Past Seminars

We introduce a novel divergence based approach, called Bregman divergence, to model sparse high dimensional problems. We also introduce a new prior which induces a new version of the (approximate) adaptive lasso in a Bayesian framework. Unlike the original adaptive lasso in which the weights should be prespecified prior to the estimation, in our approach the coefficient estimates are directly used as the weights. In addition, due to the generality of the Bregman divergence, the proposed model is easily extended to generalized linear models as well as the group lasso.

Classicalmultivariatestatistical inferencemethods(Hotelling’sT2,multivariateanalysisofvariance,multivariateregression,etc.) arebasedonthe useofL2 criterionfunction: IfZ=(z1,...,zn)T is thematrixof p-variate residuals,thenintheestimationoneminimizesthecriterionfunction 1 n nX i=1 ||zi||2= 1 n nX i=1 zT izi (or nX i=1 ||zi||2 W= nX i=1 zT iW?1zi). InthistalkweconsideralternativeL1criterionfunctionswithrelatedtests andestimates. Theunivariateconceptsofsignandrankarebasedontheorderingofthe data.Unfortunately, inthemultivariatecasetherearenonaturalorderings of thedatapoints. Anapproachutilizingobjectiveorcriterionfunctions is thereforeoftenusedtoextendtheconceptsof signandranktothemultivariatecase.Themultivariatespatialsignui,multivariatespatial(centered) rankri,andmultivariatespatialsigned-rankqi,i=1,...,nmaybeimplicitly definedusingthethreeL1criterionfunctions 1 n nX i=1 ||zi|| = 1 n nX i=1 uT izi, 1 2n nX i=1 nX j=1 ||zi?zj|| = 1 n nX i=1 rT izi,and 1 4n nX i=1 nX j=1 (||zi?zj||+||zi+zj||) = 1 n nX i=1 qT izi. Notealsothatthesign, centeredrank,andsigned-rankmaybeseenas(location) scoresTi =T(zi) correspondingto the threeobjective functions. (T(z)=zisthescorefunctionfortheregularL2 criterion.)Notethatthe firstobjectivefunction,themeandeviationoftheresiduals,isthebasisforthe socalledleastabsolutedeviation(LAD)methods; ityieldsdifferentmediantypeestimatesandsigntests intheone-sample, two-sample, c-sampleand 1f inally general linear model settings. The second objective function is the mean difference of the residuals. The second and third objective functions generate Hodges-Lehmann type estimates and rank and signed-rank tests for different location problems. In the talk, we review the theory of the multivariate spatial sign and rank methods, tests and estimates, in the one sample, several samples and, finally, multivariate linear regression cases. See M¨ott¨onen and Oja (1995), Oja and Randles (2004), Oja (2010) and Nordhausen and Oja (2011). Transformationretransformation technique is used to obtain affine invariant tests and equivariant estimates. Limiting distribution of the tests and estimates are given and their asymptotical efficiencies are discussed and compared. The theory is illustrated with examples.

In this talk, unsupervised and supervised linear dimension reduction methods based on various uses of scatter matrices are addressed. In our approach, dimension reduction is done by preprocessing the data, that is, by transforming the data to an invariant coordinate system (ICS). The unsupervised ICS functionals in Tyler et al. (2009) are based on the use of two unsupervised scatter matrix functionals while the supervised functionals such as the sliced inverse regression (SIR) utilize both unsupervised and supervised scatter matrices. See Liski et al. (2013) and Ilmonen et al. (2012). The use of several scatter functionals is discussed as well. In the so called Blind Source Separation (BSS) model, it is assumed that the observed p-variate observations are all linear combinations of p latent variables. This may be seen also as a dimension reduction problem if only few of the latent variables are relevant in data analysis. The aim is then to find an estimate for an unmixing matrix, which transforms the observed variables back to the latent variables. The most popular BSS approach is independent component analysis (ICA), and the so called AMUSE and SOBI solutions are used to find latent stationary time series in the analysis of multivariate time series. Unmixing matrix estimates are often found as ICS or SICS solutions. See e.g. Miettinen et al. (2014). Typically, dimension reduction procedures proposed in the literature are highly non-robust as the first step in the algorithms is to standardize the observations using the regular covariance matrix. In this talk, we also briefly discuss the possibility to robustify these procedures. The talk is based on joint work with several coauthors.

Data augmentation (DA) algorithms often suffer from slow convergence. The sandwich algorithm is an alternative to the DA algorithm that uses an extra simulation step at each iteration. Empirical evidence suggests that sandwich algorithm converges faster than the DA algorithm. I will present two theoretical results comparing the DA and sandwich algorithms. As an example, I will consider a widely used DA algorithm and its sandwich variant for the exploration of posterior densities associated with Bayesian mixture models.

