BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Gabriel Peyré (CNRS\, Ecole Normale Supérieure) DTSTART:20200420T120000Z DTEND:20200420T124500Z DTSTAMP:20260404T095501Z UID:OWMADS/1 DESCRIPTION:Title: Scaling Optimal Transport for High dimensional Learning\nby Gabr iel Peyré (CNRS\, Ecole Normale Supérieure) as part of One World seminar : Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nOp timal transport (OT) has recently gained lot of interest in machine learni ng. It is a natural tool to compare in a geometrically faithful way probab ility distributions. It finds applications in both supervised learning (us ing geometric loss functions) and unsupervised learning (to perform genera tive model fitting). OT is however plagued by the curse of dimensionality\ , since it might require a number of samples which grows exponentially wit h the dimension. In this talk\, I will review entropic regularization meth ods which define geometric loss functions approximating OT with a better s ample complexity. More information and references can be found on the webs ite of our book Computational Optimal Transport.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Marie-Therese Wolfram (Warwick University\, UK) DTSTART:20200420T130000Z DTEND:20200420T134500Z DTSTAMP:20260404T095501Z UID:OWMADS/2 DESCRIPTION:Title: Inverse Optimal Transport\nby Marie-Therese Wolfram (Warwick Uni versity\, UK) as part of One World seminar: Mathematical Methods for Arbit rary Data Sources (MADS)\n\n\nAbstract\nDiscrete optimal transportation pr oblems arise in various contexts in engineering\, the sciences and the soc ial sciences. Examples include the marriage market in economics or interna tional migration flows in demographics. Often the underlying cost criterio n is unknown\, or only partly known\, and the observed optimal solutions a re corrupted by noise. In this talk we discuss a systematic approach to in fer unknown costs from noisy observations of optimal transportation plans. The proposed methodologies are developed within the Bayesian framework fo r inverse problems and require only the ability to solve the forward optim al transport problem\, which is a linear program\, and to generate random numbers. We illustrate our approach using the example of international mig ration flows. Here reported migration flow data captures (noisily) the num ber of individuals moving from one country to another in a given period of time. It can be interpreted as a noisy observation of an optimal transpor tation map\, with costs related to the geographical position of countries. We use a graph-based formulation of the problem\, with countries at the n odes of graphs and non-zero weighted adjacencies only on edges between cou ntries which share a border. We use the proposed algorithm to estimate the weights\, which represent cost of transition\, and to quantify uncertaint y in these weights.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Lorenzo Rosasco (Universitá di Genova\, IT - MIT\, US) DTSTART:20200504T120000Z DTEND:20200504T124500Z DTSTAMP:20260404T095501Z UID:OWMADS/3 DESCRIPTION:Title: Efficient kernel-PCA by Nyström sampling\nby Lorenzo Rosasco (U niversitá di Genova\, IT - MIT\, US) as part of One World seminar: Mathem atical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nIn this ta lk\, we discuss and study a Nyström based approach to efficient large sca le kernel principal component analysis (PCA). The latter is a natural nonl inear extension of classical PCA based on considering a nonlinear feature map or the corresponding kernel. Like other kernel approaches\, kernel PCA enjoys good mathematical and statistical properties but\, numerically\, i t scales poorly with the sample size. Our analysis shows that Nyström sam pling greatly improves computational efficiency without incurring any loss of statistical accuracy. While similar effects have been observed in supe rvised learning\, this is the first such result for PCA. Our theoretical f indings\, which are also illustrated by numerical results\, are based on a combination of analytic and concentration of measure techniques. Our stud y is more broadly motivated by the question of understanding the interplay between statistical and computational requirements for learning.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Lars Ruthotto (Emory University\, US) DTSTART:20200518T120000Z DTEND:20200518T124500Z DTSTAMP:20260404T095501Z UID:OWMADS/4 DESCRIPTION:Title: Machine learning meets optimal transport: old solutions for new prob lems and vice versa\nby Lars Ruthotto (Emory University\, US) as part of One World seminar: Mathematical Methods for Arbitrary Data Sources (MAD S)\n\n\nAbstract\nThis talk presents new connections between optimal trans port (OT)\, which has been a critical problem in applied mathematics for c enturies\, and machine learning (ML)\, which has been receiving enormous a ttention in the past decades. In recent years\, OT and ML have become incr easingly intertwined. This talk contributes to this booming intersection b y providing efficient and scalable computational methods for OT and ML.