Date/Time : 26 Aug 2020 16:00
Venue : via Zoom
Speaker : Tong Xin, NUS

Join Zoom Meeting https://nus-sg.zoom.us/j/92076578514?pwd=aWk4Nm1Cc3htS1hGWDVpTlZ2cUgrdz09 Meeting ID: 920 7657 8514 Password: 459566 Gradient descent (GD) is known to converge quickly for convex objective functions, but it can be trapped at local minima. On the other hand, Langevin dynamics (LD) can explore the state space and find global minima, but in order to give accurate estimates, LD needs to run with a small discretization step size and weak stochastic force, which in general slows down its convergence. This paper shows that these two algorithms can “collaborate” through a simple exchange mechanism, in which they swap their current positions if LD yields a lower objective function. This idea can be seen as the singular limit of the replica-exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighborhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. We further verify our theoretical results through some numerical experiments, and observe superior performance of the proposed algorithm over running GD or LD alone. We will further explain how does replica exchange method improve LD’s spectral gap using Poincare type of inequality.

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Date/Time : 26 Aug 2020 17:00
Venue : via Zoom
Speaker : Gordon Hoi,

Join  Zoom Meeting: https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? The Exact Satisfiability problem, XSAT, is defined as the problem of finding a satisfying assignment to a formula in CNF such that there is exactly one literal in each clause assigned to be 1 and the other literals in the same clause are set to 0. If we restrict the length of each clause to be at most 3 literals, then it is known as the X3SAT problem. In this paper, we consider the problem of counting the number of satisfying assignments to the X3SAT problem, which is also known as #X3SAT. The current state of the art exact algorithm to solve #X3SAT is given by Dahlloef, Jonsson and Beigel and runs in O(1.1487^n), where n is the number of variables in the formula. In this paper, we propose an exact algorithm for the #X3SAT problem that runs in O(1.1120^n) with very few branching cases to consider, by using a result from Monien and Preis to give us a bisection width for graphs with at most degree 3. URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

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Date/Time : 26 Aug 2020 15:00
Venue : via Zoom
Speaker : Lu Ming, Sichuan University

Join Zoom Meeting https://nus-sg.zoom.us/j/99335000776?pwd=UFVJNnN1ZTJMK1MzQlpFWnVSckwrQT09 Meeting ID: 993 3500 0776 Password: 359589 Abstract: A quantum symmetric pair consists of a quantum group and its coideal subalgebra (called an i-quantum group). A quantum group can be viewed as an example of i-quantum groups associated to symmetric pairs of diagonal type. In this talk, we present a new Hall algebra construction of i-quantum groups. Our approach leads to monomial bases, PBW bases, and braid group actions for i-quantum groups. In case of symmetric pairs of diagonal type, our work reduces to a reformulation of Bridgeland’s Hall algebra realization of a quantum group, which in turn was a generalization of earlier constructions of Ringel and Lusztig for half a quantum group. This is joint work with Weiqiang Wang.

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Date/Time : 26 Aug 2020 10:00
Venue : via Zoom
Speaker : Li Qiongling , Chern Institute of Mathematics, Nankai University

Join Zoom Meeting https://nus-sg.zoom.us/j/91329086826?pwd=RFNXQzROaDhicTdKQkxkeWxpL3lhUT09 Meeting ID: 913 2908 6826 Password: 648142 We start this talk with some basic linear algebras. Consider m(A)=[A,A*] on sl(n,C). Ness studied the (normalized) square of norm |m|^2 and showed some nice properties of |m|^2 which characterize the unitary orbits inside the nilpotent orbits of sl(n,C). We generalize Ness’ results to general orbits by modifying |m|^2, motivated by the curvature formula of symmetric spaces. As geometric applications, we show some domination results in n-Fuchsian fibers in the Hitchin fibration of the moduli space of Higgs bundles over a Riemann surface. And we also show some existence and uniqueness results of equivariant minimal/maximal surfaces in certain product spaces. This is joint work with Song Dai.    

