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BEGIN:VEVENT
SUMMARY:Joan Bruna (NYU Courant)
DTSTART:20210420T121500Z
DTEND:20210420T134500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/1
DESCRIPTION:Title: Mathematical aspects of neural network approximation and learning<
/a>\nby Joan Bruna (NYU Courant) as part of Mathematics of Deep Learning\n
\n\nAbstract\nHigh-dimensional learning remains an outstanding phenomena w
here experimental evidence outpaces our current mathematical understanding
. Neural Networks provide a rich yet intricate class of functions with sta
tistical abilities to break the curse of dimensionality\, and where physic
al priors can be tightly integrated into the architecture to improve sampl
e efficiency. Despite these advantages\, an outstanding theoretical challe
nge in these models is computational\, by providing an analysis that expla
ins successful optimization and generalization in the face of existing wor
st-case computational hardness results.\n\nIn this talk\, we will describe
snippets of such challenge\, covering respectively optimization and appro
ximation. First\, we will focus on the framework that lifts parameter opti
mization to an appropriate measure space. We will overview existing result
s that guarantee global convergence of the resulting Wasserstein gradient
flows\, and present our recent results that study typical fluctuations of
the dynamics around their mean field evolution\, as well as extensions of
this framework beyond vanilla supervised learning to account for symmetrie
s in the function. Next\, we will discuss the role of depth in terms of ap
proximation\, and present novel results establishing so-called ‘depth se
paration’ for a broad class of functions. We will conclude by discussing
consequences in terms of optimization\, highlighting current and future m
athematical challenges.\n\nJoint work with: Zhengdao Chen\, Grant Rotskoff
\, Eric Vanden-Eijnden\, Luca Venturi\, Samy Jelassi and Aaron Zweig.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gitta Kutyniok (LMU Munich)
DTSTART:20210427T101500Z
DTEND:20210427T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/2
DESCRIPTION:Title: Deep Learning meets Shearlets: On the Path Towards Interpretable I
maging\nby Gitta Kutyniok (LMU Munich) as part of Mathematics of Deep
Learning\n\n\nAbstract\nPure model-based approaches are today often insuff
icient for solving complex inverse problems in medical imaging. At the sam
e time\, methods based on artificial intelligence\, in particular\, deep n
eural networks\, are extremely successful\, often quickly leading to state
-of-the-art algorithms. However\, pure deep learning approaches often negl
ect known and valuable information from the modeling world and suffer from
a lack of interpretability.\n\nIn this talk\, we will develop a conceptua
l approach by combining the model-based method of sparse regularization by
shearlets with the data-driven method of deep learning. Our solvers pay p
articular attention to the singularity structures of the data. Focussing t
hen on the inverse problem of (limited-angle) computed tomography\, we wil
l show that our algorithms significantly outperform previous methodologies
\, including methods entirely based on deep learning. Finally\, we will al
so touch upon the issue of how to interpret such algorithms\, and present
a novel\, state-of-the-art explainability method based on information theo
ry.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Möller (University of Siegen)
DTSTART:20210504T101500Z
DTEND:20210504T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/3
DESCRIPTION:Title: On the Confluence of Deep Learning and Energy Minimization Methods
for Inverse Problems\nby Michael Möller (University of Siegen) as pa
rt of Mathematics of Deep Learning\n\n\nAbstract\nMany practical applicati
ons require to infer a desired quantity from measurements that contain imp
licit information about them\, commonly resulting in ill-posed inverse rec
onstruction problems. While classical approaches formulate their solution
as the argument that minimizes a suitable cost function\, recent works dom
inate image reconstruction benchmarks using deep learning. This talk discu
sses possible ways of combining ideas from energy minimization and deep le
arning\, including algorithmic schemes that introduce learned regularity\,
networks that iteratively minimize a model based cost function\, and tech
