Publications (58)15.74 Total impact
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ABSTRACT: The main aim of this paper is to establish the applicability of a broad class of random measures, that includes the gamma process, for mixed membership modelling. We use completely random measures~(CRM) and hierarchical CRM to define a prior for Poisson processes. We derive the marginal distribution of the resultant point process, when the underlying CRM is marginalized out. Using well known properties unique to Poisson processes, we were able to derive an exact approach for instantiating a Poisson process with a hierarchical CRM prior. Furthermore, we derive Gibbs sampling strategies for hierarchical CRM models based on Chinese restaurant franchise sampling scheme. As an example, we present the sum of generalized gamma process (SGGP), and show its application in topicmodelling. We show that one can determine the powerlaw behaviour of the topics and words in a Bayesian fashion, by defining a prior on the parameters of SGGP. 
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ABSTRACT: Let $H_{n,(p_m)_{m=2,\ldots,M}}$ be a random nonuniform hypergraph of dimension $M$ on $2n$ vertices, where the vertices are split into two disjoint sets of size $n$, and colored by two distinct colors. Each nonmonochromatic edge of size $m=2,\ldots,M$ is independently added with probability $p_m$. We show that if $p_2,\ldots,p_M$ are such that the expected number of edges in the hypergraph is at least $dn\ln n$, for some $d>0$ sufficiently large, then with probability $(1o(1))$, one can find a proper 2coloring of $H_{n,(p_m)_{m=2,\ldots,M}}$ in polynomial time. The coloring algorithm presented in this paper makes use of the spectral properties of the star expansion of the hypergraph.  [Show abstract] [Hide abstract]
ABSTRACT: Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. A number of algorithms exist in the literature that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received attention in the literature as compared to the extensive studies on the guarantees of graph partitioning. For instance, consistency results of spectral clustering under the planted partition or stochastic blockmodel are wellknown (Rohe et al., 2011;Lei and Rinaldo, 2015). In this paper, we present a planted partition model for sparse random nonuniform hypergraphs that generalizes the stochastic blockmodels for graphs and uniform hypergraphs. We derive an asymptotic error bound of a spectral hypergraph partitioning algorithm under this model using matrix Bernstein inequality. To the best of our knowledge, this is the first consistency result related to partitioning nonuniform hypergraphs. 
Article: A faster algorithm for testing polynomial representability of functions over finite integer rings
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ABSTRACT: Given a function from to itself one can determine its polynomial representability by using Kempner function. In this paper we present an alternative characterization of polynomial functions over by constructing a generating set for the 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.  [Show abstract] [Hide abstract]
ABSTRACT: Lyubashevsky & Micciancio (2006) built collision resistant hash functions based on ideal lattices (in the univariate case) that in turn paved the way for the construction of other cryptographic primitives. Recently, in (Francis & Dukkipati, 2014), univariate ideal lattices have been extended to a multivariate case and its connections to Gr\"obner bases have been studied. In this paper, we show the existence of collision resistant generalized hash functions based on multivariate ideal lattices. We show that using Gr\"obner basis techniques an analogous theory can be developed for the multivariate case, opening up an area of possibilities for cryptographic primitives based on ideal lattices. For the construction of hash functions, we define an expansion factor that checks coefficient growth and determine the expansion factor for specific multivariate ideal lattices. We define a worst case problem, shortest polynomial problem w.r.t. an ideal in $\mathbb{Z}[x_1, ..., x_n]$, and prove the hardness of the problem by using certain well known problems in algebraic function fields.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we draw a connection between ideal lattices and Gr\"{o}bner bases in the multivariate polynomial rings over integers. We study extension of ideal lattices in $\mathbb{Z}[x]/\langle f \rangle$ (Lyubashevsky \& Micciancio, 2006) to ideal lattices in $\mathbb{Z}[x_1,\ldots,x_n]/\mathfrak{a}$, the multivariate case, where $f$ is a polynomial in $\mathbb{Z}[X]$ and $\mathfrak{a}$ is an ideal in $\mathbb{Z}[x_1,\ldots,x_n]$. Ideal lattices in univariate case are interpreted as generalizations of cyclic lattices. We introduce a notion of multivariate cyclic lattices and we show that multivariate ideal lattices are indeed a generalization of them. We show that the fact that existence of ideal lattice in univariate case if and only if $f$ is monic translates to short reduced Gr\"obner basis (Francis \& Dukkipati, 2014) of $\mathfrak{a}$ is monic in multivariate case. We, thereby, give a necessary and sufficient condition for residue class polynomial rings over $\mathbb{Z}$ to have ideal lattices. We also characterize ideals in $\mathbb{Z}[x_1,\ldots,x_n]$ that give rise to full rank lattices. 
Article: MacaulayBuchberger Basis Theorem for Residue Class Rings with Torsion and Border Bases over Rings
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ABSTRACT: In this paper we generalize the MacaulayBuchberger basis theorem for residue class ring $A[x_{1},\ldots,x_{n}]/ \mathfrak{a}$, in the case when it is finitely generated as an $A$module but need not necessarily a free module, where $A$ is a Noetherian ring and $\mathfrak{a}$ is an ideal. This generalization gives us an insight into the nature of generating sets that span $A$module $A[x_{1},\ldots,x_{n}]/ \mathfrak{a}$ and allows to study the concept of border bases over rings. We present a border division algorithm over rings and prove termination of the algorithm for a special class of border bases called acyclic border bases. We show the existence of such border bases and present some characterizations in this regard.  [Show abstract] [Hide abstract]
ABSTRACT: A motivation to study Gr\"{o}bner 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 Gr\"obner 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 wellordered set $\mathbb{N}$, and thereby give a method to calculate the Gr\"{o}bner basis for submodules generated by homogeneous elements.  [Show abstract] [Hide abstract]
ABSTRACT: Motivated by multidistribution divergences, which originate in information theory, we propose a notion of `multipoint' 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 multipoint (kernel) spectral clustering (MSC) method. We further emphasize on a special case of the proposed kernels, which is a multipoint extension of the linear (dotproduct) 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.  [Show abstract] [Hide abstract]
ABSTRACT: Polynomial functions over Zn, where n is a positive integer, have been characterized in (Kempner, 1921), by considering {1, X, X^2 , . . .} as a set of generators and providing suitable bounds for the coefficients of exponents of X. We provide another characterization by giving a generating set for the Zn module of polynomial functions. We also use these generators to obtain a canonical representation and extend our result to the multivariate case. 
