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Timetable (VMVW02)

Generative models, parameter learning and sparsity

Monday 30th October 2017 to Friday 3rd November 2017

Monday 30th October 2017
09:00 to 09:40 Registration
09:40 to 09:50 Welcome from Christie Marr (INI Deputy Director)
09:50 to 10:40 James Nagy (Emory University)
Spectral Computed Tomography
Co-authors: Martin Andersen (Technical University of Denmark), Yunyi Hu (Emory University)

An active area of interest in tomographic imaging is the goal of quantitative imaging, where in addition to producing an image, information about the material composition of the object is recovered. In order to obtain material composition information, it is necessary to better model of the image formation (i.e., forward) problem and/or to collect additional independent measurements. In x-ray computed tomography (CT), better modeling of the physics can be done by using the more accurate polyenergetic representation of source x-ray beams, which requires solving a challenging nonlinear ill-posed inverse problem. In this talk we explore the mathematical and computational problem of polyenergetic CT when it is used in combination with new energy-windowed spectral CT detectors. We formulate this as a regularized nonlinear least squares problem, which we solve by a Gauss-Newton scheme. Because the approximate Hessian system in the Gauss-Newton scheme is very ill-conditioned, we propose a preconditioner that effectively clusters eigenvalues and, therefore, accelerates convergence when the conjugate gradient method is used to solve the linear subsystems. Numerical experiments illustrate the convergence, effectiveness, and significance of the proposed method.
INI 1
10:40 to 11:10 Morning Coffee
11:10 to 12:00 Eldad Haber (University of British Columbia)
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INI 1
12:00 to 12:50 Christoph Brune (Universiteit Twente)
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INI 1
12:50 to 14:00 Lunch @ Wolfson Court
14:00 to 14:50 Lars Ruthotto (Emory University)
PDE-based Algorithms for Convolution Neural Network
This talk presents a new framework for image classification that exploits the relationship between the training of deep Convolution Neural Networks (CNNs) to the problem of optimally controlling a system of nonlinear partial differential equations (PDEs). This new interpretation leads to a variational model for CNNs, which provides new theoretical insight into CNNs and new approaches for designing learning algorithms. We exemplify the myriad benefits of the continuous network in three ways. First, we show how to scale deep CNNs across image resolutions using multigrid methods. Second, we show how to scale the depth of deep CNNS in a shallow-to-deep manner to gradually increase the flexibility of the classifier. Third, we analyze the stability of CNNs and present stable variants that are also reversible (i.e., information can be propagated from input to output layer and vice versa), which in combination allows training arbitrarily deep networks with limited computational resources. This is joint work with Eldad Haber (UBC), Lili Meng (UBC), Bo Chang (UBC), Seong-Hwan Jun (UBC), Elliot Holtham (Xtract Technologies)
INI 1
14:50 to 15:40 Gitta Kutyniok (Technische Universität Berlin)
Optimal Approximation with Sparsely Connected Deep Neural Networks
Despite the outstanding success of deep neural networks in real-world applications, most of the related research is empirically driven and a mathematical foundation is almost completely missing. One central task of a neural network is to approximate a function, which for instance encodes a classification task. In this talk, we will be concerned with the question, how well a function can be approximated by a neural network with sparse connectivity. Using methods from approximation theory and applied harmonic analysis, we will derive a fundamental lower bound on the sparsity of a neural network. By explicitly constructing neural networks based on certain representation systems, so-called $\alpha$-shearlets, we will then demonstrate that this lower bound can in fact be attained. Finally, we present numerical experiments, which surprisingly show that already the standard backpropagation algorithm generates deep neural networks obeying those optimal approximation rates. This is joint work with H. Bölcskei (ETH Zurich), P. Grohs (Uni Vienna), and P. Petersen (TU Berlin).
INI 1
15:40 to 16:00 Eva-Maria Brinkmann (Westfalische Wilhelms-Universitat Munster); (Westfalische Wilhelms-Universitat Munster)
Enhancing fMRI Reconstruction by Means of the ICBTV-Regularisation Combined with Suitable Subsampling Strategies and Temporal Smoothing
Based on the magnetic resonance imaging (MRI) technology, fMRI is a noninvasive functional neuroimaging method, which provides maps of the brain at different time steps, thus depicting brain activity by detecting changes in the blood flow and hence constituting an important tool in brain research.
An fMRI screening typically consists of three stages: At first, there is a short low-resolution prescan to ensure the proper positioning of the proband or patient. Secondly, an anatomical high resolution MRI scan is executed and finally the actual fMRI scan is taking place, where a series of data is acquired via fast MRI scans at consecutive time steps thus illustrating the brain activity after a stimulus. In order to achieve an adequate temporal resolution in the fMRI data series, usually only a specific portion of the entire k-space is sampled.
Based on the assumption that the full high-resolution MR image and the fast acquired actual fMRI frames share a similar edge set (and hence the sparsity pattern with respect to the gradient), we propose to use the Infimal Convolution of Bregman Distances of the TV functional (ICBTV), first introduced in \cite{Moeller_et_al}, to enhance the quality of the reconstructed fMRI data by using the full high-resolution MRI scan as a prior. Since in fMRI the hemodynamic response is commonly modelled by a smooth function, we moreover discuss the effect of suitable subsampling strategies in combination with temporal regularisation.

