Date/Time : 06 Jan 2023 14:00
Venue : S17 #05-12 (SR4)
Speaker : Tanya Veeravalli, University of Illinios, Urbana-Champaign

There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include “infinitely wide” neural nets, where the underlying nonlinearity is given by the activation function of an individual neuron. In this talk, we will bring this perspective to function representation by neural stochastic differential equations (SDEs). A neural SDE is an Itô diffusion process whose drift and diffusion matrix are elements of some parametric families. We will show that the ability of a neural SDE to realize nonlinear functions of its initial condition can be related to the problem of optimally steering a certain deterministic dynamical system between two given points in finite time. This auxiliary system is obtained by formally replacing the Brownian motion in the SDE by a deterministic control input. We derive upper and lower bounds on the minimum control effort needed to accomplish this steering; these bounds may be of independent interest in the context of motion planning and deterministic optimal control.

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Date/Time : 04 Jan 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Asbjorn Nordentoft, Université Sorbonne Paris Nord

Duke, Imamoḡlu, and Tóth have recently constructed a canonical hyperbolic orbifold whose boundary is equal to a given closed geodesic on the modular surface. Furthermore, they have shown that the projection of these hyperbolic orbifolds onto the modular surface equidistributes (as the discriminant of the closed geodesic tends to infinity). We will describe an extension to the case of modular surfaces of non-trivial genus relying on Morse coding with respect to special fundamental polygons as introduced by Kulkarni. We also obtain an equidistribution result which requires a detailed study of the homology of the modular curves. This is joint work with Peter Humphries.

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Date/Time : 22 Dec 2022 10:00
Venue : S17 #05-12 (SR4)
Speaker : Yanni Sun, City University of Hong Kong

Known as the blueprint of life, the genomic sequence contains instructions for controlling a species’ growth, development, survival, and reproduction. From the perspective of Computer Science, the primary structure of the genomic sequences is simply a string/sequence defined on a small alphabet {A, C, G, T}. Different parts of the sequence have different biological functions and thus comprise the “language of life”.

Next-generation sequencing (NGS) technologies, which produce vast amount of sequencing data for various life forms, have provided tremendous information for tackling grand challenges in various fields. NGS enables us to obtain a large amount of data to understand more about the language of life. In this talk, I will focus on my research on using deep learning models to make sense out of the BIG microbial sequencing data obtained from host-associated (such as human gut, skin) or environmental samples (such as water, soil). By using protein clusters as the tokens (“words”), we model viral genomes as a protein-based language and thus use modern language model Transformer for feature learning. In addition, to address the issues of limited labeled biological data, we use graph convolutional network (GCN), a semi-supervised learning model for leveraging both labeled and unlabeled data for feature learning in several tasks. These research projects rendered us with deeper understanding of the advantages and limitations of using deep learning models for biological sequence classification. I will share some of our observations along this line.

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Date/Time : 15 Dec 2022 10:00
Venue : S17 #04-04 (SR3)
Speaker : Ming Ha Quang, RIKEN AIP (Advanced Intelligence Project), Tokyo, Japan

Information geometry (IG) and Optimal transport (OT) have been attracting much research attention in various fields, in particular machine learning and statistics. In this talk, we present results on the generalization of IG and OT distances for finite-dimensional Gaussian measures to the setting of infinite-dimensional Gaussian measures and Gaussian processes. Our focus is on the Entropic Regularization of the 2-Wasserstein distance and the generalization of the Fisher-Rao distance and related quantities. In both settings, regularization leads to many desirable theoretical properties, including in particular dimension-independent convergence and sample complexity. All of the presented formulations admit closed form expressions that can be efficiently computed and applied practically.

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Date/Time : 05 Dec 2022 10:00
Venue : S17 #04-06 (SR1)
Speaker : Soufiane Hayou, National University of Singapore

Neural networks have achieved impressive performance in many applications such as image recognition and generation, and speech recognition. State‐of‐the‐art performance is usually achieved via a series of engineered modifications to existing neural architectures and their training procedures. However, a common feature of these systems is their large‐scale nature. Indeed, modern neural networks usually contain millions ‐ if not billions ‐ of trainable parameters, and empirical evaluations (generally) support the claim that increasing the scale of neural networks (e.g. width and depth in a multi‐layer perceptron) enhances the model performance. However, given a neural network model, it is not straightforward to address the crucial question `how do we scale the network?’. In this talk, I will discuss certain properties of large‐scale neural networks and show how we can leverage different mathematical results to build robust networks with empirically confirmed benefits.