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Date/Time : 19 Aug 2020 10:00
Venue : via Zoom
Speaker : Junho Peter Whang, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/94766702852?pwd=UG9SclBVcTVlVDFnSnRQcFJjaUw4QT09 Meeting ID: 947 6670 2852 Password: 569325 Abstract: Moduli spaces for special linear rank two local systems (with prescribed boundary traces) on a topological surface are basic objects in geometry. The study of integer points on these algebraic varieties can be traced back to 1880 work of Markoff, in the case where the surface is a one-holed torus. For general surfaces, we will show that the moduli spaces are log Calabi-Yau, and use mapping class group dynamics and differential geometric tools to establish a structure theorem for their integer points. In addition, we will also give an analysis of algebraic curves on the moduli spaces. Throughout the talk, emphasis will be put on the interplay between the dynamical, arithmetic, and topological aspects of surfaces and their local systems.  

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Date/Time : 19 Aug 2020 17:00
Venue : via Zoom
Speaker : Cristian Calude, University of Auckland

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract: We present a new ternary QRNG based on measuring located value indefinite observables with probabilities <b>1/4,1/2,1/4</b> and prove that every sequence generated is maximally unpredictable, 3-bi-immune (a stronger form of bi-immunity), and its prefixes are Borel normal. The ternary quantum random digits produced by the QRNG are algorithmically transformed into quantum random bits using an alphabetic morphism which preserves all the above properties.

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Date/Time : 19 Aug 2020 16:00
Venue : via Zoom
Speaker : Pang Tongyao , National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/97080214054?pwd=cnJtSXF0YTZWZ1U2QU5YQWpwaU1tUT09 Meeting ID: 970 8021 4054 Password: 632494 Abstract: In this talk, I will start with the classical sparse estimation, whih has been extensively studied and applied to a wide range of applications, e.g., image restoration, in the past few decades. While most of the existing methods targeted at solving a maximum a posterior (MAP) variational model with the sparsity prior, we proposed to find the minimum mean square error (MMSE) solution which provides a good balance between sparsity and unbiasedness. We develop a coordinate minimization algorithm and connect it to a variational model to establish a comprehensive convergence analysis. In recent years, deep learning has shown its advantages over the traditional sparsity prior based methods for image restoration problems. However, most deep learning methods are supervised which require a high-quality training dataset. Bringing the idea of Bayesian inference from sparse estimation to deep learning further, we are able to develop a self-supervised deep learning method that calls no other training samples but still achieves SOTA performance. The key to success of our method lies in the approximation to the MMSE estimate, which helps to reduce model bias and enables ensemble prediction scheme with lower variance.

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Date/Time : 14 Aug 2020 09:00
Venue : via Zoom
Speaker : Hong Wang , Institute for Advanced Study, Princeton

Join Zoom Meeting Meeting ID: 932 7338 0910 Password: 451089  Let $E$ be an $s$–dimensional set on the plane. How large is its distance set $\Delta(E):= \{ |x-y|, x, y\in E\}$?  Falconer conjectured in 1985 that if $s>1$, then $\Delta(E)$ should have positive Lebesgue measure.  We are going to discuss how this problem is connected to various problems in harmonic analysis, as well as different approaches toward it.

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Date/Time : 12 Aug 2020 16:00
Venue : via Zoom
Speaker : Dong Bin, Peking University

Join Zoom Meeting https://nus-sg.zoom.us/j/93373644793?pwd=Y2VrdnJRazZnRXllN2N5K1BneWE1UT09 Meeting ID: 933 7364 4793 Password: 711028 In this talk, I will start with a brief review of the dynamics and optimal control perspective on deep learning (including supervised learning, reinforcement learning, and meta-learning), especially the so-called unrolled dynamics approach and its applications in medical imaging. Then, I will present some of our recent studies on how this new approach may help us to advance CT imaging and image-based diagnosis further. Specifically, I will focus on our thoughts on how to combine the wisdom from mathematical modeling with ideas from deep learning. Such combination leads to new data-driven image reconstruction models and new data-driven scanning strategies for CT imaging, and with a potential to be generalized to other imaging modalities.

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