying different age and gender groups. To further improve the overall classification performance, weighted summation based fusion of these seven subsystems at the score level is demonstrated. Experiment results are reported on the development and test set of the 2010 Interspeech Paralinguistic Challenge aGender database. Compared to the SVM baseline system (3), which is the baseline system suggested by the challenge committee, the proposed fusion system achieves 5.6% absolute improvement in unweighted accuracy for the age task and 4.2% for the gender task on the development set. On the final test set, we obtain 3.1% and 3.8% absolute improvement, respectively.

Multi-parameter singular Radon transforms III: Real analytic surfaces

The goal of this paper is to study operators of the form,Tf(x)=ψ(x)∫f(γt(x))K(t)dt, where γ is a real analytic function defined on a neighborhood of the origin in (t,x)∈RN×Rn, satisfying γ0(x)≡x, ψ is a cutoff function supported near 0∈Rn, and K is a “multi-parameter singular kernel” supported near 0∈RN. A main example is when K is a “product kernel.” We also study maximal operators of the form,Mf(x)=ψ(x)sup0<δ1,…,δN≪1∫|t|<1|f(γδ1t1,…,δNtN(x))|dt. We show that M is bounded on Lp (1<p⩽∞). We give conditions on γ under which T is bounded on Lp (1<p<∞); these conditions hold automatically when K is a Calderón–Zygmund kernel. This is the final paper in a three part series. The first two papers consider the more general case when γ is C∞.

Multi-parameter singular radon transforms I: The L 2 theory

The purpose of this paper is to study the L 2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γ t (x))K(t)dt, where γ t (x) is a C ∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ N × ℝ n , satisfying γ 0(x) ≡ x, ψ is a C ∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ n , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L 2. The case when K is a Calderon-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderon- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L p boundedness, while the third paper deals with the special case when γ is real analytic.

On the boundedness of product kernel operators with measures

On the boundedness of product kernel operators with measures

Product kernels adapted to curves in the space

We establish L^p -boundedness for a class of operators that are given by convolution with product kernels adapted to curves in the space. The L^p bounds follow from the decomposition of the adapted kernel into a sum of two kernels with singularities concentrated respectively on a coordinate plane and along the curve. The proof of the L^p -estimates for the two corresponding operators involves Fourier analysis techniques and some algebraic tools, namely the Bernstein-Sato polynomials. As an application, we show that these bounds can be exploited in the study of L^p-L^q estimates for analytic families of fractional operators along curves in the space.

Optimal Bounds for Self-Similar Solutions to Coagulation Equations with Product Kernel

We consider mass-conserving self-similar solutions of Smoluchowski's coagulation equation with product kernel of homogeneity 2λ ∈ (0, 1). We establish rigorously that such solutions exhibit a singular behavior of the form x −(1+2λ) as x → 0. This property had been conjectured, but only weaker results had been available up to now.

Do Complex Models Increase Prediction of Complex Behaviours? Predicting Driving Ability in People with Brain Disorders

Prediction of complex behavioural tasks via relatively simple modelling techniques, such as logistic regression and discriminant analysis, often has limited success. We hypothesized that to more accurately model complex behaviour, more complex models, such as kernel-based methods, would be needed. To test this hypothesis, we assessed the value of six modelling approaches for predicting driving ability based on performance on computerized sensory-motor and cognitive tests (SMCTests™) in 501 people with brain disorders. The models included three models previously used to predict driving ability (discriminant analysis, DA; binary logistic regression, BLR; and nonlinear causal resource analysis, NCRA) and three kernel methods (support vector machine, SVM; product kernel density, PK; and kernel product density, KP). At the classification level, two kernel methods were substantially more accurate at classifying on-road pass or fail (SVM 99.6%, PK 99.8%) than the other models (DA 76%, BLR 78%, NCRA 74%, KP 81%). However, accuracy decreased substantially for all of the kernel models when cross-validation techniques were used to estimate prediction of on-road pass or fail in an independent referral group (SVM 73-76%, PK 72-73%, KP 71-72%) but decreased only slightly for DA (74-75%) and BLR (75-76%). Cross-validation of NCRA was not possible. In conclusion, while kernel-based models are successful at modelling complex data at a classification level, this is likely to be due to overfitting of the data, which does not lead to an improvement in accuracy in independent data over and above the accuracy of other less complex modelling techniques.

Simultaneous saccharification and cofermentation of lignocellulosic residues from commercial furfural production and corn kernels using different nutrient media

