arXiv subjects

Ambedkar Dukkipati

Publications and source records attributed to Ambedkar Dukkipati.

At least 73 records · Page 4Linked to original sources

arXiv2015

A Faster Algorithm For Testing Polynomial Representability Of Functions Over Finite Integer Rings

Given a function from $\mathbb{Z}_n$ to itself one can determine its polynomial representability by using Kempner function. In this paper we present an alternative characterization of polynomial functions over $\mathbb{Z}_n$ by constructing a generating set for the $\mathbb{Z}_{n}$-module of polynomial functions. This characterization results in an algorithm that is faster on average in deciding polynomial representability. We also extend the characterization to functions in several variables.

math.RA↗

arXiv2014

Gaussian Mersenne Primes of the form $x^2+dy^2$

In this paper we study Gaussian ring $\Z[i]$ with a focus on representing Gaussian Mersenne primes $G_p$ in the form $x^2+7y^2$. Interestingly when such a form exists, one can observe that, $x\equiv \pm 1\pmod{8}$ and $y\equiv 0\pmod{8}$. To prove this property of Gaussian Mersenne primes, we show that Gaussian Mersenne primes splits completely in the cyclic quartic unramified extension of $\Q(\sqrt{-14})$ and have a trivial Artin symbol in this extension. We generalize this result for $d\equiv 7\pmod{24}$. We also attempt to give an alternate proof using Artin's reciprocity law, which was earlier given by H. W. Lenstra and P. Stevenhagen to prove a similar property on ordinary Mersenne Primes.

math.NT↗

arXiv2014

Smoothed Functional Algorithms for Stochastic Optimization using q-Gaussian Distributions

Smoothed functional (SF) schemes for gradient estimation are known to be efficient in stochastic optimization algorithms, specially when the objective is to improve the performance of a stochastic system. However, the performance of these methods depends on several parameters, such as the choice of a suitable smoothing kernel. Different kernels have been studied in literature, which include Gaussian, Cauchy and uniform distributions among others. This paper studies a new class of kernels based on the q-Gaussian distribution, that has gained popularity in statistical physics over the last decade. Though the importance of this family of distributions is attributed to its ability to generalize the Gaussian distribution, we observe that this class encompasses almost all existing smoothing kernels. This motivates us to study SF schemes for gradient estimation using the q-Gaussian distribution. Using the derived gradient estimates, we propose two-timescale algorithms for optimization of a stochastic objective function in a constrained setting with projected gradient search approach. We prove the convergence of our algorithms to the set of stationary points of an associated ODE. We also demonstrate their performance numerically through simulations on a queuing model.

cs.IT↗

arXiv2014

Newton based Stochastic Optimization using q-Gaussian Smoothed Functional Algorithms

We present the first q-Gaussian smoothed functional (SF) estimator of the Hessian and the first Newton-based stochastic optimization algorithm that estimates both the Hessian and the gradient of the objective function using q-Gaussian perturbations. Our algorithm requires only two system simulations (regardless of the parameter dimension) and estimates both the gradient and the Hessian at each update epoch using these. We also present a proof of convergence of the proposed algorithm. In a related recent work (Ghoshdastidar et al., 2013), we presented gradient SF algorithms based on the q-Gaussian perturbations. Our work extends prior work on smoothed functional algorithms by generalizing the class of perturbation distributions as most distributions reported in the literature for which SF algorithms are known to work and turn out to be special cases of the q-Gaussian distribution. Besides studying the convergence properties of our algorithm analytically, we also show the results of several numerical simulations on a model of a queuing network, that illustrate the significance of the proposed method. In particular, we observe that our algorithm performs better in most cases, over a wide range of q-values, in comparison to Newton SF algorithms with the Gaussian (Bhatnagar, 2007) and Cauchy perturbations, as well as the gradient q-Gaussian SF algorithms (Ghoshdastidar et al., 2013).

math.OC↗

arXiv2014

An Algorithmic Characterization of Polynomial Functions over $Z_{p^n}$

In this paper we consider polynomial representability of functions defined over $Z_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to determine whether a given function over $Z_{p^n}$ is polynomially representable or not, and (ii) finds the polynomial if it is polynomially representable. The previous characterizations given by Kempner (1921) and Carlitz (1964) are existential in nature and only lead to an exhaustive search method, i.e., algorithm with complexity exponential in size of the input. Our characterization leads to an algorithm whose running time is linear in size of input. We also extend our result to the multivariate case.

