McGill Statistics Seminar

Date: 2025-10-03

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/82469112499

Meeting ID: 824 6911 2499

Passcode: None

Abstract:

Neural networks can be trained to classify images and achieve high levels of accuracy. However, researchers have discovered that well-targeted perturbations of an image can completely fool a trained classifier, even in cases where the modified image is visually indistinguishable from the original. This has sparked many new approaches to classification which include an adversary in the training process: such an adversary can improve robustness and generalization properties at the cost of decreased accuracy and increased training time. In this presentation, I will explore the connection between a certain class of adversarial training problems and the Bayes classification problem for binary classification. In particular, robustness can be encouraged by adding a regularizing nonlocal perimeter term, providing a strong connection to classical studies of perimeter. Borrowing tools from geometric measure theory, I will show the Hausdorff convergence of adversarially robust classifiers to Bayes classifiers as the strength of adversary decreases to 0. In this way, the theoretical results discussed in the presentation provide a rigorous comparison with the standard Bayes classification problem.

Date: 2025-09-26

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/81200178578

Meeting ID: 812 0017 8578

Passcode: None

Abstract:

In many observational studies, researchers are often interested in studying the effects of multiple exposures on a single outcome. Standard approaches for high-dimensional data such as the lasso assume the associations between the exposures and the outcome are sparse. These methods, however, do not estimate the causal effects in the presence of unmeasured confounding. In this paper, we consider an alternative approach that assumes the causal effects in view are sparse. We show that with sparse causation, the causal effects are identifiable even with unmeasured confounding. At the core of our proposal is a novel device, called the synthetic instrument, that in contrast to standard instrumental variables, can be constructed using the observed exposures directly. We show that under linear structural equation models, the problem of causal effect estimation can be formulated as an l0-penalization problem and hence can be solved efficiently using off-the-shelf software. Simulations show that our approach outperforms state-of-art methods in both low-dimensional and high-dimensional settings. We further illustrate our method using a mouse obesity dataset.

Date: 2025-09-19

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/83914219181

Meeting ID: 839 1421 9181

Passcode: None

Abstract:

Vintage factor analysis is one important type of factor analysis that aims to first find a low-dimensional representation of the original data, and then to seek a rotation such that the rotated low-dimensional representation is scientifically meaningful. The most widely used vintage factor analysis is the Principal Component Analysis (PCA) followed by the varimax rotation. Despite its popularity, little theoretical guarantee can be provided to date mainly because varimax rotation requires to solve a non-convex optimization over the set of orthogonal matrices.

Date: 2025-09-12

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/88021402798

Meeting ID: 880 2140 2798

Passcode: None

Abstract:

This work develops an intuitive concept of perfect dependence between two variables of which at least one has a nominal scale that is attainable for all marginal distributions and proposes a set of dependence measures that are 1 if and only if this perfect dependence is satisfied. The advantages of these dependence measures relative to classical dependence measures like contingency coefficients, Goodman-Kruskal’s lambda and tau and the so-called uncertainty coefficient are twofold. Firstly, they are defined if one of the variables is real-valued and exhibits continuities. Secondly, they satisfy the property of attainability. That is, they can take all values in the interval [0,1] irrespective of the marginals involved. Both properties are not shared by the classical dependence measures which need two discrete marginal distributions and can in some situations yield values close to 0 even though the dependence is strong or even perfect. Additionally, this work provide a consistent estimator for one of the new dependence measures together with its asymptotic distribution under independence as well as in the general case. This allows to construct confidence intervals and an independence test, whose finite sample performance is subsequently examine in a simulation study. Finally, we illustrate the use of the new dependence measure in two applications on the dependence between the variables country and income or country and religion, respectively.

Date: 2025-05-23

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/89626299031

Meeting ID: 896 2629 9031

Passcode: None

Abstract:

Stationary models from the GARCH class have proved to be extremely useful for forecasting volatility and measuring risk in financial time series. However, the nature of their implied copulas is opaque.

We analyse the serial dependence structure of first-order GARCH-type models in terms of the implied bivariate copulas that describe the dependence and partial dependence of pairs of variables at different lags. Our aim is to understand whether such dependence structures could be approximated with appropriately chosen bivariate copulas arranged in D-vines.