\nT he first part of the talk shows how neural networks can be used to efficie ntly approximate the optimal transport map between two densities in high d imensions. To avoid the curse-of-dimensionality\, we combine Lagrangian an d Eulerian viewpoints and employ neural networks to solve the underlying H amilton-Jacobi-Bellman equation. Our approach avoids any space discretizat ion and can be implemented in existing machine learning frameworks. We pre sent numerical results for OT in up to 100 dimensions and validate our sol ver in a two-dimensional setting. \nThe 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. W e consider continuous normalizing flows (CNF) that have emerged as one of the most promising approaches for variational inference in the ML communit y. Our numerical implementation is a discretize-optimize method whose forw ard problem relies on manually derived gradients and Laplacian of the neur al network and uses automatic differentiation in the optimization. In comm on benchmark challenges\, our method outperforms state-of-the-art CNF appr oaches by reducing the network size by 8x\, accelerate the training by 10x - 40x and allow 30x-50x faster inference.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Vincent Duval (Inria\, FR) DTSTART:20200608T130000Z DTEND:20200608T134500Z DTSTAMP:20260404T095501Z UID:OWMADS/5 DESCRIPTION:Title: Representing the solutions of total variation regularized problems\nby Vincent Duval (Inria\, FR) as part of One World seminar: Mathematic al Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nRepresenting t he solutions of total variation regularized problems\n\nThe total (gradien t) variation is a regularizer which has been widely used in inverse proble ms arising in image processing\, following the pioneering work of Rudin\, Osher and Fatemi. In this talk\, I will describe the structure the solutio ns to the total variation regularized variational problems when one has a finite number of measurements.\nFirst\, I will present a general represent ation principle for the solutions of convex problems\, then I will apply i t to the total variation by describing the faces of its unit ball.\n\nIt i s a joint work with Claire Boyer\, Antonin Chambolle\, Yohann De Castro\, Frédéric de Gournay and Pierre Weiss.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Unser (École polytechnique fédérale de Lausanne\, CH) DTSTART:20200608T120000Z DTEND:20200608T124500Z DTSTAMP:20260404T095501Z UID:OWMADS/6 DESCRIPTION:Title: Representer theorems for machine learning and inverse problems\n by Michael Unser (École polytechnique fédérale de Lausanne\, CH) as par t of One World seminar: Mathematical Methods for Arbitrary Data Sources (M ADS)\n\n\nAbstract\nRegularization addresses the ill-posedness of the trai ning problem in machine learning or the reconstruction of a signal from a limited number of measurements. The standard strategy consists in augmenti ng the original cost functional by an energy that penalizes solutions with undesirable behaviour. In this presentation\, I will present a general re presenter theorem that characterizes the solutions of a remarkably broad c lass of optimization problems in Banach spaces and helps us understand the effect of regularization. I will then use the theorem to retrieve some cl assical characterizations such as the celebrated representer theorem of ma chine leaning for RKHS\, Tikhonov regularization\, representer theorems fo r sparsity promoting functionals\, as well as a few new ones\, including a result for deep neural networks.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicolás García Trillos (University of Wisconsin-Madison\, US) DTSTART:20200615T130000Z DTEND:20200615T134500Z DTSTAMP:20260404T095501Z UID:OWMADS/7 DESCRIPTION:Title: Regularity theory and uniform convergence in the large data limit of graph Laplacian eigenvectors on random data clouds.