This is joint work with Julian Rasch, Martin Burger (both WWU Münster) and with Ville Kolehmainen (University of Eastern Finland).

[1] {Moeller_et_al} M. Moeller, E.-M. Brinkmann, M. Burger, and T. Seybold: Color Bregman TV. SIAM J. Imag. Sci. 7(4) (2014), pp. 2771-2806.
INI 1
16:00 to 16:30 Afternoon Tea
16:30 to 17:20 Joan Bruna (New York University); (University of California, Berkeley)
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INI 1
17:20 to 18:10 Justin Romberg (Georgia Institute of Technology)
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INI 1
18:10 to 19:10 Poster Session & Welcome Wine Reception at INI
Tuesday 31st October 2017
09:00 to 09:50 Peyman Milanfar (Google); (Google)
RED: Regularization by Denoising
Image denoising has reached impressive heights in performance and quality -- almost as good as it can ever get. But this isn't the only way in which tasks in image processing can exploit the image denoising engine. I will describe Regularization by Denoising (RED): using the denoising engine in defining the regularization of any inverse problem. We propose an explicit image-adaptive Laplacian-based regularization functional that makes the overall penalty defined by the denoiser clear and well-defined. With complete flexibility to choose the iterative optimization procedure for minimizing this functional, RED is capable of incorporating any image denoising algorithm as a regularizer, treat general inverse problems very effectively, and is guaranteed to converge to a globally optimal result. I will show examples of its utility, including state-of-the-art results in image deblurring and super-resolution problems. (Joint work with Yaniv Romano and Michael Elad) https://arxiv.org/pdf/1611.02862.pdf
INI 1
09:50 to 10:40 Ozan Öktem (KTH - Royal Institute of Technology); (Karolinska Institute)
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INI 1
10:40 to 11:10 Morning Coffee
11:10 to 12:00 Lior Horesh (IBM Research)
Accelerated Free-Form Model Discovery of Interpretable Models using Small Data
The ability to abstract the behavior of a system or a phenomenon and distill it into a consistent mathematical model is instrumental for a broad range of applications. Historically, models were manually derived in a first principles fashion. The first principles approach often offers the derivation of interpretable models of remarkable levels of universality using little data. Nevertheless, their derivation is time consuming and relies heavily upon domain expertise. Conversely, with the rising pervasiveness of data-driven approaches, the rapid derivation and deployment of models has become a reality. Scalability is gained through dependence upon exploitable structure (functional form). Such structures, in turn, yield non-interpretable models, require Big Data for training, and provide limited predictive power outside the training set span. In this talk, we will introduce an accelerated model discovery approach that attempts to bridge between the two conducts, to enable the discovery of universal, interpretable models, using Small Data. To accomplish that, the proposed algorithm searches for free-form symbolic models, where neither the structure nor the set of operator primitives are predetermined. The discovered models are provably globally optimal, promoting superior predictive power for unseen input. Demonstration of the algorithm in re-discovery of some fundamental laws of science will be provided, and references to on-going work in the discovery of new models for, hitherto, unexplainable phenomena.
INI 1
12:00 to 12:50 Martin Benning (University of Cambridge)
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INI 1
12:50 to 14:00 Lunch @ Wolfson Court
14:00 to 14:50 Alfred Hero (University of Michigan)
The tensor graphical lasso (Teralasso)
Co-authors: Kristjian Greenewald (Harvard University), Shuheng Zhou (University of Michigan), Alfred Hero (University of Michigan)

We propose a new ultrasparse graphical model for representing multiway data based on a Kronecker sum representation of the process inverse covariance matrix. This statistical model decomposes the inverse covariance into a linear Kronecker sum representation with sparse Kronecker factors.