niques that aim at learning suitable regularizers. For the latter\, I will
highlight recent advances and future challenges in the design of such par
ameterized regularizers as well as the solution of the bi-level optimizati
on problems resulting from their training.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enrique Zuazua (FAU Erlangen-Nürnberg)
DTSTART:20210511T101500Z
DTEND:20210511T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/4
DESCRIPTION:Title: Neural Differential Equations\, Control and Machine Learning\n
by Enrique Zuazua (FAU Erlangen-Nürnberg) as part of Mathematics of Deep
Learning\n\n\nAbstract\nWe discuss Neural Ordinary Differential Equations
(NODEs) from a control theoretical perspective to address some of the main
challenges in Machine Learning and\, in particular\, data classification
and Universal Approximation. More precisely\, we adopt the perspective of
the simultaneous control of systems of NODEs. For instance\, in the contex
t of classification\, each item to be classified corresponds to a differen
t initial datum for the Cauchy problem of the NODE. And all the solutions
corresponding the data under consideration need to be driven to the corres
ponding target by means of the same control. We present a genuinely nonlin
ear and constructive method\, allowing to estimate the complexity of the c
ontrol strategies we develop. The very nonlinear nature of the activation
functions governing the nonlinear dynamics of NODEs under consideration pl
ays a key role. It allows deforming half of the phase space while the othe
r half remains invariant\, a property that classical models in mechanics d
o not fulfill. This very property allows to build elementary controls indu
cing specific dynamics and transformations whose concatenation\, along wit
h properly chosen hyperplanes\, allows achieving our goals in finitely man
y steps. We also present the counterparts in the context of the control of
neural transport equations\, establishing a link between optimal transpor
t and deep neural networks.\n\nThis is a joint work Domènec Ruiz-Balet.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Ruthotto (Emory University)
DTSTART:20210518T121500Z
DTEND:20210518T134500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/5
DESCRIPTION:Title: An Introduction to Generative Modeling\nby Lars Ruthotto (Emor
y University) as part of Mathematics of Deep Learning\n\n\nAbstract\nDeep
generative models (DGM) are neural networks with many hidden layers traine
d to approximate complicated\, high-dimensional probability distributions
from a finite number of samples. When trained successfully\, we can use th
e DGMs to estimate the likelihood of each observation and to create new sa
mples from the underlying distribution. Developing DGMs has become one of
the most hotly researched fields in artificial intelligence in recent year
s. The literature on DGMs has become vast and is growing rapidly.\nSome ad
vances have even reached the public sphere\, for example\, the recent succ
esses in generating realistic-looking images\, voices\, or movies\; so-cal
led deep fakes.\n\nDespite these successes\, several mathematical and prac
tical issues limit the broader use of DGMs: given a specific dataset\, it
remains challenging to design and train a DGM and even more challenging to
find out why a particular model is or is not effective. To help students
contribute to this field\, this talk provides an introduction to DGMs and
provides a concise mathematical framework for modeling the three most popu
lar approaches: normalizing flows (NF)\, variational autoencoders (VAE)\,
and generative adversarial networks (GAN). We illustrate the advantages an
d disadvantages of these basic approaches using numerical experiments. Our
goal is to enable and motivate the reader to contribute to this prolifera
ting research area. Our presentation also emphasizes relations between gen
erative modeling and optimal transport.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeff Calder (University of Minnesota)
DTSTART:20210525T141500Z
DTEND:20210525T154500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/6
DESCRIPTION:Title: Random walks and PDEs in graph-based learning\nby Jeff Calder
(University of Minnesota) as part of Mathematics of Deep Learning\n\n\nAbs
tract\nI will discuss some applications of random walks and PDEs in graph-
based learning\, both for theoretical analysis and algorithm development.