BackgroundAs the supply of starch grain and sugar cane, currently the main feedstocks for bioethanol production, become limited, lignocelluloses will be sought as alternative materials for bioethanol production. Production of cellulosic ethanol is still cost-inefficient because of the low final ethanol concentration and the addition of nutrients. We report the use of simultaneous saccharification and cofermentation (SSCF) of lignocellulosic residues from commercial furfural production (furfural residue, FR) and corn kernels to compare different nutritional media. The final ethanol concentration, yield, number of live yeast cells, and yeast-cell death ratio were investigated to evaluate the effectiveness of integrating cellulosic and starch ethanol.ResultsBoth the ethanol yield and number of live yeast cells increased with increasing corn-kernel concentration, whereas the yeast-cell death ratio decreased in SSCF of FR and corn kernels. An ethanol concentration of 73.1 g/L at 120 h, which corresponded to a 101.1% ethanol yield based on FR cellulose and corn starch, was obtained in SSCF of 7.5% FR and 14.5% corn kernels with mineral-salt medium. SSCF could simultaneously convert cellulose into ethanol from both corn kernels and FR, and SSCF ethanol yield was similar between the organic and mineral-salt media.ConclusionsStarch ethanol promotes cellulosic ethanol by providing important nutrients for fermentative organisms, and in turn cellulosic ethanol promotes starch ethanol by providing cellulosic enzymes that convert the cellulosic polysaccharides in starch materials into additional ethanol. It is feasible to produce ethanol in SSCF of FR and corn kernels with mineral-salt medium. It would be cost-efficient to produce ethanol in SSCF of high concentrations of water-insoluble solids of lignocellulosic materials and corn kernels. Compared with prehydrolysis and fed-batch strategy using lignocellulosic materials, addition of starch hydrolysates to cellulosic ethanol production is a more suitable method to improve the final ethanol concentration.

Effect of Fermentation on Chemical Composition and Nutritional Quality of Extruded and Fermented Soya Products

The effect of fermentation on chemical composition, amino acid profile and protein digestibility of extruded soya protein products was investigated. The soya protein products (low moisture extruded soya product, medium moisture extruded soya product, high moisture extruded soya product, soya meal and soya kernel) were fermented with Bacillus natto from small and large brands of soya bean cultivar. The protein content (%) ranged from 38.20 to 62.98 with the highest content in the high moisture extruded protein product fermented with Bacillus natto from large brand of soya bean cultivar. Contents of carbohydrates ranged from 14.77 to 29.08 while those of crude fibre, fat and ash were generally low. Fermentation improved protein digestibility in the raw soya meal and kernel than in the unfermented extruded and extruded fermented products. Extrusion reduced the nitrogen solubility index of the extruded samples. The contents of amino acids except arginine increased in all the samples after fermentation. Trypsin inhibitor activity decreased from 27.33 TIU/g in the unfermented soya sample to between 1.58 and 18.5 TIU/g after fermentation, whereas there was a decrease in phytic acid content only in the fermented soya meal and low moisture extruded soya sample.

Asymptotics of Self-similar Solutions to Coagulation Equations with Product Kernel

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(\xi,\eta)= (\xi \eta)^{\lambda}$ with $\lambda \in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2\lambda} g(x)$ is bounded above and below as $x \to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_{\lambda} x^{-1+2\lambda} g(x)$ in the limit $\lambda \to 0$. It turns out that $h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x)$ as $x \to 0$. As $x$ becomes larger $h$ develops peaks of height $1/\lambda$ that are separated by large regions where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x \to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.

Characterization and sequence prediction of structural variations in α-helix

BackgroundThe structure conservation in various α-helix subclasses reveals the sequence and context dependent factors causing distortions in the α-helix. The sequence-structure relationship in these subclasses can be used to predict structural variations in α-helix purely based on its sequence. We train support vector machine(SVM) with dot product kernel function to discriminate between regular α-helix and non-regular α-helices purely based on the sequences, which are represented with various overall and position specific propensities of amino acids.ResultsWe characterize the structural distortions in five α-helix subclasses. The sequence structure correlation in the subclasses reveals that the increased propensity of proline, histidine, serine, aspartic acid and aromatic amino acids are responsible for the distortions in regular α-helix. The N-terminus of regular α-helix prefers neutral and acidic polar amino acids, while the C-terminus prefers basic polar amino acid. Proline is preferred in the first turn of regular α-helix , while it is preferred to produce kinked and curved subclasses. The SVM discriminates between regular α-helix and the rest with precision of 80.97% and recall of 88.05%.ConclusionsThe correlation between structural variation in helices and their sequences is manifested by the performance of SVM based on sequence features. The results presented here are useful for computational design of helices. The results are also useful for prediction of structural perturbations in helix sequence purely based on its sequence.

Kernel density estimation on the torus

Kernel density estimation for multivariate, circular data has been formulated only when the sample space is the sphere, but theory for the torus would also be useful. For data lying on a d-dimensional torus ( d ⩾ 1 ) , we discuss kernel estimation of a density, its mixed partial derivatives, and their squared functionals. We introduce a specific class of product kernels whose order is suitably defined in such a way to obtain L 2-risk formulas whose structure can be compared to their Euclidean counterparts. Our kernels are based on circular densities; however, we also discuss smaller bias estimation involving negative kernels which are functions of circular densities. Practical rules for selecting the smoothing degree, based on cross-validation, bootstrap and plug-in ideas are derived. Moreover, we provide specific results on the use of kernels based on the von Mises density. Finally, real-data examples and simulation studies illustrate the findings.

Multi-parameter singular Radon transforms

The purpose of this announcement is to describe a development given in a series of forthcoming papers by the authors that concern operators of the form \[ f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt, \] where $\gamma_t(x)=\gamma(t,x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \mathbb{R}^N\times \mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $K(t)$ is a singular supported near $t=0$, and $\psi$ is a cutoff function supported near $x=0$. This note concerns the case when $K$ is a product kernel. The goal is to give conditions on $\gamma$ such that the above operator is bounded on $L^p$ for $1<p<\infty$. Associated maximal functions are also discussed. The single-parameter case when $K$ is a Calder\'on-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger. The theory here extends these results to the multi-parameter context and also deals effectively with the case when $\gamma$ is real-analytic.