arXiv subjects

Kinjal Basu

Publications and source records attributed to Kinjal Basu.

52 records · Page 3Linked to original sources

arXiv2019

Conversational AI : Open Domain Question Answering and Commonsense Reasoning

Our research is focused on making a human-like question answering system which can answer rationally. The distinguishing characteristic of our approach is that it will use automated common sense reasoning to truly "understand" dialogues, allowing it to converse like a human. Humans often make many assumptions during conversations. We infer facts not told explicitly by using our common sense. Incorporating commonsense knowledge in a question answering system will simply make it more robust.

cs.AI↗

arXiv2019

Optimal Convergence for Stochastic Optimization with Multiple Expectation Constraints

In this paper, we focus on the problem of stochastic optimization where the objective function can be written as an expectation function over a closed convex set. We also consider multiple expectation constraints which restrict the domain of the problem. We extend the cooperative stochastic approximation algorithm from Lan and Zhou [2016] to solve the particular problem. We close the gaps in the previous analysis and provide a novel proof technique to show that our algorithm attains the optimal rate of convergence for both optimality gap and constraint violation when the functions are generally convex. We also compare our algorithm empirically to the state-of-the-art and show improved convergence in many situations.

math.ST↗

arXiv2017

Large-Scale Quadratically Constrained Quadratic Program via Low-Discrepancy Sequences

We consider the problem of solving a large-scale Quadratically Constrained Quadratic Program. Such problems occur naturally in many scientific and web applications. Although there are efficient methods which tackle this problem, they are mostly not scalable. In this paper, we develop a method that transforms the quadratic constraint into a linear form by sampling a set of low-discrepancy points. The transformed problem can then be solved by applying any state-of-the-art large-scale quadratic programming solvers. We show the convergence of our approximate solution to the true solution as well as some finite sample error bounds. Experimental results are also shown to prove scalability as well as improved quality of approximation in practice.

stat.ML↗

arXiv2017

Permutation p-value approximation via generalized Stolarsky invariance

It is common for genomic data analysis to use $p$-values from a large number of permutation tests. The multiplicity of tests may require very tiny $p$-values in order to reject any null hypotheses and the common practice of using randomly sampled permutations then becomes very expensive. We propose an inexpensive approximation to $p$-values for two sample linear test statistics, derived from Stolarsky's invariance principle. The method creates a geometrically derived set of approximate $p$-values for each hypothesis. The average of that set is used as a point estimate $\hat p$ and our generalization of the invariance principle allows us to compute the variance of the $p$-values in that set. We find that in cases where the point estimate is small the variance is a modest multiple of the square of the point estimate, yielding a relative error property similar to that of saddlepoint approximations. On a Parkinson's disease data set, the new approximation is faster and more accurate than the saddlepoint approximation. We also obtain a simple probabilistic explanation of Stolarsky's invariance principle.

math.ST↗

arXiv2017

Constrained Multi-Slot Optimization for Ranking Recommendations

Ranking items to be recommended to users is one of the main problems in large scale social media applications. This problem can be set up as a multi-objective optimization problem to allow for trading off multiple, potentially conflicting objectives (that are driven by those items) against each other. Most previous approaches to this problem optimize for a single slot without considering the interaction effect of these items on one another. In this paper, we develop a constrained multi-slot optimization formulation, which allows for modeling interactions among the items on the different slots. We characterize the solution in terms of problem parameters and identify conditions under which an efficient solution is possible. The problem formulation results in a quadratically constrained quadratic program (QCQP). We provide an algorithm that gives us an efficient solution by relaxing the constraints of the QCQP minimally. Through simulated experiments, we show the benefits of modeling interactions in a multi-slot ranking context, and the speed and accuracy of our QCQP approximate solver against other state of the art methods.

stat.ML↗

arXiv2017

Quasi-Monte Carlo for an Integrand with a Singularity along a Diagonal in the Square

Quasi-Monte Carlo methods are designed for integrands of bounded variation, and this excludes singular integrands. Several methods are known for integrands that become singular on the boundary of the unit cube $[0,1]^d$ or at isolated possibly unknown points within $[0,1]^d$. Here we consider functions on the square $[0,1]^2$ that may become singular as the point approaches the diagonal line $x_1=x_2$, and we study three quadrature methods. The first method splits the square into two triangles separated by a region around the line of singularity, and applies recently developed triangle QMC rules to the two triangular parts. For functions with a singularity `no worse than $|x_1-x_2|^{-A}$ for $0<A<1$ that method yields an error of $O( (\log(n)/n)^{(1-A)/2})$. We also consider methods extending the integrand into a region containing the singularity and show that method will not improve up on using two triangles. Finally, we consider transforming the integrand to have a more QMC-friendly singularity along the boundary of the square. This then leads to error rates of $O(n^{-1+ε+A})$ when combined with some corner-avoiding Halton points or with randomized QMC, but it requires some stronger assumptions on the original singular integrand.

