Seminars 2015

The canonical basis theory is initially constructed by Lusztig using perverse sheaves. This basis has many remarkable properties such as integrality and positivity of structure constants. It has a profound impact not only on Lie theory, but also on other areas, such as cluster algebras, quiver Hecke algebras and categorications of quantum groups. In this talk, I will provide a construction of the canonical basis for half parts of quantum super-algebras and two-parameter quantum algebras and their modified forms etc. I will also talk about positivity of the canonical basis of quantum algebras which is conjectured by Lusztig in 1990s. Furthermore, I will briefly review recent progress on the canonical basis theory and its relation to Schur-Weyl duality.

Theoretical results usually assume that the collected data follow a specific model without errors. However, the estimation error associated with the model, especially in a high-dimensional setting, poses practical concerns. In this talk, I will discuss different types of portfolio optimization problems with estimation errors, particularly for the high-dimensional setting that the number of asset is (much) larger than the number of observations. This talk will first reveal the degeneracies of an estimated optimal portfolio with the use of the traditional plug-in empirical mean and covariance matrix. To resolve such degeneracies, we propose a constrained l1 minimization approach to directly estimate the effective parameters appearing in the optimal portfolio solutions. The proposed approach works well even for the high-dimensional setting. Similar to the Dantzig Selector, the estimator is efficiently implemented with linear programming and the resulting portfolio is called the linear programming optimal (LPO) portfolio. Intensive empirical studies show that the LPO portfolio outperforms the equally weighted portfolio and the estimated optimal portfolios using shrinkage or other competitive estimators. The idea and the advantages of the LPO approach will be discussed in this talk. I will also address the connection between the LPO approach and the main results of my other works.

Remark: The related paper is awarded the First Prize of Best Student Research Paper in INFORMS Financial Section.

I will introduce a single-index model for the intensity of an inhomogeneous spatial point process, relating the intensity function to an unknown function ρ of a linear combination of a p-dimensional spatial covariate process. Such a model extends and generalizes the commonly used log-linear model. I will describe an estimating procedure for ρ and for the coefficient parameters β. Consistency and asymptotic normality of estimates of β can be achieved under some regularity assumptions.
I will show results from a simulation study showing the effectiveness of the procedure and from fitting the model to a dataset of fast food restaurant locations in New York City.

This paper is concerned with the problems of interaction screening and nonlinear classification in high-dimensional setting. We propose a two-step procedure, IIS-SQDA, where in the first step an innovated interaction screening (IIS) approach based on transforming the original $p$-dimensional feature vector is proposed, and in the second step a sparse quadratic discriminant analysis (SQDA) is proposed for further selecting important interactions and main effects and simultaneously conducting classification. Our IIS approach screens important interactions by examining only $p$ features instead of all two-way interactions of order $O(p^2)$. Our theory shows that the proposed method enjoys sure screening property in interaction selection in the high-dimensional setting of $p$ growing exponentially with the sample size. In the selection and classification step, we establish a sparse inequality on the estimated coefficient vector for QDA and prove that the classification error of our procedure can be upper-bounded by the oracle classification error plus some smaller order term. Extensive simulation studies and real data analysis show that our proposal compares favorably with existing methods in interaction selection and high-dimensional classification.

Deep learning is a neural network technique that gained great prominence in recent years for recognizing faces (Facebook), translating speech (Microsoft) and identifying cat videos (Google). Besides furnishing more intelligent tools, deep learning and neural networks also provide powerful strategies for designing efficient sensor networks -- the nervous system of smart cities. In the first half of the talk, I will give a brief introduction to deep learning, and explore some interesting ideas for overcoming challenges in sensor networks, such as missing data imputation, low-complexity data compression, unsupervised outlier detection and multimodal sensor fusion. Before deep learning, however, neural networks were largely unpopular due to issues with overfitting, local minima and hyperparameters. The main culprits? Singularities. Just as the classical laws of physics break down near a black hole, the classical laws of learning also break down near statistical singularities. Neural networks are highly singular statistical models, and Sumio Watanabe proved in 2000 (using Hironaka's theorem on the resolution of singularities) that the optimal conditions for learning are dictated by the structure of singularities in the model. In the second half of the talk, we discuss how deep learning seems to overcome these singularities, without explicit regularization but by exploiting sampling strategies such as contrastive divergence and minimum probability flow.