Date: 2025-04-04

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/81100654212

Meeting ID: 811 0065 4212

Passcode: None

Abstract:

We study the effect of normalization on the layers of deep neural networks. A given layer $i$ with $N_{i}$ hidden units is normalized by $1/N_{i}^{\gamma_{i}}$ with $\gamma_{i}\in[1/2,1]$. We study the effect of the choice of the $\gamma_{i}$ on the statistical behavior of the neural network’s output (such as variance) as well as on the test accuracy and generalization properties of the architecture. We find that in terms of variance of the neural network’s output and test accuracy the best choice is to choose the $\gamma_{i}$’s to be equal to one, which is the mean-field scaling. We also find that this is particularly true for the outer layer, in that the neural network’s behavior is more sensitive in the scaling of the outer layer as opposed to the scaling of the inner layers. The mechanism for the mathematical analysis is an asymptotic expansion for the neural network’s output. An important practical consequence of the analysis is that it provides a systematic and mathematically informed way to choose the learning rate hyperparameters. Such a choice guarantees that the neural network behaves in a statistically robust way as the number of hidden units $N_i$ grow. Time permitting, I will discuss applications of these ideas to design of deep learning algorithms for scientific problems including solving high dimensional partial differential equations (PDEs), closure of PDE models and reinforcement learning with applications to financial engineering, turbulence and more.

Date: 2025-03-28

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/85849766730

Meeting ID: 858 4976 6730

Passcode: None

Abstract:

I will provide a personalized account of a sequence of problems, that I have worked on over the years, beginning with string counts in Bernoulli sequences and transiting to multivariate discrete models. As a starting point, we consider independent Bernoulli trials with varying success probabilities 1/k for the kth trial, the sum of the products of two consecutive occurrences,  and  the problem of establishing that the sum is distributed Poisson with mean equal to 1.  We will explain how this finding connects to cycles in random permutations, records for continuous random variables, the Hoppe-Polya urn, and the classical Montmort matching problem.

Date: 2025-03-14

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/88555780651

Meeting ID: 885 5578 0651

Passcode: None

Abstract:

We present a framework for solving linear inverse problems that is computationally tractable and has mathematical certificates. To this end, we interpret the ground truth of a linear inverse problem as a random vector with unknown distribution. We solve for a distribution which is close to a prior P (guessed or data-driven) measured in the KL-divergence while also having an expectation that yields high fidelity with the given data that defines the problem. After reformulation this yields a strictly convex, finite dimensional optimization problem whose regularizer, the MEM functional, is paired in duality with the log-moment generating function of the prior P. We exploit this computationally via Fenchel-Rockafellar duality. When no obvious guess for P is available, we use data to generate an empirical prior. Using techniques from variational analysis and stochastic optimization, we show that, and at what rate, the solution of the empirical problems converge (as the sample size grows) to the solution of the problem with known prior.

Date: 2025-02-28

Time: 15:30-16:30 (Montreal time)

Location: Online, retransmitted in Burnside 1104

https://mcgill.zoom.us/j/89838224036

Meeting ID: 898 3822 4036

Passcode: None

Abstract:

Artificial intelligence is arguably the hottest topic in science. Computer science and engineering currently set the agenda in this field, sidelining mathematics to a large extent. This talk, however, will highlight that mathematics has a lot to offer. We will introduce mathematical guarantees that provide deep insights into the inner workings of AI, and we will show how statistical principles can make AI more efficient. More generally, we will discuss the role of mathematics, especially statistics, in AI and data science.

Date: 2025-02-07

Time: 15:30-16:30 (Montreal time)

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/81032144286

Meeting ID: 810 3214 4286

Passcode: None

Abstract:

The copula of a random vector with unknown marginals can be estimated non-parametrically by the empirical copula, akin to the empirical distribution. However, the asymptotic analysis of the empirical copula is made considerably more involved than that of the empirical distribution by the use of pseudo-observations, involving the marginal empirical distribution functions. In particular, it is still unknown whether the empirical copula evaluated at a non-rectangular set is asymptotically normally distributed. In this work, sufficient conditions under which this is the case are identified. The result is extended to a Donsker theorem for the empirical copula indexed by an infinite collection of non-rectangular sets. Some aspects of the proof involving geometric measure theory will be discussed. Based on ongoing joint work with Axel Bücher, Johan Segers and Stanislav Volgushev.