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.Comment: 34 pages, 8 figure

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.Comment: 29 Pages, 13 figure

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' $\gamma_{m,j}$, 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 $\limsup_{m \rightarrow \infty} \sum_{j=1}^m \gamma_{m,j} < \infty$ and polynomial tractability holds iff $\limsup_{m \rightarrow \infty} \frac{\sum_{j=1}^m \gamma_{m,j}}{\log(m + 1 )} < \infty$. We also show that weak tractability holds iff $\lim_{m \rightarrow \infty} \frac{\sum_{j=1}^m \gamma_{m,j}}{m} = 0$. 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.Comment: 20 Page

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.Comment: Accepted at NIPS 2017. arXiv admin note: substantial text overlap with arXiv:1602.0439

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.Comment: 10 pages, 2 figures, KDD'16 Submitted Versio

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.Comment: 12 Pages, 6 figure

A/B Testing in Dense Large-Scale Networks: Design and Inference

Design of experiments and estimation of treatment effects in large-scale networks, in the presence of strong interference, is a challenging and important problem. Most existing methods' performance deteriorates as the density of the network increases. In this paper, we present a novel strategy for accurately estimating the causal effects of a class of treatments in a dense large-scale network. First, we design an approximate randomized controlled experiment by solving an optimization problem to allocate treatments in the presence of competition among neighboring nodes. Then we apply an importance sampling adjustment to correct for any leftover bias (from the approximation) in estimating average treatment effects. We provide theoretical guarantees, verify robustness in a simulation study, and validate the scalability and usefulness of our procedure in a real-world experiment on a large social network.Comment: NeurIPS 202

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

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

Uniqueness of Galilean Conformal Electrodynamics and its Dynamical Structure

We investigate the existence of action for both the electric and magnetic sectors of Galilean Electrodynamics using Helmholtz conditions. We prove the existence of unique action in magnetic limit with the addition of a scalar field in the system. The check also implies the non existence of action in the electric sector of Galilean electrodynamics. Dirac constraint analysis of the theory reveals that there are no local degrees of freedom in the system. Further, the theory enjoys a reduced but an infinite dimensional subalgebra of Galilean conformal symmetry algebra as global symmetries. The full Galilean conformal algebra however is realized as canonical symmetries on the phase space. The corresponding algebra of Hamilton functions acquire a state dependent central charge.Comment: 27 pages, no figure