Graph-based learning is a field within machine learning that uses similari
ties between datapoints to create efficient representations of high-dimens
ional data for tasks like semi-supervised classification\, clustering and
dimension reduction. There has been considerable interest recently in semi
-supervised learning problems with very few labeled examples (e.g.\, 1 lab
el per class). The widely used Laplacian regularization is ill-posed at lo
w label rates and gives very poor classification results. In the first par
t of the talk\, we will use the random walk interpretation of the graph La
placian to precisely characterize the lowest label rate at which Laplacian
regularized semi-supervised learning is well-posed. At lower label rates\
, where Laplace learning performs poorly\, we will show how our random wal
k analysis leads to a new algorithm\, called Poisson learning\, that is pr
obably more stable and informative than Laplace learning. We will conclude
with some applications of Poisson learning to image classification and me
sh segmentation of broken bone fragments of interest in anthropology.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Pock (University of Graz)
DTSTART:20210601T101500Z
DTEND:20210601T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/7
DESCRIPTION:Title: Learning with energy-based models\nby Thomas Pock (University
of Graz) as part of Mathematics of Deep Learning\n\n\nAbstract\nIn this ta
lk\, I will show how to use learning techniques to significantly improve e
nergy-based models. I will start by showing that even for the simplest mod
els such as total variation\, one can greatly improve the accuracy of the
numerical approximation by learning the "best" discretization within a cla
ss of consistent discretizations. Then I will move forward to more express
ive models and show how they can be learned in order to give state-of-the
art performance for image reconstruction problems\, such as denoising\, su
perresolution\, MRI and CT. Finally\, I will show how energy based models
for image labeling such as Markov random fields can be used in the framewo
rk of deep learning.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Papadakis (University of Bordeaux)
DTSTART:20210608T101500Z
DTEND:20210608T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/8
DESCRIPTION:Title: On the learning of Wasserstein generative models\nby Nicolas P
apadakis (University of Bordeaux) as part of Mathematics of Deep Learning\
n\n\nAbstract\nThe problem of WGAN (Wasserstein Generative Adversarial Net
work) learning is an instance of optimization problems where one wishes to
find\, among a parametric class of distributions\, the one which is close
st to a target distribution in terms of an optimal transport (OT) distance
. Applying a gradient-based algorithm for this problem requires to express
the gradient of the OT distance with respect to one of its argument\, whi
ch can be related to the solutions of the dual problem (Kantorovich potent
ials). The first part of this talk aims at finding conditions that ensure
the existence of such gradient. After discussing regularity issues that ma
y appear with discrete target measures\, we will show that regularity prob
lems are avoided when using entropy-regularized OT and/or considering the
semi-discrete formulation of OT. Then\, we will see how these gradients ca
n be exploited in a stable way to address some imaging problems where the
target discrete measure is reasonably large. Using OT distances between mu
lti-scale patch distributions\, this allows to estimate a generative convo
lutional network that can synthesize an exemplar texture in a faithful and
efficient way.\nThis is a joint work with Antoine Houdard\, Arthue Leclai
re and Julien Rabin.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Cremers (TU Munich)
DTSTART:20210615T101500Z
DTEND:20210615T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/9
DESCRIPTION:Title: Self-supervised Learning for 3D Shape Analysis\nby Daniel Crem
ers (TU Munich) as part of Mathematics of Deep Learning\n\n\nAbstract\nWhi
le neural networks have swept the field of computer vision and replaced cl
assical methods in most areas of image analysis and beyond\, extending the
ir power to the domain of 3D shape analysis remains an important open chal
lenge. In my presentation\, I will focus on the problems of shape matching
\, correspondence estimation and shape interpolation and develop suitable
deep learning approaches to tackle these challenges. In particular\, I wil
l focus on the difficult problem of computing correspondence and interpola
tion for pairs of shapes from different classes — say a human and a hors
e — where traditional isometry assumptions no longer hold.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yixing Huang (FAU Erlangen-Nürnberg)