Under the assumption that the multiway observations are matrix-normal the l1 sparsity regularized log-likelihood function is convex and admits significantly faster statistical rates of convergence than other sparse matrix normal algorithms such as graphical lasso or Kronecker graphical lasso.

We specify a scalable composite gradient descent method for minimizing the objective function and analyze both the statistical and the computational convergence ratesm, showing that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to the global minimizer. We will illustrate the method on several real multiway datasets, showing that we can recover sparse graphical structures in high dimensional data.

Related Links
INI 1
14:50 to 15:40 Francis Bach (INRIA Paris - Rocquencourt); (ENS - Paris)
Breaking the Curse of Dimensionality with Convex Neural Networks

We consider neural networks with a single hidden layer and non-decreasing positively homogeneous activation functions like the rectified linear units. By letting the number of hidden units grow unbounded and using classical non-Euclidean regularization tools on the output weights, they lead to a convex optimization problem and we provide a detailed theoretical analysis of their generalization performance, with a study of both the approximation and the estimation errors. We show in particular that they are adaptive to unknown underlying linear structures, such as the dependence on the projection of the input variables onto a low-dimensional subspace. Moreover, when using sparsity-inducing norms on the input weights, we show that high-dimensional non-linear variable selection may be achieved, without any strong assumption regarding the data and with a total number of variables potentially exponential in the number of observations. However, solving this convex optimization pro blem in infinite dimensions is only possible if the non-convex subproblem of addition of a new unit can be solved efficiently. We provide a simple geometric interpretation for our choice of activation functions and describe simple conditions for convex relaxations of the finite-dimensional non-convex subproblem to achieve the same generalization error bounds, even when constant-factor approximations cannot be found. We were not able to find strong enough convex relaxations to obtain provably polynomial-time algorithms and leave open the existence or non-existence of such tractable algorithms with non-exponential sample complexities.