Article: To go deep or wide in learning?
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ABSTRACT: To achieve acceptable performance for AI tasks, one can either use sophisticated feature extraction methods as the first layer in a twolayered supervised learning model, or learn the features directly using a deep (multilayered) model. While the first approach is very problemspecific, the second approach has computational overheads in learning multiple layers and finetuning of the model. In this paper, we propose an approach called wide learning based on arccosine 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.  [Show abstract] [Hide abstract]
ABSTRACT: 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.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we extend the idea of comprehensive Gr\"{o}bner 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 Gr\"{o}bner bases and comprehensive border bases w.r.t. a term order and give an algorithm to compute such comprehensive border bases from comprehensive Gr\"{o}bner bases. 
Conference Paper: Generative Maximum Entropy Learning for Multiclass Classification
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ABSTRACT: 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 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 multiclass case, we propose a completely new approach by considering a multidistribution divergence: we replace Jeffreys divergence by JensenShannon (JS) divergence to discriminate conditional densities of multiple classes. In order to reduce computational complexity, we employ a modified JensenShannon divergence (JSGM), based on AMGM 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. We give a similar theoretical justification for JSGM − divergence for multiclass case. 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.  [Show abstract] [Hide abstract]
ABSTRACT: We present the first qGaussian smoothed functional (SF) estimator of the Hessian and the first Newtonbased stochastic optimization algorithm that estimates both the Hessian and the gradient of the objective function using qGaussian 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 qGaussian 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 qGaussian 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 qvalues, in comparison to Newton SF algorithms with the Gaussian (Bhatnagar, 2007) and Cauchy perturbations, as well as the gradient qGaussian SF algorithms (Ghoshdastidar et al., 2013).  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we extend the characterization of $\mathbb{Z}[x]/\ < f \ >$, where $f \in \mathbb{Z}[x]$ to be a free $\mathbb{Z}$module to multivariate polynomial rings over any commutative Noetherian ring, $A$. The characterization allows us to extend the Gr\"obner basis method of computing a $\Bbbk$vector space basis of residue class polynomial rings over a field $\Bbbk$ (MacaulayBuchberger Basis Theorem) to rings, i.e. $A[x_1,\ldots,x_n]/\mathfrak{a}$, where $\mathfrak{a} \subseteq A[x_1,\ldots,x_n]$ is an ideal. We give some insights into the characterization for two special cases, when $A = \mathbb{Z}$ and $A = \Bbbk[\theta_1,\ldots,\theta_m]$. As an application of this characterization, we show that the concept of Border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free $A$module.  [Show abstract] [Hide abstract]
ABSTRACT: Gröbner basis detection (GBD) is defined as follows: given a set of polynomials, decide whether there exists–and if “yes” find–a term order such that the set of polynomials is a Gröbner basis. This problem was proposed by Gritzmann and Sturmfels (1993) [12] and it was shown to be NPhard by Sturmfels and Wiegelmann. We investigate the computational complexity of this problem when the given set of polynomials are the generators of a zerodimensional ideal. Further, we propose the Border basis detection (BBD) problem which is formulated as follows: given a set of generators of an ideal, decide whether the set of generators is a border basis of the ideal with respect to some order ideal. We analyse the complexity of this problem and prove it to be NPcomplete.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we study constrained maximum entropy and minimum divergence optimization problems, in the cases where integer valued sufficient statistics exists, using tools from computational commutative algebra. We show that the estimation of parametric statistical models in this case can be transformed to solving a system of polynomial equations. We give an implicit description of maximum entropy models by embedding them in algebraic varieties for which we give a Gröbner basis method to compute it. In the cases of minimum KLdivergence models we show that implicitization preserves specialization of prior distribution. This result leads us to a Gröbner basis method to embed minimum KLdivergence models in algebraic varieties.  [Show abstract] [Hide abstract]
ABSTRACT: The importance of the qGaussian family of distributions lies in its powerlaw nature, and its close association with Gaussian, Cauchy and uniform distributions. This class of distributions arises from maximization of a generalized information measure. We use the powerlaw nature of the qGaussian distribution to improve upon the smoothing properties of Gaussian and Cauchy distributions. Based on this, we propose a Smoothed Functional (SF) scheme for gradient estimation using qGaussian distribution. Our work extends the class of distributions that can be used in SF algorithms by including the qGaussian distributions, which encompass the above three distributions as special cases. Using the derived gradient estimates, we propose twotimescale algorithms for optimization of a stochastic objective function with gradient descent method. We prove that the proposed algorithms converge to a local optimum. Performance of the algorithms is shown by simulation results on a queuing model.
Publication Stats
89  Citations  
15.74  Total Impact Points  
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Institutions

20092012

Indian Institute of Science
 Department of Computer Science and Automation
Bengalūru, Karnataka, India