Confidentiality and Authentication are two of the most important goals of cryptography. Confidentiality is achieved through encryption schemes whereas message authentication is achieved through signature schemes in public key setting. Simple patching of encryption and signature schemes may not provide confidentiality and authentication both. For example, "encrypt & sign" and "encrypt then sign" do not give desired security but "sign then encrypt" provides so. In the year 1997, Zheng introduced the notion of signcryption scheme whose motivation was to provide confidentiality and authentication at the same time with lower computational cost and lower communication overhead than "Sign then Encrypt" approach. In this talk , we shall discuss some existing paradigms to construct signcryption schemes and then investigate their security in different security models.

We study minimization problems with respect to a one-parameter family of generalized divergences (denoted I (PQ)). These I-divergences are a generalization of the so-called KullbackLeibler divergence (KL-divergence). Just like KL-divergence, these I-divergences also behave like squared Euclidean distance and satisfy the Pythagorean property. This talk is about the usefulness of these geometric properties in robust statistics. The talk is organized in three parts. In the rst part, we study minimization of I (PQ) as the rst argument varies over a family of probability distributions that satisfy linear statistical constraints. Such a constraint set is called a linear family. This minimization problem generalizes the maximum Renyi or Tsallis entropy principle of statistical physics. The structure of the minimizing probability distribution naturally suggests a statistical model of power-law probability distributions, which we call an-power-law family. This is analogous to the exponential family that arises when relative entropy is minimized subject to the same linear statistical constraints. In the second part, we study minimization of I (PQ) over the second argument. This minimization is generally on parametric families such as the exponential family or the-power-law family, and is of interest in robust statistical estimation. In the third part, we show an orthogonality relationship between an-power-law family and an associated linear family. As a consequence of this, the minimization of I over the second argument on an-power-law family can be shown to be equivalent to a minimization of I over the rst argument on a linear family. The latter turns out to be a simpler problem of minimization of a quasi-convex objective function subject to linear constraints.

There is a need to model asymmetric or skewed spatial process with applications in finance, economics, hydrology and ecology. To this aim, we introduce new classes of stationary spatial processes with asymmetric, sub-Gaussian marginal distributions using the idea of expectiles. This is related to the concept of a Double Normal distribution and an Asymmetric Gaussian distribution. We derive theoretical properties of the proposed processes. Moreover, we use the proposed spatial processes to formulate a spatial regression model for point-referenced data where the spatially correlated errors have skewed marginal distribution. We introduce a Bayesian computational procedure for model fitting and inference for this class of spatial regression models. We compare the performance of the proposed method with the traditional Gaussian process-based spatial regression through simulation studies and by applying it to a data set on air pollution in California.

We present the rst deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph G = (VE), with V = n and E=m, in o( m) time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a (2 + ) approximation in O(logn 2) amortized time per update. For maximum matching, we show how to maintain a (3+ ) approximation in O(min( n m1 3 2)) amortized time per update, and a (4+ ) approximation in O(m1 3 2) worst-case time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld from STOC2010. Joint work with Monika Henzinger and Guiseppe F. Italiano. Appeared in ACM-SIAM Symposium on Discrete Algorithms (SODA) 2015.

In this talk our aim is to present some necessary and su cient conditions for the starlikeness of Bessel functions of the rst kind and their derivatives of the rst, second and third order by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some Mittag-Le er expansions for the derivatives of Bessel functions of the rst kind, as well as some results on the zeros of these functions. Our aim is also to present necessary and su cient conditions for the close-to-convexity of some special combinations of Bessel functions of the rst kind by using some newly discovered Mittag-Le er expansions for Bessel functions of the rst kind; as well as a necessary and su cient condition for the close-to-convexity of a cross product of Bessel and modi ed Bessel functions of the rst kind, and their derivatives by using the newly discovered power series and innite product representation of this cross-product, and a slightly modi ed version of a result of Lorch on the monotonicity of the zeros of the cross product with respect to the order.

Self-similar curves played important role in the development of fractal geometry since the times of H.von Koch, J.Peano, V. Sierpinsky and P. Levy. The author will discuss most known examples of self-similar curves, geometric properties of self-similar curves such as their rigidity and projection properties, nite presentability, their deformations. The talk will contain last results about self-similar dendrites and self-similar Jordan arcs in space.