Related links: http://jmlr.org/papers/volume18/14-546/14-546.pdf
 - JMLR paper
INI 1
15:40 to 16:00 Jonas Adler (KTH - Royal Institute of Technology)
Learned forward operators: Variational regularization for black-box models
In inverse problems, correct modelling of the forward model is typically one of the most important components to obtain good reconstruction quality. Still, most work is done on highly simplified forward models. For example, in Computed Tomography (CT), the true forward model, given by the solution operator for the radiative transport equation, is typically approximated by the ray-transform. The primary reason for this gross simplification is that the higher quality forward models are both computationally costly, and typically do not have an adjoint of the derivative of the forward operator that can be feasibly evaluated. The community is not un-aware of this miss-match, but the work has been focused on “the model is right, lets fix the data”. We instead propose going the other way around by using machine learning in order to learn a mapping from the simplified model to the complicated model using deep neural networks. Hence instead of learning how to correct complicated data so that it matches a simplified forward model, we accept that the data is always right and instead correct the forward model. We then use this learned forward operator, which is given as a composition of a simplified forward operator and a convolutional neural network, as a forward operator in a classical variational regularization scheme. We give a theoretical argument as to why correcting the forward model is more stable than correcting the data and provide numerical examples in Cone Beam CT reconstruction.
INI 1
16:00 to 16:30 Afternoon Tea
16:30 to 17:20 Julianne Chung (Virginia Polytechnic Institute and State University)
Advancements in Optimal Design and Hybrid Iterative Methods for Inverse Problems
In this talk, we consider two approaches to regularization. First, we describe a framework for solving inverse problems that incorporates probabilistic information in the form of training data. We provide theoretical results for the underlying Bayes risk minimization problem and discuss efficient approaches for solving the associated empirical Bayes risk minimization problem. Second, for very large-scale problems, we describe hybrid iterative approaches based on extensions of the Golub-Kahan bidiagonalization, where Tikhonov regularization is applied to the projected subproblem rather than the original problem. We show that hybrid methods have many benefits including avoiding semiconvergence behavior and being able to estimate the regularization parameter during the iterative process. Results from image processing will be presented.
INI 1
17:20 to 18:10 Andreas Hauptmann (University College London)
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INI 1
Wednesday 1st November 2017
09:00 to 09:50 Mila Nikolova (CNRS (Centre national de la recherche scientifique)); (ENS de Cachan)
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INI 1
09:50 to 10:40 Xavier Bresson (Nanyang Technological University)
Convolutional Neural Networks on Graphs
Convolutional neural networks have greatly improved state-of-the-art performances in computer vision and speech analysis tasks, due to its high ability to extract multiple levels of representations of data. In this talk, we are interested in generalizing convolutional neural networks from low-dimensional regular grids, where image, video and speech are represented, to high-dimensional irregular domains, such as social networks, telecommunication networks, or words' embedding. We present a formulation of convolutional neural networks on graphs in the context of spectral graph theory, which provides the necessary mathematical background and efficient numerical schemes to design fast localized convolutional filters on graphs. Numerical experiments demonstrate the ability of the system to learn local stationary features on graphs.
INI 1
10:40 to 11:10 Morning Coffee
11:10 to 12:00 Julie Delon (Université Paris Descartes)
High-Dimensional Mixture Models For Unsupervised Image Denoising (HDMI)
This work addresses the problem of patch-based image denoising through the unsupervised learning of a probabilistic high-dimensional mixture models on the noisy patches. The model, named HDMI, proposes a full modeling of the process that is supposed to have generated the noisy patches. To overcome the potential estimation problems due to the high dimension of the patches, the HDMI model adopts a parsimonious modeling which assumes that the data live in group-specific subspaces of low dimensionalities. This parsimonious modeling allows in turn to get a numerically stable computation of the conditional expectation of the image which is applied for denoising. The use of such a model also permits to rely on model selection tools to automatically determine the intrinsic dimensions of the subspaces and the variance of the noise. This yields a blind denoising algorithm that demonstrates state-of-the-art performance, both when the noise level is known and unknown. Joint work with Charles Bouveyron and Antoine Houdard.
INI 1
12:00 to 12:50 Bangti Jin (University College London)
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INI 1
12:50 to 14:00 Lunch @ Wolfson Court
14:00 to 17:00 Free Afternoon
Thursday 2nd November 2017
09:00 to 09:50 Richard Baraniuk (Rice University)
A Probabilistic Theory of Deep Learning
A grand challenge in machine learning is the development of computational algorithms that match or outperform humans in perceptual inference tasks that are complicated by nuisance variation. For instance, visual object recognition involves the unknown object position, orientation, and scale in object recognition while speech recognition involves the unknown voice pronunciation, pitch, and speed. Recently, a new breed of deep learning algorithms have emerged for high-nuisance inference tasks that routinely yield pattern recognition systems with near- or super-human capabilities. But a fundamental question remains: Why do they work? Intuitions abound, but a coherent framework for understanding, analyzing, and synthesizing deep learning architectures has remained elusive. We answer this question by developing a new probabilistic framework for deep learning based on the Deep Rendering Model: a generative probabilistic model that explicitly captures latent nuisance variation. By relaxing the generative model to a discriminative one, we can recover two of the current leading deep learning systems, deep convolutional neural networks and random decision forests, providing insights into their successes and shortcomings, a principled route to their improvement, and new avenues for exploration. This is joint work with Ankit Patel and Tan Nguyen of Rice University..
INI 1
09:50 to 10:40 Pierre Weiss (Université de Toulouse)
Generating sampling patterns in MRI
In this work I will describe a few recent results for the generation of sampling patterns in MRI. In the first part of my talk, I will provide mathematical models describing the sampling problem in MRI. This will allow me to show that the traditional way mathematicians look at an MRI scanner is usually way too idealized and that important ingredients are currently missing in the theories. The mathematical modelling shows that a natural way to generate a pattern consists in projecting a density onto a set of admissible measures. I will then describe two projection algorithms. The first one is based on a distance defined through a convolution mapping the measures to L^2, while the second is based on the L^2 transportation distance. After describing a few original applications of this formalism, I will show how it allows to significantly improve scanning times in MRI systems with real in vivo experiments. An outcome of this work is that compressed sensing, as it stands, only allows for moderate acceleration factors, while other ideas that take advantage of all the degrees of freedom of an MRI scanner yield way more significant improvements.
INI 1
10:40 to 11:10 Morning Coffee
11:10 to 12:00 Anders Hansen (University of Cambridge)
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INI 1
12:00 to 12:50 Josiane Zerubia (INRIA Sophia Antipolis)
Stochastic geometry for automatic object detection and tracking