\nby Nicolás Garc ía Trillos (University of Wisconsin-Madison\, US) as part of One World se minar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstrac t\nGraph Laplacians are omnipresent objects in machine learning that have been used in supervised\, unsupervised and semi supervised settings due to their versatility in extracting local and global geometric information fr om data clouds. In this talk I will present an overview of how the mathema tical theory built around them has gotten deeper and deeper\, layer by lay er\, since the appearance of the first results on pointwise consistency in the 2000’s\, until the most recent developments\; this line of research has found strong connections between PDEs built on proximity graphs on da ta clouds and PDEs on manifolds\, and has given a more precise mathematica l meaning to the task of “manifold learning”. In the first part of the talk I will highlight how ideas from optimal transport made some of the initial steps\, which provided L2 type error estimates between the spectra of graph Laplacians and Laplace-Beltrami operators\, possible. In the sec ond part of the talk\, which is based on recent work with Jeff Calder and Marta Lewicka\, I will present a newly developed regularity theory for gra ph Laplacians which among other things allow us to bootstrap the L2 error estimates developed through optimal transport and upgrade them to uniform convergence and almost C^{0\,1} convergence rates. The talk can be seen as a tale of how a flow of ideas from optimal transport\, PDEs\, and in gene ral\, analysis\, has made possible a finer understanding of concrete objec ts popular in data analysis and machine learning.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Michaël Fanuel (KU Leuven\, BE) DTSTART:20200504T130000Z DTEND:20200504T134500Z DTSTAMP:20260404T095501Z UID:OWMADS/8 DESCRIPTION:Title: Diversity sampling in kernel method\nby Michaël Fanuel (KU Leuv en\, BE) as part of One World seminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nA well-known technique for large scale kernel methods is the Nyström approximation. Based on a subset of landmar ks\, it gives a low rank approximation of the kernel matrix\, and is known to provide a form of implicit regularization. We will discuss the impact of sampling diverse landmarks for constructing the Nyström approximation in supervised and unsupervised problems. In particular\, three methods wil l be considered: uniform sampling\, leverage score sampling and Determinan tal Point Processes (DPP). The implicit regularization due the diversity o f the landmarks will be made explicit by numerical simulations and analyse d further in the case of DPP sampling by some theoretical results.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Francis Bach (Inria\, FR) DTSTART:20200518T130000Z DTEND:20200518T134500Z DTSTAMP:20260404T095501Z UID:OWMADS/9 DESCRIPTION:Title: On the convergence of gradient descent for wide two-layer neural net works\nby Francis Bach (Inria\, FR) as part of One World seminar: Math ematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nMany sup ervised learning methods are naturally cast as optimization problems. For prediction models which are linear in their parameters\, this often leads to convex problems for which many guarantees exist. Models which are non-l inear in their parameters such as neural networks lead to non-convex optim ization problems for which guarantees are harder to obtain. In this talk\, I will consider two-layer neural networks with homogeneous activation fun ctions where the number of hidden neurons tends to infinity\, and show how qualitative convergence guarantees may be derived. I will also highlight open problems related to the quantitative behavior of gradient descent for such models. (Based on joint work with Lénaïc Chizat\, https://arxiv.or g/abs/1805.09545\, https://arxiv.org/abs/2002.04486)\n\nPlease note that t his is a joint talk with the One World Optimization Seminar.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrea Braides (University of Rome Tor Vergata) DTSTART:20200615T120000Z DTEND:20200615T124500Z DTSTAMP:20260404T095501Z UID:OWMADS/10 DESCRIPTION:Title: Continuum limits of interfacial energies on (sparse and) dense grap hs\nby Andrea Braides (University of Rome Tor Vergata) as part of One World seminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\ nAbstract\nI review some results on the convergence of energies defined on graphs. My interest in such energies comes from models in Solid Mechanics (where the bonds in the graph represent the relevant atomistic interactio ns) or Statistical Physics (Ising systems)\, but the nodes of the graph ca n also be thought as a collection of data on which the bonds describe some relation between the data.\nThe typical objective is an approximate (simp lified) continuum description of problems of minimal cut as the number N o f the nodes of the graphs diverges.