cs.SC↗

arXiv2014

Tropical Grassmannian and Tropical Linear Varieties from phylogenetic trees

In this paper we study tropicalization of Grassmannian and linear varieties. In particular, we study the tropical linear spaces cor- responding to the phylogenetic trees. We prove that corresponding to each subtree of the phylogenetic tree there is a point on the tropical grassmannian. We deduce a necessary and sufficient condition for it to be on the facet of the tropical linear space.

math.CO↗

arXiv2014

Gröbner Basis Theory for Modules over Polynomial Rings over Fields with Valuation

A motivation to study Gröbner theory for fields with valuations comes from tropical geometry, for example, they can be used to compute tropicalization of varieties \citep{maclagan2009introduction}. The computational aspect of this theory was first studied in (Chen \& Maclagan, 2013). In this paper, we generalize this Gröbner basis theory to free modules over polynomial rings over fields with valuation. As the valuation of coefficients is also taken into account while defining the initial term, we do not necessarily get a monomial order. To overcome this problem we have to resort to other techniques like the use of ecart function where the codomain is the well-ordered set $\mathbb{N}$, and thereby give a method to calculate the Gröbner basis for submodules generated by homogeneous elements.

math.AC↗

arXiv2014

Spectral Clustering with Jensen-type kernels and their multi-point extensions

Motivated by multi-distribution divergences, which originate in information theory, we propose a notion of `multi-point' kernels, and study their applications. We study a class of kernels based on Jensen type divergences and show that these can be extended to measure similarity among multiple points. We study tensor flattening methods and develop a multi-point (kernel) spectral clustering (MSC) method. We further emphasize on a special case of the proposed kernels, which is a multi-point extension of the linear (dot-product) kernel and show the existence of cubic time tensor flattening algorithm in this case. Finally, we illustrate the usefulness of our contributions using standard data sets and image segmentation tasks.

cs.LG↗

arXiv2014

To go deep or wide in learning?

To achieve acceptable performance for AI tasks, one can either use sophisticated feature extraction methods as the first layer in a two-layered supervised learning model, or learn the features directly using a deep (multi-layered) model. While the first approach is very problem-specific, the second approach has computational overheads in learning multiple layers and fine-tuning of the model. In this paper, we propose an approach called wide learning based on arc-cosine kernels, that learns a single layer of infinite width. We propose exact and inexact learning strategies for wide learning and show that wide learning with single layer outperforms single layer as well as deep architectures of finite width for some benchmark datasets.

cs.LG↗

arXiv2013

Generative Maximum Entropy Learning for Multiclass Classification

Maximum entropy approach to classification is very well studied in applied statistics and machine learning and almost all the methods that exists in literature are discriminative in nature. In this paper, we introduce a maximum entropy classification method with feature selection for large dimensional data such as text datasets that is generative in nature. To tackle the curse of dimensionality of large data sets, we employ conditional independence assumption (Naive Bayes) and we perform feature selection simultaneously, by enforcing a `maximum discrimination' between estimated class conditional densities. For two class problems, in the proposed method, we use Jeffreys ($J$) divergence to discriminate the class conditional densities. To extend our method to the multi-class case, we propose a completely new approach by considering a multi-distribution divergence: we replace Jeffreys divergence by Jensen-Shannon ($JS$) divergence to discriminate conditional densities of multiple classes. In order to reduce computational complexity, we employ a modified Jensen-Shannon divergence ($JS_{GM}$), based on AM-GM inequality. We show that the resulting divergence is a natural generalization of Jeffreys divergence to a multiple distributions case. As far as the theoretical justifications are concerned we show that when one intends to select the best features in a generative maximum entropy approach, maximum discrimination using $J-$divergence emerges naturally in binary classification. Performance and comparative study of the proposed algorithms have been demonstrated on large dimensional text and gene expression datasets that show our methods scale up very well with large dimensional datasets.