In this talk, we combine the methods from probability theory and stochastic geometry to put forward new solutions to the multiple object detection and tracking problem in high resolution remotely sensed image sequences. First, we present a spatial marked point process model to detect a pre-defined class of objects based on their visual and geometric characteristics. Then, we extend this model to the temporal domain and create a framework based on spatio-temporal marked point process models to jointly detect and track multiple objects in image sequences. We propose the use of simple parametric shapes to describe the appearance of these objects. We build new, dedicated energy based models consisting of several terms that take into account both the image evidence and physical constraints such as object dynamics, track persistence and mutual exclusion. We construct a suitable optimization scheme that allows us to find strong local minima of the proposed highly non-convex energy. As the simulation of such models comes with a high computational cost, we turn our attention to the recent filter implementations for multiple objects tracking, which are known to be less computationally expensive. We propose a hybrid sampler by combining the Kalman filter with the standard Reversible Jump MCMC. High performance computing techniques are also used to increase the computational efficiency of our method. We provide an analysis of the proposed framework. This analysis yields a very good detection and tracking performance at the price of an increased complexity of the models. Tests have been conducted both on high resolution satellite and UAV image sequences
INI 1
12:50 to 14:00 Lunch @ Wolfson Court
14:00 to 14:50 Mario Figueiredo (Universidade de Lisboa); (Instituto Superior Técnico, Lisboa)
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INI 1
14:50 to 15:40 Marcelo Pereyra (Heriot-Watt University)
Bayesian model selection in imaging inverse problems
Most imaging inverse problems can be solved with different models operating under different assumptions about the data observation processes and the unknown image. As a result of these differences, the inferences about the unknown image may also be potentially very different. This talk presents a Bayesian model selection methodology to compare imaging models objectively, for scenarios with no ground truth available, and with a focus on convex problems. The proposed methodology is based on a new class of proximal Markov chain Monte Carlo algorithms. These algorithms scale very efficiently to high dimensions, have simple theoretical conditions for convergence, and are straightforward to apply to most convex problems that are currently solved by proximal optimisation. The proposed methodology is demonstrated by performing model selection in some challenging mathematical imaging problems related to image reconstruction.
INI 1
15:40 to 16:00 Pol del Aguila Pla (KTH - Royal Institute of Technology)
Cell Detection by Functional Inverse Diffusion and Group Sparsity
Biological assays in which particles generated by cells bind to a surface and can be imaged to reveal the cells' location and secretion properties are ubiquitous in biochemical, pharmacological and medical research. In this talk, I will first describe the physics of the problem, a 3D radiation-diffusion-adsorption-desorption PDE system, and present our novel parametrization of its solution in terms of convolutional operators. Secondly, I will present the functional optimization problem with group-sparsity regularization we propose to invert the problem and explain the physical, mathematical and heuristic reasoning behind our choice of regularizer. Thirdly, I will present the proofs needed to apply the accelerated proximal gradient algorithm to our problem, and justify why we chose to formulate the algorithm in the original function spaces that characterize the physical problem. Finally, I will present the details of the discretization we used to implement the resulting algorithm, and show its final performance both in simulated and real data.
INI 1
16:00 to 16:30 Afternoon Tea
16:30 to 17:20 Claire Boyer (Université Pierre et Marie Curie Paris)
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INI 1
17:20 to 18:10 Pierre Vanergheynst (EPFL - Ecole Polytechnique Fédérale de Lausanne)
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INI 1
19:30 to 22:00 Formal Dinner at Corpus Christi College
Friday 3rd November 2017
09:00 to 09:50 Irene Waldspurger (Université Paris-Dauphine); (INRIA Paris - Rocquencourt)
Alternating projections for phase retrieval with random sensing vectors
Phase retrieval consists in reconstructing an unknown element in a complex vector space from the modulus of linear measurements. The first reconstruction algorithms for which theoretical guarantees could be proven relied on convexification techniques. It has only recently been realized that similar guarantees hold for non-convex local search methods, that are faster and simpler, provided that their starting point is carefully chosen. We will explain how to establish these guarantees for the most well-known local search method: alternating projections. We will also discuss the role of the initialization method.
INI 1
09:50 to 10:40 Martin Holler (University of Graz)
Analysis and applications of structural-prior-based total variation regularization for inverse problems
INI 1
10:40 to 11:10 Morning Coffee
11:10 to 12:00 Raymond Chan (Chinese University of Hong Kong)
A Nuclear-norm Model for Multi-Frame Super-resolution Reconstruction
In this talk, we give a new variational approach to obtain super-resolution images from multiple low-resolution image frames extracted from video clips. First the displacement between the low-resolution frames and the reference frame are computed by an optical flow algorithm. The displacement matrix is then decomposed into product of two matrices corresponding to the integer and fractional displacement matrices respectively. The integer displacement matrices give rise to a non-convex low-rank prior which is then convexified to give the nuclear-norm regularization term. By adding a standard 2-norm data fidelity term to it, we obtain our proposed nuclear-norm model. Alternating direction method of multipliers can then be used to solve the model. Comparison of our method with other models on synthetic and real video clips shows that our resulting images are more accurate with less artifacts. It also provides much finer and discernable details. Joint work with Rui Zhao. Research supported by HKRGC.
INI 1
12:00 to 12:50 Mihaela Pricop-jeckstadt (Technische Universität Dresden)
Inverse pattern identification based on longitudinal clustering with applications to behavioural science
In this talk we consider an example-driven approach for identifying ability patterns from data partitions based on learning behaviour in the water maze experiment.  A modification of the k-means algorithm for longitudinal data as introduced in [1], based on the Mahalanobis distance metric (see e.g. [2]), is used to identify clusters that maximize the covariance between the psychological information and the learning variable (see [3]). Stability of the partition as well as reproducibility of the ability pattern is evaluated based on simulations, and the algorithm is applied to a data set originating in the psychological  tests for German fight pilots. 
References:1. Genolini, C.; Ecochard, R.; Benghezal, M. et al., ''kmlShape: An Efficient Method to Cluster Longitudinal Data (Time-Series) According to Their Shapes'', PLOS ONE,  Vol. 11, 2016.2. Sung, K.K. and Poggio, T., ''Example-based learning for view-based human face detection'',  IEEE Transactions on pattern analysis and machine intelligence, Vol. 20, 39-51, 1998. 3. Rosipal, R. and Kraemer, N. , ''Overview and recent advances in partial least squares'',  Subspace, latent structure and feature selection,  Book Series: Lecture Notes in Computer Science,  Vol. 3940, 34-51, 2006.
INI 1
12:50 to 14:00 Lunch @ Wolfson Court
14:00 to 14:50 Robert Plemmons (Wake Forest University)
Sparse Recovery Algorithms for 3D Imaging using Point Spread Function Engineering
Co-authors: Chao Wang (Mathematics, Chinese University of Hong Kong), Raymond Chan (Mathematics, Chinese University of Hong Kong), Sudhakar Prasad (Physics, University of New Mexico)