We consider two-server queues as an example of multi-server queues and attempt to understand how large delays happen in steady-state. In particular, we focus on the case where the incoming job sizes are heavy-tailed. We shall discuss the tail asymptotics in various regimes. Speci cally, we shall discuss in detail our work on the half-loaded regime where the arrival rate is equal to the service rate, in which we make the following interesting observations: When the incoming jobs have nite variance, there are basically two types of e ects that dominate the tail asymptotics. While the quantitative distinction between these two manifests itself only in the slowly varying components, the two e ects arise from qualitatively very di erent phenomena (arrival of one extremely big job (or) two big jobs). Then there is a phase transition that occurs when the incoming jobs have innite variance. In that case, only one of these e ects dominate the tail asymptotics, the one involving arrival of one extremely big job.
The talk assumes only basic probability and not any prior knowledge on queuing systems.

In any environment, the main objective is to provide a set of best alternatives for given criteria. The decision maker provides some necessary and basic information about each criterion and the alternatives that helps in identifying the relation between them. The problems of this kind can be dealt with Multi Criteria Decision Making or Multi Criteria Decision Aid (MCDA) techniques. The main aim of MCDA is to account for several views and provide some tools for the Decision Maker (DM) in solving complex decision problems. Here, the presentation focuses on various MCDA techniques like ELECTRE TRI method, Goal Programming and Analytical Hierarchy Process (AHP) along with this software oriented approach is also addressed.

Brodskii and Milman proved that there is a point in C(K), the set of all Chebyshev center of K, which is fixed by every surjective isometry from K into K whenever K is a nonempty weakly compact convex subset having normal structure in a Banach space. Motivated by this result, Lim et al. raised the following question namely "does there exist a point in C(K) which is fixed by every isometry from K into K?". In fact, Lim et al. proved that "if K is a nonempty weakly compact convex subset of a uniformly convex Banach space, then the Chebyshev center of K is fixed by every isometry T from K into K". In this talk, we discuss some partial answers to the aforementioned question.

This will be a couple of expository talks for the summer students at ISI-Chennai. The famous Jordan curve theorem states that every simple closed curve in the plane separates the plane into two components: a bounded "interior" component and an unbounded "exterior" component. Though this theorem seems intuitively obvious at first sight, there are subtle topological issues. We shall first prove this theorem using only rudimentary graph theory. We shall then present Schoenflies's generalization of the Jordan curve theorem that says that the two components determined by a simple closed curve are in fact homeomorphic to the interior and exterior of the unit-circle.
We will present some applications of the Jordan--Schoenflies to the classification surfaces. We will prove that every surface admits a triangulation which is a key step in the classification of surfaces.

The availability of big and high dimensional data poses new challenges to the arena of statistics. It is well established that methods of classical statistics are no more adequate to deal with these kinds of data. For couple of decades statisticians all over the world tried to come up with pertinent and applicable statistical methods for analysis of such data. In this presentation I will discuss some of the salient tools to analyse high dimensional data effectively.

Using a panel of establishments from the Annual Survey of Industries (ASI), I study the impact of the 1991 trade liberalization episode in India on the employment share of women. Contrary to the predictions of a taste-based discrimination model, I find that establishments exposed to larger output tariff reductions and import competition reduced the share of female workers. I also find that input tariff reductions neither raised nor reduced female employment share. The negative association between output tariff reductions and female employment appears to be driven by two factors. First, establishments facing larger output tariff declines engaged in more skill-upgrading which worked against women (who are less skilled in terms of measured education). Second, establishments facing larger tariff declines increased the number of shifts per worker. Since women in India are prohibited by law from working long hours and night shifts, this hours-constraint appears to have reduced relative employment of women. I find this effect to be particularly large among ``big and private'' establishments.

It is known from earlier result of Gordon-Jaksic-Molchanov-Simon, that spectrum of the Random Schrodinger Operators with Unbounded Potentials is pure point. Recently we obtain the eigenvalue statistics for this model and is turn out that the statistics is Poisson. Also if time permit I want to give an short overview about random Schrodinger operators (Anderson Model).
This is a joint work with Anish Mallick.

We study the set of non-Lebesgue points for Sobolev functions defined on metric measure spaces. In this talk, I will first give some definitions and preliminary results. I will discuss the relation between capacity and Hausdorff measure of a set in R^n and also in metric measure spaces. Then I will mention the results about Lebesgue points for functions in Sobolev space N^{1,p}(X), where X is a metric measure space and present the ''improvement'' we have got in the case when p equals the ''dimension'' of the space X. I will also talk about generalized Lebesgue points for functions in Hajlasz-Sobolev spaces M^{1,p}(X), 0

Subordinated stochastic processes X(T(t)) are obtained by time changing a parent process X(t) with a positive non-decreasing stochastic process T(t). The process X(T(t)) is said to be subordinated to the process X(t) and the process T(t) is called the directing process. Subordinated processes demonstrate interesting probabilistic properties and have applications in f inance, economics, statistical physics and fractional calculus. The aim of this talk is to discuss the concept of subordinated processes and to explore the applications of these processes in different fields.