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Optimization and Data Analysis - Oberseminar

Upcoming Talk

There is no talk on 22 July 2019.

Contacts

PersonSorted ascending E-Mail Office
Carlos Améndola Cerón carlos.amendolaemattum.de MI 02.08.038
Felix Krahmer felix.krahmeremattum.de MI 02.10.039
Massimo Fornasier massimo.fornasierematma.tum.de MI 02.10.058
Peter Massopust massopustematma.tum.de MI 02.10.038
Sandro Belz sandro.belzematma.tum.de MI 02.10.052

All Scheduled Talks

Mo, 29.04.2019 14:00 - 15:30
Speaker Anthea Monod (Columbia University)
Title Tropical Statistics for Phylogenetic Trees
Abstract Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric structure results in a non-Euclidean phylogenetic tree space, which poses computational and statistical limitations. We propose a novel framework constructed from tropical geometry for the statistical analysis of evolutionary biological processes represented by sets of phylogenetic trees. Our structure allows for the definition of probability measures, expectations, variances, and other fundamental statistical quantities. In addition, our setting exhibits analytic, geometric, and topological properties that are desirable for rigorous theoretical treatment in probability and statistics, as well as increased computational efficiency over the current state-of-the-art. We demonstrate our approach and compare against the current standard on seasonal influenza data. This is joint work with Bo Lin (Georgia Tech), Qiwen Kang (University of Kentucky), and Ruriko Yoshida (Naval Postgraduate School).
Room 02.08.011

Mo, 06.05.2019 14:00 - 15:30
Speaker Max Pfeffer (MPI-MiS Leipzig)
Title Riemannian Optimization for Constrained Low Rank Matrix Problems
Abstract Low rank matrix factorizations have found many applications in recent years. Often, one is interested in minimizing a function over the manifold of low rank matrices in order to find a data sparse or well structured solution. This can be done using Riemannian optimization, which is the generalization of gradient related algorithms to smooth Riemannian manifolds. In theory, additional constraints on the components, like sparsity or nonnegativity, can be formulated. However, this significantly complicates the procedure and rigorous results are sparse. The talk will give an introduction into Riemannian optimization for constrained low rank problems and illustrate some of the difficulties.
Room 02.08.011

Mo, 13.05.2019 14:00 - 15:30
Speaker Massimo Fornasier (TUM)
Title Spatially Inhomogeneous Evolutionary Games and Applications
Abstract We present a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. We conclude with a couple of open problems: the first one is the learning of the game payoff functional from the observation of the dynamics; the second problem is about a novel particle swarm (global) algorithm for global nonconvex optimization based on an evolutionary game. We sketch the motivations for the success of this method and we explain the open problems for a rigorous analysis of convergence.
Room 02.08.011

Mo, 20.05.2019 14:00 - 15:30
Speaker Hui Huang (TUM)
Title Self-organized dynamics in bounded domains
Abstract Collective behaviour is widely observed in many biological contexts, such as flock of birds, school of fish, or aggregation of bacteria. This talk will discuss the mathematical models for collective behavior over domains with boundaries.
Room 02.08.011

Mo, 27.05.2019 14:00 - 15:30
Speaker Mauro Bonafini (TUM)
Title A convex approach to the Steiner problem
Abstract Connected 1-dimensional structures play a crucial role in very different areas like discrete geometry (graphs, networks, spanning, and Steiner trees), structural mechanics (crack formation and propagation), and inverse problems (defects identification, contour segmentation), etc. The modeling of these structures is a key problem both from the theoretical and the numerical points of view. Most of the difficulties encountered in studying such 1-dimensional objects are related to the fact that they are not canonically associated to standard mathematical quantities. In this seminar we focus on the Steiner Tree Problem as a prototypical example of problems involving such 1-dimensional optimal networks. We introduce a convex relaxation of the problem in a fairly general context (arbitrary dimension and even manifold ambients) and we show through calibration type arguments that minimizers of the relaxed energy are generally (but not always) convex combinations of Steiner Minimal Trees. We then present an extensive numerical investigation of the relaxed framework in the two and three dimensional case and in the surface scenario.
Room 02.08.011

Mo, 03.06.2019 14:00 - 15:30
Speaker Peter Massopust (TUM)
Title What are iterated function systems and what can you do with them?
Abstract We give a rudimentary introduction to iterated function systems and highlight some of their properties and applications. For the latter we present the collage theorem and fractal compression, fractal functions, and wavelets.
Room 02.08.011

Mo, 17.06.2019 14:00 - 15:30
Speaker Felix Krahmer (TUM)
Title Sparse Harmonic Transforms: A New Class of Sublinear-time Algorithms for Learning Functions of Many Variables
Abstract In this talk we will discuss fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification (UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases. More concretely, let B be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality #B=N. Herein we will develop methods that rapidly approximate any function f that is sparse in the BOPB, that is, f of the form f(x)=∑b∈Scb⋅b(x) with S⊂B of cardinality #S=s is much less than N. Our method has a runtime of just (slogN)O(1), uses only (slogN)O(1) function evaluations on a fixed and nonadaptive grid, and not more than (slogN)O(1) bits of memory. We emphasize that nothing about S or any of the coefficients cb is assumed in advance other than that S⊂B has #S=s. Both S and its related coefficients cb will be learned from the given function evaluations by the developed method. For s≪N, the runtime (slogN)O(1) will be less than what is required to simply enumerate the elements of the basis B; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as sublinear-time. This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size Ω(N) during the solution process.Besides these improved theoretical guarantees, also the empirical performance of our method improves over previous approaches. In particular, the enhanced memory efficiency allows us to tackle much larger problem sizes than in previous works. This is joint work with Bosu Choi (Michigan State University/UT Austin) and Mark Iwen (Michigan State University)
Room 02.08.011

Mo, 24.06.2019 14:00 - 15:30
Speaker Hanna Veselovska (TU Braunschweig)
Title Prony-type polynomials and their common zeros
Abstract click here
Room 02.08.011

Mo, 01.07.2019 14:00 - 15:30
Speaker Dominik Stöger (TUM)
Title The convex geometry of blind deconvolution
Abstract Blind deconvolution problems have been studied for several decades. Recently, a new viewpoint has been introduced. Namely one assumes that the convolved signals are contained in random, but known subspaces. Ahmed, Recht, and Romberg proposed to use convex optimization in the space of lifted matrix representations. Their theoretical analysis, based on the Golfing Scheme, yields seemingly suboptimal error bounds and proof artifacts. We present a novel analysis based on Mendelson’s small-ball method, which shows near-optimal error bounds for large noise-levels.
Room 02.08.011

Mo, 15.07.2019 14:00 - 15:30
Speaker Richard Huber (KFU)
Title Convergence of Pixel-Driven Projection Methods
Abstract Tomography is an important part of the medical diagnostic process, as it allows for non-invasive visualization of a patient interior. Such reconstruction corresponds to inversion of the Radon transform, and concrete computation requires discretization of the Radon transform and its adjoint -- the backprojection. Thus, over the years a number of discretization methods have been proposed, however, little rigorous analysis of these methods was done. These methods include the pixel-driven approach, which is known to create oscillations along some projection angles and thus has gained little attention in spite of its numerically very efficient structure. We offer a rigorous mathematical interpretation of the approach, allowing to show convergence -- including rates -- in the operator norm when a suitable choice of discretization parameters is used. In particular, these suitable choices do not contain the standard practice of considering constant ratio between the size of image pixels and detector pixels, thus suggesting that the standard way this method is used is not mathematically justified as it is not supported by the theory.
Room 02.08.011

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