McGill Statistics Seminar

Date: 2019-01-25

Time: 16:00-17:00

Location: BURN 920

Abstract:

We have witnessed a lot of exciting development of data science in recent years. From the perspective of optimization, many modern data-science problems involve some basic ``non’’-properties that lack systematic treatment by the current approaches for the sake of the computation convenience. These non-properties include the coupling of the non-convexity, non-differentiability and non-determinism. In this talk, we present rigorous computational methods for solving two typical non-problems: the piecewise linear regression and the feed-forward deep neural network. The algorithmic framework is an integration of the first order non-convex majorization-minimization method and the second order non-smooth Newton methods. Numerical experiments demonstrate the effectiveness of our proposed approach. Contrary to existing methods for solving non-problems which provide at best very weak guarantees on the computed solutions obtained in practical implementation, our rigorous mathematical treatment aims to understand properties of these computed solutions with reference to both the empirical and the population risk minimizations.

Date: 2019-01-18

Time: 15:30-16:30

Location: BURN 1205

Abstract:

Non-singularity of the information matrix plays a key role in model identification and the asymptotic theory of statistics. For many statistical models, however, this condition seems virtually impossible to verify. An example of such models is a class of mixture models associated with multi-path change-point problems (MCP) which can model longitudinal data with change points. The MCP models are similar in nature to mixture-of-experts models in machine learning. The question then arises as to how often the non-singularity assumption of the information matrix fails to hold. We show that

Date: 2019-01-11

Time: 15:30-16:30

Location: BURN 1205

Abstract:

Cross-validation (CV) is perhaps the most widely used tool for tuning supervised machine learning algorithms in order to achieve better generalization error rate. In this paper, we focus on leave-one-out cross-validation (LOOCV) for the support vector machine (SVM) and related algorithms. We first address two wide-spreading misconceptions on LOOCV. We show that LOOCV, ten-fold, and five-fold CV are actually well-matched in estimating the generalization error, and the computation speed of LOOCV is not necessarily slower than that of ten-fold and five-fold CV. We further present a magic CV theory with a surprisingly simple recipe which allows users to very efficiently tune the SVM. We then apply the magic CV theory to demonstrate a straightforward way to prove the Bayes risk consistency of the SVM. We have implemented our algorithms in a publicly available R package magicsvm, which is much faster than the state-of-the-art SVM solvers. We demonstrate our methods on extensive simulations and benchmark examples.

Date: 2018-11-23

Time: 15:30-16:30

Location: BURN 1104

Abstract:

This is not a research talk. Rather, the goal is to address the topic of the talk title through a 2017 multi-authored paper published in Nature Human Behaviour. The Nature article proposes that the standard cut-off significance level of .05 should be replaced by a cut-off level of .005 when new discoveries are being claimed. The authors attribute the high proportion of irreducible results in the literature that accompany claimed new discoveries, in part, to the low-bar cut-off of .05. Their fix is built around the Bayes factor. I will begin with a brief presentation of the difference between the frequentist and Bayesian approaches to statistical inference, and lead into p-values vs Bayes factors for hypothesis testing before discussing the Nature article itself. It is hoped that the talk will provoke thought about the way we do statistics.

Date: 2018-11-16

Time: 15:30-16:30

Location: BURN 1104

Abstract:

The median residual lifetime function is a statistical quantity which describes the future point in time at which the probability of current survival has dropped by 50%. In deriving an estimator for the median residual lifetime function for length-biased data, the added features of left-truncation and right-censoring must be taken into account.

In this talk, we give a brief description of length-biased failure time data and show that by using a particular non-parametric estimator for the survival function that it is possible to derive the asymptotically most-efficient non-parametric estimator for the median residual lifetime function. We give some details on the proof of the asymptotic results and examine the performance of the estimator using simulated data. We also apply the proposed estimator to the Canadian Study of Health and Aging data set to study the median residual lifetime function of patients with dementia.

Date: 2018-11-09

Time: 15:30-16:30

Location: BURN 1104

Abstract:

Density estimation lies at the intersection of statistics, theoretical computer science, and machine learning. We review some old and new results on the sample complexities (also known as minimax convergence rates) of estimating densities of high-dimensional distributions, in particular mixtures of Gaussians and Ising models.

Based on joint work with Hassan Ashtiani, Shai Ben-David, Luc Devroye, Nick Harvey, Christopher Liaw, Yani Plan, and Tommy Reddad.

Date: 2018-11-02

Time: 15:30-16:30

Location: BURN 1104

Abstract:

The rigorous mathematical treatment of random fragmentation-coalescent models in the literature is difficult to find, and perhaps for good reason. We examine two different types of random fragmentation-coalescent models which produce somewhat unexpected results.

The first concerns an agent-based model in which, with a rate that depends on the configuration of the system, agents coalesce into clusters that also fragment into their individual constituent membership. We consider the large-scale, long-term behaviour of this system in a similar spirit to recent use of such models to characterise the evolution of terrorist cells. Under appropriate assumptions we find an unusual behaviour; the system displays stabilisation with clusters that only contain an odd number of individuals.

Date: 2018-10-26

Time: 15:30-16:30

Location: BURN 1104

Abstract:

In this talk, I will first briefly introduce my research on object oriented data analysis with application to neuroimaging studies. I will then talk about a detailed example on imaging genetics. In this project, we develop a high-dimensional matrix linear regression model to correlate 2D imaging responses with high-dimensional genetic covariates. We propose a fast and efficient screening procedure based on the spectral norm to deal with the case that the dimension of scalar covariates is much larger than the sample size. We develop an efficient estimation procedure based on the nuclear norm regularization, which explicitly borrows the matrix structure of coefficient matrices. We examine the finite-sample performance of our methods using simulations and a large-scale imaging genetic dataset from the Alzheimer’s Disease Neuroimaging Initiative study.

Date: 2018-10-19

Time: 15:30-16:30

Location: BURN 1104

Abstract:

Optimal transport plays an increasingly relevant and useful role in the theory and application of mixture model based clustering and inference. In this talk I will describe some recent progress in characterizing the convergence behavior of mixing distributions when one fits a mixture model to the data. This theory hinges on the relationship between the space of mixture densities, which is endowed with variational or Hellinger distance, and the space of mixing measures endowed with optimal transport distance metrics. Next, I will introduce an optimal transport based technique for the problem of multilevel clustering, which aims to simultaneously partition data in each group and discover grouping patterns among groups in a potentially large hierarchically structured corpus of data. Our method involves a joint optimization formulation over several spaces of discrete probability measures. We propose a number of variants of this problem, which admit fast optimization algorithms, by exploiting the connection to the problem of finding Wasserstein barycenters. Some theoretical and experimental results will be presented.

Date: 2018-10-05

Time: 15:30-16:30

Location: BURN 1104

Abstract:

In this talk, we discuss how sufficient dimension reduction can be used to aid causal inference. We propose a new matching approach based on the reduced covariates obtained from sufficient dimension reduction. Compared with the original covariates and the propensity scores, which are commonly used for matching in the literature, the reduced covariates are estimable nonparametrically and are effective in imputing the missing potential outcomes. Under the ignorability assumption, the consistency of the proposed approach requires a weaker common support condition than the one we often assume for propensity score-based methods. We develop asymptotic properties, and conduct simulation studies as well as real data analysis to illustrate the proposed approach.