Imaging and localizing point sources with high accuracy in a 3D volume is an important but challenging task. For example, super-resolution 3D single molecule localization is an area of intense interest in biology (cell imaging, folding, membrane behavior, etc.), in chemistry (spectral diffusion, molecular distortions, etc.), and in physics (structures of materials, quantum optics, etc.). We consider here the high-resolution imaging problem of 3D point source image recovery from 2D data using methods based on point spread function (PSF) design. The methods involve a new technique, recently patented by S. Prasad, for applying rotating point spread functions with a single lobe to obtain depth from defocus. The amount of rotation of the PSF encodes the depth position of the point source. The distribution of point sources is discretized on a cubical lattice where the indexes of nonzero entries represent the 3D locations of point sources. The values of these entries are the point source fluxes. Finding the locations and fluxes is a large-scale sparse 3D inverse problem and we have developed solution algorithms based on sparse recovery using non-convex optimization. Applications to high-resolution single molecule localization microscopy are described, as well as localization of space debris using a space-based telescope. Sparse recovery optimization methods, including the Continuous Exact L0 (CEL0) algorithm, are used in our numerical experiments.
INI 1
14:50 to 15:40 Jeff Calder (University of Minnesota)
The weighted p-Laplacian and semi-supervised learning
Semi-supervised learning refers to machine learning algorithms that make use of both labeled data and unlabeled data for learning tasks. Examples include large scale nonparametric regression and classification problems, such as predicting voting preferences of social media users, or classifying medical images. In today's big data world, there is an abundance of unlabeled data, while labeled data often requires expert labeling and is expensive to obtain. This has led to a resurgence of semi-supervised learning techniques, which use the topological or geometric properties of large amounts of unlabeled data to aid the learning task. In this talk, I will discuss some new rigorous PDE scaling limits for existing semisupervised learning algorithms and their practical implications. I will also discuss how these scaling limits suggest new ideas for fast algorithms for semi-supervised learning. 
INI 1
15:40 to 16:10 Afternoon Tea
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons