Date/Time : 03 Mar 2021 10:00
Venue : via Zoom
Speaker : Xiuyuan Cheng, Duke University

Join Zoom Meeting https://nus-sg.zoom.us/j/82863583302?pwd=eHlCQkg3UWs5NDNwNXZXUytQSkNUQT09&from=addon Meeting ID: 828 6358 3302 Password: 860921 Abstract:  The recent success of generative adversarial network (GAN) and variational learning suggests that training a classification network may work well in differentiating two distributions from data samples, known as the classical two-sample problem. An important consideration here is the geometric properties of data distributions. In this talk, we focus on the situation where data samples lie on or near to low-dimensional manifolds embedded in a possibly high-dimensional space. After reviewing some background knowledge of kernel-based two-sample methods, especially the kernel Maximum Mean Discrepancy (MMD) test, we introduce a neural network test using the logit function of a trained classifier motivated by the D-net in GAN. The classification logit function provides the “witness function” under the MMD framework, namely the test function indicating where the two distributions differ and thus is of central application interest. Theoretically, with Lipschitz regularization of the network functions, both the approximation and estimation error of the neural network method are proved to only scale with the intrinsic dimensionality of the data. We discuss applications to biomedical datasets, particularly the differential abundance analysis of single-cell RNA sequencing data. Joint work with Alexander Cloninger (UCSD) and Kluger’s lab (Yale School of Medicine).

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Date/Time : 17 Feb 2021 10:00
Venue : via Zoom
Speaker : Zhizhen Zhao, UIUC

Join Zoom Meeting https://nus-sg.zoom.us/j/82194201907?pwd=aDJnUTlqM0lXaXl0ZW5ORUsvTDE0QT09 Meeting ID: 821 9420 1907 Password: 953495 Abstract: In this talk, I will introduce a new unsupervised learning framework for data points that lie on or close to a smooth manifold naturally equipped with a group action. In many applications, such as cryo-electron microscopy image analysis and shape analysis, the dataset of interest consists of images or shapes of potentially high spatial resolution, and admits a natural group action that needs to be quotient out before useful information is revealed. For such datasets, we can evaluate group-invariant distance and optimal alignment between data points. From noisy observations and pairwise relations, our goal is to robustly learn the nonlinear geometrical structure of the data and improve the nearest neighbor identification. In the first part, I will focus on a specific application–cryo-electron microscopy single particle class averaging. Cryo-electron microscopy single particle reconstruction is an entirely general technique for 3D structure determination of macromolecular complexes. However, because the images are taken at low electron dose, it is extremely hard to visualize the individual particle with low contrast and high noise level. I will introduce the multi-frequency class averaging (MFCA) algorithm to classify images of similar views and discuss the associated spectral properties of the MFCA matrices that ensure the stability of the algorithm. In the second part, I will extend the discussion to general manifold and compact Lie group structures. The key idea is to define robust similarity measures for the data based on the cycle consistency and algebraic relations of the group transformations under multiple irreducible representations.

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Date/Time : 10 Feb 2021 16:00
Venue : via Zoom
Speaker : Vincent Tan , National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/82921718900?pwd=VHVEeWszTUZWT0o5KzQrWXVPbXRxZz09&from=addon Meeting ID: 829 2171 8900 Password: 968735 Abstract: We consider the classical problem of learning tree-structured graphical models but with the twist that the observations are corrupted in independent noise. For the case in which the noise is identically distributed, we derive the exact asymptotics via the use of probabilistic tools from the theory of strong large deviations. Our results strictly improve those of Bresler and Karzand (2020) and Nikolakakis et al. (2019) and demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds. When the noise is non-identically distributed, Katiyar et al. (2020) showed that although the exact tree structure cannot be recovered, one can recover a “partial” tree structure; that is, one that belongs to the equivalence class containing the true tree. We propose Symmetrized Geometric Averaging (SGA), a statistically robust algorithm for partial tree recovery. We provide error exponent analyses and extensive numerical results on a variety of trees to show that the sample complexity of SGA is significantly better than the algorithm of Katiyar et al. (2020). SGA can be readily extended to Gaussian models and is shown via numerical experiments to be similarly superior.

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Date/Time : 04 Feb 2021 16:00
Venue : via Zoom
Speaker : Zhou Xiang, City University of Hong Kong

Join Zoom Meeting https://nus-sg.zoom.us/j/81177164345?pwd=Z1BiMWx1dWI1bU1xdlN5WFl3b3ZJdz09 Meeting ID: 811 7716 4345 Password: 311396 Abstract:  Optimal control problem is typically cast as a nonlinear Hamiltonian-Jacobi-Bellman PDE problem which the value function satisfies. In policy iteration, the nonlinear HJB equation is split into alternative iteration between a linear HJB equation with a given control (policy evaluation) and an point-wise optimization for control variable (policy improvement), but the latter is in fact determined only by the gradient of value function. In this talk, under smoothness assumption, we derive the PDE system for the (vector-valued) gradient of the value function (value-gradient function), which is closed and linear and enjoys a nice component-decoupling property. This PDE system of value-gradient can be solved by the method of characteristics as the linear HJB equation: one curve of characteristics will produce the data for both value and value-gradient. Supplemented by this additional value-gradient data, the value function is then computed by minimizing the sum of two mean square errors between the data and the parametric function approximations. We show by a few numerical examples the improvement of both robustness and accuracy when such value-gradient is taken into account. This is joint work with A. Bensoussan and P. Yam and JY Han.

