Scientific

Generative models such as Generative Adversarial Nets (GANs), Variational Autoencoders and Normalizing Flows have been very successful in the unsupervised learning task of generating samples from a high-dimensional probability distribution. However, the task of conditioning a high-dimensional distribution from limited empirical samples has attracted less attention in the literature but it is a central problem in Bayesian inference and supervised learning. In this talk we will discuss some ideas in this direction by viewing generative modelling as a measure transport problem. In particular, we present a simple recipe using block-triangular maps and monotonicity constraints that enables standard models such as the original GAN to perform conditional sampling. We demonstrate the effectiveness of our method on various examples ranging from synthetic test sets to image in-painting and function space inference in porous medium flow.

The intrinsic infinite-dimensional nature of functional data creates a bottleneck in the application of traditional classifiers to functional settings. These classifiers are generally either unable to generalize to infinite dimensions or have poor performance due to the curse of dimensionality. To address this concern, we propose building a distance-weighted discrimination (DWD) classifier on scores obtained by projecting data onto one specific direction. We choose this direction by minimizing, over a reproducing kernel Hilbert space, an empirical risk function containing the DWD classifier loss function. Our proposed classifier avoids overfitting and enjoys the appealing properties of DWD classifiers. We further extend this framework to accommodate functional data classification problems where scalar covariates are involved. In contrast to previous work, we establish a non-asymptotic estimation error bound on the relative misclassification rate. Through simulation studies and a real-world application, we demonstrate that the proposed classifier performs favourably relative to other commonly used functional classifiers in terms of prediction accuracy in finite-sample settings.

In this presentation, we discuss the Variational AutoEncodeur (VAE): a latent variable model emerging from the machine learning community. To begin, we introduce the theoretical foundations of the model and its relationship with well-established statistical models. Then, we discuss how we used VAEs to solve two widely different problems. First, we tackled a classic statistical problem, survival analysis, and then a classic machine learning problems, image analysis and image generation. We conclude with a short discussion of our latest research project where we establish a new metric for the evaluation or regularization of latent variable models such a Gaussian Mixture Models and VAEs.​

We develop a general framework for identifying phase reduced equations for finite populations of coupled oscillators that is valid far beyond the weak coupling approximation. This strategy represents a general extension of the theory from [Wilson and Ermentrout, Phys. Rev. Lett 123, 164101 (2019)] and yields coupling functions that are valid to higher-order accuracy in the coupling strength for arbitrary types of coupling (e.g., diffusive, gap-junction, chemical synaptic). These coupling functions can be used to understand the behavior of potentially high-dimensional, nonlinear oscillators in terms of their phase differences. The proposed formulation accurately replicates nonlinear bifurcations that emerge as the coupling strength increases and is valid in regimes well beyond those that can be considered using classic weak coupling assumptions. We demonstrate the performance of our approach through two examples. First, we use diffusively coupled complex Ginzburg-Landau (CGL) model and demonstrate that our theory accurately predicts bifurcations far beyond the range of existing coupling theory. Second, we use a realistic conductance-based model of a thalamic neuron and show that our theory correctly predicts asymptotic phase differences for non-weak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak coupling theories can not.

Speaker Biography

Youngmin Park, Ph.D., is currently a PIMS Postdoc at the University of Manitoba under the supervision of Prof. Stéphanie Portet. He received his PhD in Mathematics from the University of Pittsburgh in 2018, where he applied dynamical systems methods to problems in neuroscience. His first postdoc involved auditory neuroscience research at the University of Pennsylvania in the Department of Otorhinolaryngology, before moving on to his next postdoc researching molecular motor dynamics in the Department of Mathematics at Brandeis University. He is now at Manitoba, continuing to apply dynamical systems methods to biological questions related to molecular motor transport and neural oscillators.

Large systems of interacting particles (or agents) are widely used to investigate self-organization and collective behavior. They frequently appear in modeling phenomena such as biological swarms, crowd dynamics, self-assembly of nanoparticles and opinion formation. Similar particle models are also used in metaheuristics, which provide empirically robust solutions to tackle hard optimization problems with fast algorithms. In this talk I will start with introducing some generic particle models and their underlying mean-field equations. Then we will focus on a specific particle model that belongs to the class of Consensus-Based Optimization (CBO) methods, and we show that it is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning.

Speaker Biography

Hui Huang, Ph.D., is currently a PIMS Postdoc at the University of Calgary under the supervision of Prof. Jinniao Qiu. Before moving to Calgary, he worked as a postdoctoral researcher in the Chair for Applied Numerical Analysis at the Technical University of Munich, Germany. Prior to being at TUM he was an Alan Mekler Postdoctoral Fellow in the Department of Mathematics at Simon Fraser University. In 2017, he received his PhD in Mathematics from Tsinghua University. His doctoral dissertation was conducted in consultations with Prof. Jian¬-Guo Liu from Duke University, where he studied as a joint PhD student from 2014 to 2016. His research has been focused on complex dynamical systems and their related kinetic equations.

Read more about Hui Huang on the PIMS Medium blog.