arXiv subjects

Soumyabrata Pal

Publications and source records attributed to Soumyabrata Pal.

41 records · Page 3Linked to original sources

arXiv2019

Same-Cluster Querying for Overlapping Clusters

Overlapping clusters are common in models of many practical data-segmentation applications. Suppose we are given $n$ elements to be clustered into $k$ possibly overlapping clusters, and an oracle that can interactively answer queries of the form "do elements $u$ and $v$ belong to the same cluster?" The goal is to recover the clusters with minimum number of such queries. This problem has been of recent interest for the case of disjoint clusters. In this paper, we look at the more practical scenario of overlapping clusters, and provide upper bounds (with algorithms) on the sufficient number of queries. We provide algorithmic results under both arbitrary (worst-case) and statistical modeling assumptions. Our algorithms are parameter free, efficient, and work in the presence of random noise. We also derive information-theoretic lower bounds on the number of queries needed, proving that our algorithms are order optimal. Finally, we test our algorithms over both synthetic and real-world data, showing their practicality and effectiveness.

cs.LG↗

arXiv2019

High Dimensional Discrete Integration over the Hypergrid

Recently Ermon et al. (2013) pioneered a way to practically compute approximations to large scale counting or discrete integration problems by using random hashes. The hashes are used to reduce the counting problem into many separate discrete optimization problems. The optimization problems then can be solved by an NP-oracle such as commercial SAT solvers or integer linear programming (ILP) solvers. In particular, Ermon et al. showed that if the domain of integration is $\{0,1\}^n$ then it is possible to obtain a solution within a factor of $16$ of the optimal (a 16-approximation) by this technique. In many crucial counting tasks, such as computation of partition function of ferromagnetic Potts model, the domain of integration is naturally $\{0,1,\dots, q-1\}^n, q>2$, the hypergrid. The straightforward extension of Ermon et al.'s method allows a $q^2$-approximation for this problem. For large values of $q$, this is undesirable. In this paper, we show an improved technique to obtain an approximation factor of $4+O(1/q^2)$ to this problem. We are able to achieve this by using an idea of optimization over multiple bins of the hash functions, that can be easily implemented by inequality constraints, or even in unconstrained way. Also the burden on the NP-oracle is not increased by our method (an ILP solver can still be used). We provide experimental simulation results to support the theoretical guarantees of our algorithms.

cs.DS↗

arXiv2018

The Geometric Block Model

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model generalizes the random geometric graphs in the same way that the well-studied stochastic block model generalizes the Erdos-Renyi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancement in community detection. While being a topic of fundamental theoretical interest, our main contribution is to show that many practical community structures are better explained by the geometric block model. We also show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. Indeed, even in the regime where the average degree of the graph grows only logarithmically with the number of vertices (sparse-graph), we show that this algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model. We simulate our results on both real and synthetic datasets to show superior performance of both the new model as well as our algorithm.

cs.SI↗

arXiv2016

Polynomials and Second Order Linear Recurrences

One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling, based on previously developed machinery, Hilbert's question whether there exists a general algorithm for checking the solvability in integers of any diophantine equation. In this paper we describe techniques to prove the nonexistence of polynomials in two variables for some simple generalizations of the Fibonacci sequence (explicit diophantine representation of Fibonacci numbers were known from Jones' polynomial whose positive values have the same range as that of Fibonacci numbers), and we believe similar techniques exist for the primes. In this paper we mainly show the following results: (1) using one of the many techniques known for solving the Pell's equation, namely the solution in an extended number system, we prove the existence and explicitly find the polynomials for the recurrences of the form $e(n)=ae(n-1)+e(n-2)$ with starting values of 0 and 1 in particular, and for any arbitrary starting values, in the process defining a concept of fundamental starting numbers, (2) we prove a few identities that seem to be quite interesting and useful, (3) we use these identities in a novel way to generate systems of equations of certain rank deficiency using which we disprove for the first time the existence of any polynomial in 2 variables for the generalized recurrence of the form $e(n)=ae(n-1)+be(n-2)$

math.CO↗

arXiv2015

Prime Power Divisibility,Periodicity and Other Properties of Some Second Order Recurrences

Wall published a paper in 1960 on the Fibonacci sequence where he derived many results concerning the period and prime power divisibility modulo m. His periodicity results have been generalized to second order linear recurrences. Here we study the sequences generated by such recurrences, with starting values of {0,1}: among other things, we derive new prime power divisibility results, derive the period by new methods, establish new identities, show derivations involving powers of matrices generated by these general recurrences, etc.

math.CO↗