math.NA↗

arXiv2016

Asymptotic Normality of Scrambled Geometric Net Quadrature

In a very recent work, Basu and Owen (2015) propose the use of scrambled geometric nets in numerical integration when the domain is a product of $s$ arbitrary spaces of dimension $d$ having a certain partitioning constraint. It was shown that for a class of smooth functions, the integral estimate has variance $O( n^{-1 -2/d} (\log n)^{s-1})$ for scrambled geometric nets, compared to $O(n^{-1})$ for ordinary Monte Carlo. The main idea of this paper is to develop on the work by Loh (2003), to show that the scrambled geometric net estimate has an asymptotic normal distribution for certain smooth functions defined on products of suitable subsets of $\mathbb{R}^d$.

math.ST↗

arXiv2016

Large scale multi-objective optimization: Theoretical and practical challenges

Multi-objective optimization (MOO) is a well-studied problem for several important recommendation problems. While multiple approaches have been proposed, in this work, we focus on using constrained optimization formulations (e.g., quadratic and linear programs) to formulate and solve MOO problems. This approach can be used to pick desired operating points on the trade-off curve between multiple objectives. It also works well for internet applications which serve large volumes of online traffic, by working with Lagrangian duality formulation to connect dual solutions (computed offline) with the primal solutions (computed online). We identify some key limitations of this approach -- namely the inability to handle user and item level constraints, scalability considerations and variance of dual estimates introduced by sampling processes. We propose solutions for each of the problems and demonstrate how through these solutions we significantly advance the state-of-the-art in this realm. Our proposed methods can exactly handle user and item (and other such local) constraints, achieve a $100\times$ scalability boost over existing packages in R and reduce variance of dual estimates by two orders of magnitude.

stat.AP↗

arXiv2015

Transformations and Hardy-Krause variation

Using a multivariable Faa di Bruno formula we give conditions on transformations $τ:[0,1]^m\to\mathcal{X}$ where $\mathcal{X}$ is a closed and bounded subset of $\mathbb{R}^d$ such that $f\circτ$ is of bounded variation in the sense of Hardy and Krause for all $f\in C^d(\mathcal{x})$. We give similar conditions for $f\circτ$ to be smooth enough for scrambled net sampling to attain $O(n^{-3/2+ε})$ accuracy. Some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition. Some other transformations due to Fang and Wang (1993) satisfy the first but not the second condition. We provide transformations for the simplex that makes $f\circτ$ smooth enough to fully benefit from scrambled net sampling for all $f$ in a class of generalized polynomials. We also find sufficient conditions for the Rosenblatt-Hlawka-Mück transformation in $\mathbb{R}^2$ and for importance sampling to be of bounded variation in the sense of Hardy and Krause.

stat.CO↗

arXiv2015

Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers $c, s \geq 1$, let $M_3(c, s)$ be the least integer such that any set of at least $M_3(c, s)$ points in the plane, no three on a line and colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. The case $s=0$, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that $M_3(1, 0)=3$, $M_3(2, 0)=9$ and $M_3(c, 0)=\infty$, for $c\geq 3$. In this paper we extend these results when $c \geq 2$ and $s \geq 1$. We prove that the least integer $λ_3(c)$ such that $M_3(c, λ_3(c))< \infty$ satisfies: $$\left\lfloor\frac{c-1}{2}\right\rfloor \leqλ_3(c)\leq c-2,$$ where $c \geq 2$. Moreover, the exact values of $M_3(c, s)$ are determined for small values of $c$ and $s$. We also conjecture that $λ_3(4)=1$, and verify it for sufficiently large Horton sets.

math.CO↗

arXiv2015

Quasi-Monte Carlo tractability of high dimensional integration over products of simplices

Quasi-Monte Carlo (QMC) methods for high dimensional integrals over unit cubes and products of spheres are well-studied in literature. We study QMC tractability of integrals of functions defined over the product of $m$ copies of the simplex $T^d \subset \mathbb{R}^{d}$. The domain is a tensor product of $m$ reproducing kernel Hilbert spaces defined by `weights' $γ_{m,j}$, for $j = 1,2, \ldots, m$. Similar to the results on the unit cube in $m$ dimensions, and the product of $m$ copies of the $d$-dimensional sphere, we prove that strong polynomial tractability holds iff $\limsup_{m \rightarrow \infty} \sum_{j=1}^m γ_{m,j} < \infty$ and polynomial tractability holds iff $\limsup_{m \rightarrow \infty} \frac{\sum_{j=1}^m γ_{m,j}}{\log(m + 1 )} < \infty$. We also show that weak tractability holds iff $\lim_{m \rightarrow \infty} \frac{\sum_{j=1}^m γ_{m,j}}{m} = 0$. The proofs employ Sobolev space techniques and weighted reproducing kernel Hilbert space techniques for the simplex and products of simplices as domain. Properties of orthogonal polynomials on a simplex are also used extensively.