DTSTART:20210622T101500Z
DTEND:20210622T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/10
DESCRIPTION:Title: Deep Learning for Computed Tomography Image Reconstruction from I
nsufficient Data\nby Yixing Huang (FAU Erlangen-Nürnberg) as part of
Mathematics of Deep Learning\n\n\nAbstract\nComputed tomography (CT) image
reconstruction from insufficient data is a severely ill-posed inverse pro
blem. Conventional methods solely have very limited performance to address
this problem. Deep learning has achieved impressive results in solving va
rious inverse problems. \nHowever\, the robustness of deep learning method
s is still a concern for clinical applications due to the following two ch
allenges: a) With limited access to sufficient training data\, a learned d
eep learning model may not generalize well to unseen data\; b) Deep learni
ng models are sensitive to noise. Therefore\, the quality of images proces
sed by neural networks only may be inadequate. In this talk\, we investiga
te the robustness of deep learning in CT image reconstruction first. Since
learning-based images with incorrect structures are likely not consistent
with measured projection data\, we propose a data consistent reconstructi
on (DCR) method to improve their image quality\, which combines the advant
ages of conventional methods and deep learning: \nFirst\, a prior image is
generated by deep learning. Afterwards\, unmeasured data are inpainted by
forward projection of the prior image. \nFinally\, a final image is recon
structed by a conventional method\, integrating data consistency for measu
red data and learned prior information for missing data. The DCR method is
demonstrated in two\nscenarios: image reconstruction from limited-angle d
ata and truncated data.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carola-Bibiane Schönlieb (University of Cambridge)
DTSTART:20210629T101500Z
DTEND:20210629T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/11
DESCRIPTION:Title: Deeply learned regularisation for inverse problems\nby Carola
-Bibiane Schönlieb (University of Cambridge) as part of Mathematics of De
ep Learning\n\n\nAbstract\nInverse problems are about the reconstruction o
f an unknown physical quantity from indirect measurements. In imaging\, th
ey appear in a variety of places\, from medical imaging\, for instance MRI
or CT\, to remote sensing\, for instance Radar\, to material sciences and
molecular biology\, for instance electron microscopy. Here\, imaging is a
tool for looking inside specimen\, resolving structures beyond the scale
visible to the naked eye\, and to quantify them. It is a mean for diagnosi
s\, prediction and discovery.\nMost inverse problems of interest are ill-p
osed and require appropriate mathematical treatment for recovering meaning
ful solutions. Classically\, inversion approaches are derived almost concl
usively in a knowledge driven manner\, constituting handcrafted mathematic
al models. Examples include variational regularization methods with Tikhon
ov regularisation\, the total variation and several sparsity-promoting reg
ularizers such as the L1 norm of Wavelet coefficients of the solution. Whi
le such handcrafted approaches deliver mathematically rigorous and computa
tionally robust solutions to inverse problems\, they are also limited by o
ur ability to model solution properties accurately and to realise these ap
proaches in a computationally efficient manner.\nRecently\, a new paradigm
has been introduced to the regularisation of inverse problems\, which der
ives regularised solutions to inverse problems in a data driven way. Here\
, the inversion approach is not mathematically modelled in the classical s
ense\, but modelled by highly over-parametrised models\, typically deep ne
ural networks\, that are adapted to the inverse problems at hand by approp
riately selected (and usually plenty of) training data. Current approaches
that follow this new paradigm distinguish themselves through solution acc
uracies paired with computational efficieny that were previously unconceiv
able.\nIn this talk I will provide a glimpse into such deep learning appro
aches and some of their mathematical properties. I will finish with open p
roblems and future research perspectives.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andreas Bärmann/Kevin Aigner (FAU Erlangen-Nürnberg)
DTSTART:20210706T101500Z
DTEND:20210706T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/12
DESCRIPTION:Title: Online Learning for Optimization Problems with Unknown or Uncerta
in Cost Functions\nby Andreas Bärmann/Kevin Aigner (FAU Erlangen-Nür
nberg) as part of Mathematics of Deep Learning\n\n\nAbstract\nThe first pa
rt of the talk begins by recapitulating several basic algorithms and resul