Local smoothing testing based on multivariate nonparametric regression estimation is one of the main model checking methodologies in the literature. However, the relevant tests suffer from typical curse of dimensionality, resulting in slow convergence rates to their limits under the null hypothesis and less deviation from the null hypothesis under alternative hypotheses. This problem prevents tests from maintaining the significance level well and makes tests less sensitive to alternative hypotheses. In this talk, a model-adaption concept in lack-of-fit testing is introduced and a dimension-reduction model-adaptive test procedure is proposed for parametric single-index models. The test behaves like a local smoothing test, as if the model was univariate. It is consistent against any global alternative hypothesis and can detect local alternative hypotheses distinct from the null hypothesis at a fast rate that existing local smoothing tests can achieve only when the model is univariate. Simulations are conducted to examine the performance of our methodology. An analysis of real data is shown for illustration. The method can be readily extended to global smoothing methodology and other testing problems.

Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study the statistical properties of directed network models. In this talk, we give for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary or non-binary weighted edges. We establish the uniform consistency and the asymptotic normality of the maximum likelihood estimator, when the number of parameters grows. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy.
Joint work with Chenlei Leng and Ji Zhu

In estimating the integrated volatility using high-frequency data, it is well documented that the presence of the microstructure noise causes big challenge. In this paper, as demonstrated by the motivational simulation study, the common feature of multiple observations brings an additional problem to the estimation of the integrated volatility. It becomes one more source of bias in addition to the microstructure noise. In this paper, we propose a multiplicity-adjusted and noise-corrected pre-averaging estimator which is proved to be consistent and have asymptotic normal distribution. Our approach is also easily extended to the case when the latent process has jumps. Extensive comparisons with empirical procedures in dealing with microstructure noise and/or multiple transactions show that our newly proposed estimator is superior over others. Yet surprisingly, in some cases, our estimator performs even better than the ideal estimator which assumes the transaction times within a single time stamp are observable. Simulation studies justify our theory and we also implement our estimator to some real data sets.

In this talk, we present an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. The proposed error indicators are L2-based and can be inexpensively computed which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.

Efficient processing of massive data sets has become increasingly important over the last two decades. The data streaming model, in particular, the turnstile streaming model, has attracted a lot of attention. In this model, an underlying vector x in R^n is presented as a long sequence of positive and negative updates to its coordinates. A randomized algorithm seeks to approximate a function f(x) with constant probability while only making a single pass over this sequence of updates and using a small amount of space. I shall give a brief introduction to the basic concepts and notions as well as typical approaches, then present the sparse recovery problem in greater detail (where f(x) returns the largest coordinates of x). In the end I shall discuss some future projects and directions.

We report some new results obtained jointly with F. Gotze. We obtain the non-improvable estimates of the rate of convergence of an expected spectral distribution function of Wigner random matrix to the semi-circular law under moment restrictions on the distributions of matrix entries. We prove as well the optimal (up to power of loga-rithmic factor) bounds of Lpnorm of Kolmogorov distance between an empirical spectral distribution function of Wigner random matrix and the semi-circular law under the same assumption about matrix entries distribution.

Cure models are special survival models that have wide applications in health research when evaluating survival times from subjects when some of them may be cured. There are numerous cure models developed to deal with complex data structures. However, there is little study on model diagnostic methods for cure models. In this talk, we will present some residual methods that we developed recently for mixture cure models. We demonstrate numerically that the proposed methods perform better than the standard model diagnostic methods for survival models in checking the fit of mixture cure models. The proposed methods are applied to some existing studies to examine the mixture cure models that were considered.

In the first part of this talk, I will recall some classical asymptotic aspects of the class group: in the CM situation, in Iwasawa theory, in tame extensions, etc.
In the second part, I will present a recent joint work with Hajir (UMASS- USA) concerning the behaviour of the exponent of the p-part of the class group in some unramified p-extensions.