cs.IT↗

arXiv2013

Comprehensive Border Bases for Zero Dimensional Parametric Polynomial Ideals

In this paper, we extend the idea of comprehensive Gröbner bases given by Weispfenning (1992) to border bases for zero dimensional parametric polynomial ideals. For this, we introduce a notion of comprehensive border bases and border system, and prove their existence even in the cases where they do not correspond to any term order. We further present algorithms to compute comprehensive border bases and border system. Finally, we study the relation between comprehensive Gröbner bases and comprehensive border bases w.r.t. a term order and give an algorithm to compute such comprehensive border bases from comprehensive Gröbner bases.

cs.SC↗

arXiv2013

Minimum Description Length Principle for Maximum Entropy Model Selection

Model selection is central to statistics, and many learning problems can be formulated as model selection problems. In this paper, we treat the problem of selecting a maximum entropy model given various feature subsets and their moments, as a model selection problem, and present a minimum description length (MDL) formulation to solve this problem. For this, we derive normalized maximum likelihood (NML) codelength for these models. Furthermore, we prove that the minimax entropy principle is a special case of maximum entropy model selection, where one assumes that complexity of all the models are equal. We apply our approach to gene selection problem and present simulation results.

cs.IT↗

arXiv2013

On Power-law Kernels, corresponding Reproducing Kernel Hilbert Space and Applications

The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. This generalization is based on distributions, arising out of maximization of a generalized information measure known as nonextensive entropy that is very well studied in statistical mechanics. We prove that the proposed kernels are positive definite, and provide some insights regarding the corresponding Reproducing Kernel Hilbert Space (RKHS). We also study practical significance of both kernels in classification and regression, and present some simulation results.

cs.LG↗

arXiv2013

A Preprocessor Based on Clause Normal Forms and Virtual Substitutions to Parallelize Cylindrical Algebraic Decomposition

The Cylindrical Algebraic Decomposition (CAD) algorithm is a comprehensive tool to perform quantifier elimination over real closed fields. CAD has doubly exponential running time, making it infeasible for practical purposes. We propose to use the notions of clause normal forms and virtual substitutions to develop a preprocessor for CAD, that will enable an input-level parallelism. We study the performance of CAD in the presence of the preprocessor by extensive experimentation. Since parallelizability of CAD depends on the structure of given prenex formula, we introduce some structural notions to study the performance of CAD with the proposed preprocessor.

cs.DM↗

arXiv2012

On Shore and Johnson properties for a Special Case of Csiszár f-divergences

The importance of power-law distributions is attributed to the fact that most of the naturally occurring phenomenon exhibit this distribution. While exponential distributions can be derived by minimizing KL-divergence w.r.t some moment constraints, some power law distributions can be derived by minimizing some generalizations of KL-divergence (more specifically some special cases of Csiszár f-divergences). Divergence minimization is very well studied in information theoretical approaches to statistics. In this work we study properties of minimization of Tsallis divergence, which is a special case of Csiszár f-divergence. In line with the work by Shore and Johnson (IEEE Trans. IT, 1981), we examine the properties exhibited by these minimization methods including the Pythagorean property.

cs.IT↗

arXiv2012

q-Gaussian based Smoothed Functional Algorithm for Stochastic Optimization

The q-Gaussian distribution results from maximizing certain generalizations of Shannon entropy under some constraints. The importance of q-Gaussian distributions stems from the fact that they exhibit power-law behavior, and also generalize Gaussian distributions. In this paper, we propose a Smoothed Functional (SF) scheme for gradient estimation using q-Gaussian distribution, and also propose an algorithm for optimization based on the above scheme. Convergence results of the algorithm are presented. Performance of the proposed algorithm is shown by simulation results on a queuing model.

eess.SY↗

arXiv2011

An Algebraic Characterization of Rainbow Connectivity

The use of algebraic techniques to solve combinatorial problems is studied in this paper. We formulate the rainbow connectivity problem as a system of polynomial equations. We first consider the case of two colors for which the problem is known to be hard and we then extend the approach to the general case. We also give a formulation of the rainbow connectivity problem as an ideal membership problem.

cs.DM↗

arXiv2011

Tropical Algebraic approach to Consensus over Networks

In this paper we study the convergence of the max-consensus protocol. Tropical algebra is used to formulate the problem. Necessary and sufficient conditions for convergence of the max-consensus protocol over fixed as well as switching topology networks are given.

math.OC↗