ts in online learning\, in particular the multiplicative weights method an
d online gradient descent. Based on these algorithms\, we demonstrate how
to learn the objective function of a decision-maker while only observing t
he problem input data and the decision-maker’s corresponding decisions o
ver multiple rounds. Our approach works for linear objectives over arbitra
ry feasible sets for which we have a linear optimization oracle. The two e
xact algorithms we present – based on multiplicative weights updates and
online gradient descent respectively – converge at a rate of $O(1/\\sqr
t T)$ and thus allow taking decisions which are essentially as good as tho
se of the observed decision-maker already after relatively few observation
s. We show the effectiveness and possible applications of our methods in a
broad computational study. This is joint work with Alexander Martin\, Seb
astian Pokutta and Oskar Schneider.\n\nIn the second part of the talk\, we
consider the robust treatment of stochastic optimization problems involvi
ng random vectors with unknown discrete probability distributions. With th
is problem class\, we demonstrate the basic concepts of data-driven optimi
zation under uncertainty. Furthermore\, we introduce a new iterative appro
ach that uses scenario observations to learn more about the uncertainty ov
er time. This means our solutions become less and less conservative\, inte
rpolating between distributionally robust and stochastic optimization. We
achieve this by solving the distributionally robust optimization problem o
ver time via an online-learning approach while iteratively updating the am
biguity sets. We provide a regret bound for the quality of the obtained so
lutions that converges at a rate of $O(\\log T/T)$ and illustrate the effe
ctiveness of our procedure by numerical experiments. Our proposed algorith
m is able to solve the online learning problem significantly faster than e
quivalent reformulations. This is joint work with Kristin Braun\, Frauke L
iers\, Sebastian Pokutta\, Oskar Schneider\, Kartikey Sharma and Sebastian
Tschuppik.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne Koelewijn (FAU Erlangen-Nürnberg)
DTSTART:20210713T101500Z
DTEND:20210713T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/13
DESCRIPTION:Title: Biomechanics meets Deep Learning\nby Anne Koelewijn (FAU Erla
ngen-Nürnberg) as part of Mathematics of Deep Learning\n\n\nAbstract\nBio
mechanics is the study of human movement. Until recently\, artificial inte
lligence (AI) or deep learning was hardly used in biomechanics research\,
but instead it was mainly based on physical models and experiments. Howeve
r\, recently deep learning has also become increasingly important in the f
ield of biomechanics. This talk will discuss different ways how biomechani
cs and deep learning can be combined to improve research outcomes in movem
ent analysis. In the first part of the talk\, we start with a general intr
oduction into movement analysis\, and discuss more traditional methods tha
t are used in the field. Mainly\, we will cover how gait simulations can b
e created by solving trajectory optimization problems\, since here many be
nefits of adding AI/deep learning can be identified. In the second part of
the talk\, we will discuss the combination of biomechanics and deep learn
ing. First\, we will discuss different ways to improve biomechanics models
with deep learning\, and highlight one example regarding energy expenditu
re models. Finally\, we will discuss how gait simulations can be used to i
mprove outcomes of deep learning models\, by creating larger datasets.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jevgenija Rudzusika (KTH Stockholm)
DTSTART:20210720T101500Z
DTEND:20210720T114500Z
DTSTAMP:20260404T095134Z
UID:MathDeep/14
DESCRIPTION:Title: Accelerated Forward-Backward Optimization using Deep Learning
\nby Jevgenija Rudzusika (KTH Stockholm) as part of Mathematics of Deep Le
arning\n\n\nAbstract\nWe propose several deep-learning accelerated optimiz
ation solvers with convergence guarantees. We use ideas from the analysis
of accelerated forward-backward schemes like FISTA\, but instead of the cl
assical approach of proving convergence for a choice of parameters\, such
as a step-size\, we show convergence whenever the update is chosen in a sp
ecific set. Rather than picking a point in this set using some predefined
method\, we train a deep neural network to pick the best update. Finally\,
we show that the method is applicable to several cases of smooth and non-
smooth optimization and show superior results to established accelerated s
olvers.\n
LOCATION:https://stable.researchseminars.org/talk/MathDeep/14/
END:VEVENT
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