math.NA↗

arXiv2014

Low discrepancy constructions in the triangle

Most quasi-Monte Carlo research focuses on sampling from the unit cube. Many problems, especially in computer graphics, are defined via quadrature over the unit triangle. Quasi-Monte Carlo methods for the triangle have been developed by Pillands and Cools (2005) and by Brandolini et al. (2013). This paper presents two QMC constructions in the triangle with a vanishing discrepancy. The first is a version of the van der Corput sequence customized to the unit triangle. It is an extensible digital construction that attains a discrepancy below 12/sqrt(N). The second construction rotates an integer lattice through an angle whose tangent is a quadratic irrational number. It attains a discrepancy of O(log(N)/N) which is the best possible rate. Previous work strongly indicated that such a discrepancy was possible, but no constructions were available. Scrambling the digits of the first construction improves its accuracy for integration of smooth functions. Both constructions also yield convergent estimates for integrands that are Riemann integrable on the triangle without requiring bounded variation.

math.NA↗

arXiv2012

A spatio-spectral hybridization for edge preservation and noisy image restoration via local parametric mixtures and Lagrangian relaxation

This paper investigates a fully unsupervised statistical method for edge preserving image restoration and compression using a spatial decomposition scheme. Smoothed maximum likelihood is used for local estimation of edge pixels from mixture parametric models of local templates. For the complementary smooth part the traditional L2-variational problem is solved in the Fourier domain with Thin Plate Spline (TPS) regularization. It is well known that naive Fourier compression of the whole image fails to restore a piece-wise smooth noisy image satisfactorily due to Gibbs phenomenon. Images are interpreted as relative frequency histograms of samples from bi-variate densities where the sample sizes might be unknown. The set of discontinuities is assumed to be completely unsupervised Lebesgue-null, compact subset of the plane in the continuous formulation of the problem. Proposed spatial decomposition uses a widely used topological concept, partition of unity. The decision on edge pixel neighborhoods are made based on the multiple testing procedure of Holms. Statistical summary of the final output is decomposed into two layers of information extraction, one for the subset of edge pixels and the other for the smooth region. Robustness is also demonstrated by applying the technique on noisy degradation of clean images.

stat.ME↗

arXiv2012

Spline Smoothing for Estimation of Circular Probability Distributions via Spectral Isomorphism and its Spatial Adaptation

Consider the problem when $X_1,X_2,..., X_n$ are distributed on a circle following an unknown distribution $F$ on $S^1$. In this article we have consider the absolute general set-up where the density can have local features such as discontinuities and edges. Furthermore, there can be outlying data which can follow some discrete distributions. The traditional Kernel Density Estimation methods fail to identify such local features in the data. Here we device a non-parametric density estimate on $S^1$, by the use of a novel technique which we term as Fourier Spline. We have also tried to identify and incorporate local features such as support, discontinuity or edges in the final density estimate. Several new results are proved in this regard. Simulation studies have also been performed to see how our methodology works. Finally a real life example is also shown.

stat.ME↗

arXiv2012

Tests for exponentiality against NBUE alternatives: a Monte Carlo comparison

Testing of various classes of life distributions has been addressed in the literature for more than 45 years. In this paper, we consider the problem of testing exponentiality (which essentially implies no ageing) against positive ageing which is captured by the fairly large class of new better than used in expectation (NBUE) distributions. These tests of exponentiality against NBUE alternatives are discussed and compared. The empirical size of the tests is obtained by simulations. Power comparisons for different popular alternatives are done using Monte Carlo simulations. These comparisons are made for both small and large sample sizes. The paper concludes with a discussion in which suggestions are made regarding the choices of the test when a particular alternative is suspected.

stat.CO↗

arXiv2012

The exact null distribution of the generalized Hollander-Proschan type test for NBUE alternatives

In this note we derive the exact null distribution for the test statistic proposed by Anis and Mitra (2011) for testing exponentiality against NBUE alternatives. As a special case, we obtain the exact null distribution for the test statistic proposed by Hollander and Proschan (1975). Selected critical values for different size are tabulated for these two statistics. Some remarks concerning the benefits of using the exact distribution are made.

math.ST↗