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Date/Time : 01 Dec 2022 16:00
Venue : Zoom
Speaker : Daren Wei, Hebrew University of Jerusalem

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https://nus-sg.zoom.us/j/82637054452?pwd=ZkpEVllwN3g1YUNYRHIwMXF3TmUzdz09

Meeting ID: 826 3705 4452
Passcode: 332211

Unipotent flows have some striking rigidity properties: for instance, every orbit is recurrent, and every orbit closure as well as every invariant measure is homogeneous. In particular, the isomorphism rigidity theorem tells us that a measurable isomorphism between two flows in the class of unipotent flows on quotients of semisimple groups implies an algebraic isomorphism between their corresponding groups and lattices. Moreover, such systems always have zero entropy and thus they cannot be distinguished by classical entropy invariants.

We extend the isomorphism rigidity for unipotent flows to its time changes. More precisely, one parameter unipotent flows on quotients of semisimple groups fall into two categories: 1. unipotent flows that are time changes of linear irrational flows on T^2 and hence are all time changes of each other; 2. The measurable isomorphism between their time changes implies the much stronger (algebraic) equivalence as above. We also show that the complexity of time changes of unipotent flows can be described explicitly in terms of the corresponding adjoint action, and associated Jordan block‐like structures. Moreover, this is also true for the complexity of high rank abelian unipotent actions.

This is based on joint works with Adam Kanigowski, Philipp Kunde, Elon Lindenstrauss and Kurt Vinhage.

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Date/Time : 01 Dec 2022 10:00
Venue : Zoom
Speaker : Yusheng Luo, Stony Brook University

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https://nus-sg.zoom.us/j/82637054452?pwd=ZkpEVllwN3g1YUNYRHIwMXF3TmUzdz09

Meeting ID: 826 3705 4452
Passcode: 332211

The Mandelbrot set is a fundamental object in the field of complex dynamics. It has been thoroughly studied with much success, and we know a lot about its combinatorial, topological and geometrical structures. The natural generalization of the Mandelbrot set in higher degree is the connectedness locus. It is much more challenging to understand their structures as they are of higher dimensions. Very few of the one‐dimensional tools that we use to study the Mandelbrot set carry over to this general setting.

In this talk, I will start with a gentle introduction of the Mandelbrot and its higher dimensional analogues. I will then discuss some tools that I developed to navigate these higher dimensional Mandelbrot sets. In particular, I will explain how self‐bumps can be constructed and how these tools allow us to understand polynomials with zero core entropy. I will also discuss many exciting new directions to explore motivated by the parallel theories for Teichmuller spaces.

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Date/Time : 30 Nov 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ji Hui, National University of Singapore

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https://nus-sg.zoom.us/j/85864142504?pwd=aUUyaFRETE5hYit6WlYyMGhhdDZYUT09

Meeting ID: 858 6414 2504
Passcode: 764936

Imaging has been vital in advancing science, medicine, engineering, and industrial applications. Today’s imaging systems increasingly rely on computing power and algorithms to indirectly form images (visual interpretation) from only partial and noisy measurements. Most image reconstruction tasks are about solving a challenging linear or non-linear ill-posed inverse problem. For the past few decades, we have witnessed phenomenal developments in mathematical theories, models, and algorithms for solving inverse imaging problems. This talk will start by reviewing our works on prevailing model-driven mathematical approaches in computational imaging with manually-crafted model/transforms, including wavelets and sparsity-based regularization. Then we will describe how to advance these studies into the era of data-driven approaches that directly learn intrinsic structures from image data. Finally, we will discuss the connection between these data-driven models/algorithms and deep convolutional neural networks, which led to further advances in self-supervised deep learning for computational imaging.

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Date/Time : 25 Nov 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Slawomir Solecki, Cornell University

One of the areas of interest of Descriptive Set Theory is dynamics of Polish groups, that is, groups carrying a group topology that is separable and completely metrizable. Such groups, like for example, the unitary group of the infinite dimensional Hilbert space or the homeomorphism group of the unit interval, are not, in general, locally compact. Therefore, in studying their dynamics, classical methods relying on Haar measure are not available. They can often be replaced by descriptive set theoretic tools.

I will describe the descriptive set theoretic background and indicate how this point of view lead to a recent answer to an old question in Ergodic Theory. The question lies within a long-established theme, going back to the work of Halmos and Rokhlin, of investigating generic measure preserving transformations.

To be more specific: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$—the group of all Lebesgue measurable functions from $[0,1]$ to the circle. Glasner and Weiss asked if this was the case. I will sketch a negative answer to this question that rests on an analysis of unitary representations of the non-locally compact Polish group $L^0$.

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