\nIf the graphs are sparse (i.e. the nu mber of bonds is much less than the total number of pairs of nodes as N go es to infinity)\, often (more precisely when we have some control on the r ange or on the decay of the interactions) such minimal-cut problems transl ate into minimal-perimeter problems for sets or partitions on the continuu m. This description is easily understood for periodic lattice systems\, bu t carries on also for random distributions of nodes. In the case of a (loc ally) uniform Poisson distribution\, actually the limit minimal-cut proble ms are described by more regular energies than in the periodic-lattice cas e. \nWhen we relax the hypothesis on the range of interactions\, the descr iption of the limit of sparse graphs becomes more complex\, as it depends subtly on geometric characteristics of the graph\, and is partially unders tood. Some easy examples show that\, even though for the continuum limit w e still remain in a similar analytical environment\, the description as (s harp) interfacial energies can be lost in this case\, and more “diffuse ” interfaces must be taken into account.\nIf instead we consider dense s equences of graphs (i.e.\, the number of bonds is of the same order as the total number of pairs as N goes to infinity) then a completely different limit environment must be used\, that of graphons (which are abstract limi ts of graphs)\, for which sophisticated combinatoric results can be used. We can re-read the existing notion of convergence of graphs to graphons as a convergence of the related cut functionals to non-local energies on a s imple reference parameter set. This convergence provides an approximate de scription of the corresponding minimal-cup problems.\nWorks in collaborati on with Alicandro\, Cicalese\, Piatnitski and Solci (sparse graphs) and Ce rmelli and Dovetta (dense graphs).\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Jana de Wiljes (Universität Potsdam\, DE) DTSTART:20200629T120000Z DTEND:20200629T124500Z DTSTAMP:20260404T095501Z UID:OWMADS/11 DESCRIPTION:Title: Sequential learning for decision support under uncertainty\nby Jana de Wiljes (Universität Potsdam\, DE) as part of One World seminar: M athematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nIn ma ny applicational areas there is a need to determine a control variable tha t optimizes a pre-specified objective. This problem is particularly challe nging when knowledge on the underlying dynamics is subject to various sour ces of uncertainty. A scenario such as that arises for instance in the c ontext of therapy individualization to improve the efficacy and safety of medical treatment. Mathematical models describing the pharmacokinetics and pharmacodynamics of a drug together with data on associated biomarkers ca n be leveraged to support decision-making by predicting therapy outcomes. We present a continuous learning strategy which follows a novel sequential Monte Carlo tree search approach and explore how the underlying uncertain ties reflect in the approximated control variable.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Björn Sprungk (TU Freiberg\, DE) DTSTART:20200629T130000Z DTEND:20200629T134500Z DTSTAMP:20260404T095501Z UID:OWMADS/12 DESCRIPTION:Title: Noise-level robust Monte Carlo methods for Bayesian inference with infomative data\nby Björn Sprungk (TU Freiberg\, DE) as part of One W orld seminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\n Abstract\nThe Bayesian approach to inverse problems provides a rigorous fr amework for the incorporation and quantification of uncertainties in measu rements\, parameters and models. However\, sampling from or integrating w. r.t. the resultung posterior measure can become computationally challengin g. In recent years\, a lot of effort has been spent on deriving dimension- independent methods and to combine efficient sampling strategies with mult ilevel or surrogate methods in order to reduce the computational burden of Bayesian inverse problems.\nIn this talk\, we are interested in designing numerical methods which are robust w.r.t. the size of the observational n oise\, i.e.\, methods which behave well in case of concentrated posterior measures. The concentration of the posterior is a highly desirable situati on in practice\, since it relates to informative or large data. However\, it can pose as well a significant computational challenge for numerical me thods based on the prior or reference measure. We propose to employ the La place approximation of the posterior as the base measure for numerical int egration in this context. The Laplace approximation is a Gaussian measure centered at the maximum a-posteriori estimate (MAPE) and with covariance m atrix depending on the Hessian of the log posterior density at the MAPE. W e discuss convergence results of the Laplace approximation in terms of the Hellinger distance and analyze the efficiency of Monte Carlo methods base d on it. In particular\, we show that Laplace-based importance sampling an d quasi-Monte-Carlo as well as Laplace-based Metropolis-Hastings algorithm s are robust w.r.t. the concentration of the posterior for large classes o f posterior distributions and integrands whereas prior-based Monte Carlo s ampling methods are not.\n LOCATION:https://stable.researchseminars.org/talk/OWMADS/12/ END:VEVENT END:VCALENDAR