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Date/Time : 27 Jan 2021 16:00
Venue : via Zoom
Speaker : Subhroshekhar Ghosh, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/86838713064?pwd=SG5JNGpXN2owVnd5R1lJa0g5SEtFQT09 Meeting ID: 868 3871 3064 Password: 519115 Abstract: The classical paradigm of randomness in the sciences in general, and data science in particular, is the model of independent and identically distributed (i.i.d.) random variables. In fact, going beyond i.i.d. is often considered a difficulty and a challenge to be overcome. In this talk, we will explore a different perspective, wherein stochastic systems with constraints in fact help to understand fundamental problems in the math of data. Our constrained systems are well-motivated from statistical physics, including models like the random critical points and determinantal probability measures. These will be used to shed important light on natural questions of relevance in understanding data, including the problems of constrained likelihood maximization and dimension reduction. En route, we will explore connections to spiked random matrix models in the PCA setting and novel asymptotics for the log likelihood in the context of the classical Wilk’s theorem. Based on multiple joint works with Philippe Rigollet (MIT) and Sanjay Chaudhuri (NUS). No special background assumptions will be made.

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Date/Time : 20 Jan 2021 16:00
Venue : via Zoom
Speaker : Marko Weber, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/82724882064?pwd=RFdxUWF4REdhblJKWi9vazJzUW9TQT09&from=addon Meeting ID: 827 2488 2064 Password: 167307 Abstract: We explicitly solve an equilibrium model in which several long-lived agents with heterogenous absolute risk-aversion and discount rates make consumption and investment decisions, trading an asset that pays a dividend whose growth rate randomly fluctuates over time. We analyze equilibrium consumption shares, stock price, risk-free rate, and optimal trading strategies. We introduce a novel technique to study the market equilibrium in an incomplete model.

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Date/Time : 13 Jan 2021 09:00
Venue : via Zoom
Speaker : Lars Ruthotto, Emory university

Join Zoom Meeting https://nus-sg.zoom.us/j/83743056445?pwd=QTFIeE9RZVh1M3FvNDhNNENlcFh0Zz09 Meeting ID: 837 4305 6445 Password: 384731 Abstract: In recent years, OT and ML have become increasingly intertwined. This talk presents new avenues for solving high-dimensional optimal transport (OT) problems using machine learning (ML). The first part of the talk shows how neural networks can be used to efficiently approximate the optimal transport map between two densities in high dimensions. To avoid the curse-of-dimensionality, we combine Lagrangian and Eulerian viewpoints and employ neural networks to solve the underlying Hamilton-Jacobi-Bellman equation. Our approach avoids any space discretization and can be implemented in existing machine learning frameworks. We present numerical results for OT in up to 100 dimensions and validate our solver in a two-dimensional setting. The second part of the talk shows how optimal transport theory can improve the efficiency of training generative models and density estimators, which are critical in machine learning. We consider continuous normalizing flows (CNF) that have emerged as one of the most promising approaches for variational inference in the ML community. Our numerical implementation is a discretize-optimize method whose forward problem relies on manually derived gradients and Laplacian of the neural network and uses automatic differentiation in the optimization. In common benchmark challenges, our method outperforms state-of-the-art CNF approaches by 1-2 orders of magnitude during training and inference.

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Date/Time : 11 Nov 2020 16:00
Venue : via Zoom
Speaker : Cai Zhenning, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/83949543700?pwd=MVBqSlBTaUtwRmxWRndta0NZZ2tFZz09 Meeting ID: 839 4954 3700 Password: 380740 Abstract: The complex Langevin (CL) method is a numerical approach to alleviate the numerical sign problem in the computation of path integrals in lattice field theories. Mathematically, it is a simple numerical tool to compute a wide class of high-dimensional and oscillatory integrals with the form of an ensemble average. The method was developed in 1980s. However, after it was proposed, it had very few applications due to its subtle nature. It is often observed that the CL method converges but the limiting result is incorrect. Less than one decade ago, an improvement of the CL method, called gauge cooling, was proposed, after which the CL method acquired much more attention and was later successfully applied to a number of fields including finite density quantum chromodynamics, superstring theory, and the spin-orbit coupling. In this talk, I will explain the basic idea of the CL method, the reason of its failure, the mechanism and new development of gauge cooling, and the current limitation of the method.

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Date/Time : 04 Nov 2020 16:00
Venue : via Zoom
Speaker : Katharina Schratz, Sorbonne Université

Join Zoom Meeting  https://nus-sg.zoom.us/j/88145005740?pwd=RjVlcmtKRU5iSmV0KysrY0FNYjdZdz09 Meeting ID: 881 4500 5740  Password: 233926  Abstract: In recent years, a large toolbox of numerical schemes for dispersive equations has been established, based on different discretisation techniques such as discretising the variation of constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low-regularity and/or with high oscillations. Classical schemes fail indeed to capture the oscillatory nature of the solution, a fact that leads to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of these new schemes is to tackle and deeply embed the underlying structure of resonances into the numerical discretisation. As in the continuous case, these resonances are central to structure preservation and provide the new schemes with strong geometric properties at low regularity.

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