2025 Winter

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.

Date: 2025-01-31

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

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/88929152266

Meeting ID: 889 2915 2266

Passcode: None

Abstract:

Generating realistic extremes from an observational dataset is crucial when trying to estimate the risks associated with the occurrence of future extremes, possibly of greater magnitude than those already observed. Generative approaches from the machine learning community are not applicable to extreme samples without careful adaptation. On the other hand, asymptotic results from extreme value theory provide a theoretical framework for modeling multivariate extreme events, through the notion of multivariate regular variation. Bridging these two fields, this presentation details a variational autoencoder approach for sampling multivariate distributions with heavy tails, i.e., distributions likely to exhibit extremes of particularly large intensities.

Date: 2025-01-17

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

Location: In person, Burnside 1104

https://mcgill.zoom.us/j/82903352833

Meeting ID: 829 0335 2833

Passcode: None

Abstract:

The analysis of multivariate count data is fundamental in various fields. An appropriate model must be able to be flexible enough for inducing correlation, but also simple for inference and interpretation. One such model is the Pólya Splitting model, which randomly decomposes the sum of a discrete vector into its components. This simple approach offers several compelling properties. However, it imposes the constraint that the dependency structure must be identical across all components. To overcome this limitation, a generalization of this model called Tree Pólya splitting is proposed. For this new model, the splitting process is represented by a tree structure, allowing for more flexibility. In this seminar, we will define the Tree Pólya Splitting model and explore various properties, including marginal distributions, factorial moments, and the dependency structure.