## Tuesday, April 30, 2013

### Nuit Blanche in Review (April 2013)

In this past month, we've had:
• Eleven implementations made available by their authors and soon to be added to the Reproducible Research Page. Let us note the appearance of an implementation toward compressive hardware calibration, one on nonlinear compressive sensing, one on phase transition determination, one on the connection between the cross and bouquet approach and a greedy algorithm and finally field reconstruction issues.
• Four Sunday Morning Insights including a review. Guess which one got the most attention ?
• Four meetings announcements
• Two posts on hardware development, mostly using current camera technology.
• Six posts of featured Papers and preprints
• Four Nuit Blanche Reader's reviews and Around the webs
• one job announcement, one dataset and one op-ed.

### Big data: theoretical and practical challenges May 14-15, 2013, Paris

Francis Bach just let me know of this upcoming meeting on Big data: theoretical and practical challenges on May 14-15, 2013 in Paris. From the meeting's page:

Science and business are becoming increasingly driven by large amounts of data. This creates new practical and theoretical challenges, at the interface between statistics and computer science. The goal of the workshop is to bring together world-leaders in all aspects of large-scale inference, including computer scientists, statisticians and mathematicians, and explore the challenges and opportunities offered by Big data analysis.
The workshop will be held on May 14-15, 2013, at the Institut Henri Poincaré, located downtown Paris (http://www.ihp.fr/), is organized by Francis Bach and Michael Jordan, and is funded by the Fondation de Sciences Mathematiques de Paris and INRIA.
Invited speakers:

• Alexandre d'Aspremont, CNRS - Ecole Polytechnique
• Francis Bach, INRIA - ENS
• Leon Bottou, Microsoft Research
• Alfred Hero, University of Michigan
• Chris Holmes, Oxford University
• Michael Jordan, U.C. Berkeley
• Lester Mackey, Stanford University
• Michael Mahoney, Stanford University
• Eric Moulines, Telecom Paristech
• Sonia Petrone, Università Bocconi
• Ion Stoica, U.C. Berkeley
• Martin Wainwright, U.C. Berkeley

## Monday, April 29, 2013

### Slides: Fête Parisienne in Computation, Inference and Optimization: A Young Researchers' Forum

Thanks to Francis Bach, we now have the slides of the presentation made at the "Fête Parisienne in Computation, Inference and Optimization: A Young Researchers' Forum­" he co-organized and that we mentioned earlier. From the program page, here are the attendant slides:

PROGRAM
9-9.30
Welcome
9.30-10.05
Yee Whye Teh (Oxford University)
Fast MCMC sampling for Markov jump processes and extensions
10.05-10.40
Francois Caron (INRIA Bordeaux - Sud-Ouest)
Bayesian nonparametric models for bipartite graphs
10.40-11.10
pause cafe
11.10-11.45
Nicolas Chopin (CREST - ENSAE)
EP-ABC: Expectation-Propagation for Likelihood-Free Inference
11.45-12.20
Igor Pruenster (University of Torino & Collegio Carlo Alberto)
On some distributional properties of Gibbs-type priors
12.20-12.55
Sylvain Arlot (CNRS - Ecole Normale Supérieure)
Optimal model selection with V-fold cross-validation: how should V be chosen?
12.55-14.30
Dejeuner - Session Poster
14.30-15.05
Aurélien Garivier (Institut Mathématique de Toulouse, Université Paul Sabatier)
Dynamic resource allocation as an estimation problem
15.05-15.40
Aarti Singh (Carnegie Mellon University)
Optimal convex optimization under Tsybakov noise through reduction toactive learning
15.40-16.10
Pause cafe
16.10-16.45
Zaid Harchaoui (INRIA Grenoble - Rhône-Alpes)
Large-scale learning with conditional gradient algorithms
16.45-17.20
Peter Richtarik (University of Edinburgh)
Mini-batch primal and dual methods for support vector machines
17.20-17.55
Guillaume Obozinski (Ecole des Ponts - Paristech)­
Convex relaxation for Combinatorial Penalties
POSTERS
Djalel Benbouzid, "Fast classification using sparse decision DAGs"
Pierre Chiche, "Central kernels for compact groups"
Pierre Gaillard, "Mirror descent meets fixed-share (and feels no regret)"
Edouard Grave, "Hidden Markov tree model for semantic class induction"
Emilie Kaufmann, "Improved bandit algorithms: Go Bayesian!"
Azadeh Khaleghi, "Temporal Clustering of Highly Dependent Data"
Simon Lacoste-Julien, "Block-Coordinate Frank-Wolfe for Structural SVMs"
Aurore Lomet, "Model selection in block clustering by the ICL"
Emile Richard, "Intersecting singularities for multi-structured estimation"
Sylvain Robbiano, "Ranking Ordinal data and aggregation of bipartite ranking rules"

......
Thanks Francis !

### Compressive Light Field Photography Using Overcomplete Dictionaries And Optimized Projections

After watching Saturday's video on the work at Duke on Hyperspectral video imaging, I was not expecting the item Kshitij Marwah just sent me:

Dear Igor,
Have been a great fan of your blog. There is some recent work I did on a new practical light field camera architecture using compressive sensing that, thanks to its great resolution, is very competitive to LYTRO. Though still a research prototype I wanted to draw your attention to it. This is the first practical and working light field camera based on CS techniques.This technique allows conversion of today's DSLRs into light field cameras as well so that people do not have to go and buy a new camera for refocusing and 3D.
Let me know what you think. Thanks.
regards,

Wow!  thanks Kshitij. Here is the paper and accompanying videos from the project page:

Light field photography has gained a significant research interest in the last two decades; today, commercial light field cameras are widely available. Nevertheless, most existing acquisition approaches either multiplex a low-resolution light field into a single 2D sensor image or require multiple photographs to be taken for acquiring a high-resolution light field. We propose a compressive light field camera architecture that allows for higher-resolution light fields to be recovered than previously possible from a single image. The proposed architecture comprises three key components:light field atoms as a sparse representation of natural light fields, an optical design that allows for capturing optimized 2D light field projections, and robust sparse reconstruction methods to recover a 4D light field from a single coded 2D projection. In addition, we demonstrate a variety of other applications for light field atoms and sparse coding techniques, including 4D light field compression and denoising.

## Sunday, April 28, 2013

### Sunday Morning Insight: Compressive Sensing, What is it good for ?

When Francesca invited me to give a talk at Supelec this past week, I was reminded that I really would have liked to have seen an expository presentation of compressive sensing that was not using always the same figures. In particular, Compressive sensing is always presented as some sort of convex programming / L_1 minimization issue which is certainly interesting but that doesn't allow for an exposition of the fact that, nowadays, the use of belief propagation or greedy solvers is not entirely relevant to that picture anymore. After the presentation I made at Supelec this week, I added a few things that I felt I did not have time to talk about. Without further due, here is an "improved" version of that presentation.

Thank you again to Francesca, Elsa and Pierre for the invitation, this exercise straightened some of my thoughts. I am going to add this presentation to the Learning Compressive Sensing page.

Join the CompressiveSensing subreddit or the Google+ Community and post there !
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.

## Saturday, April 27, 2013

### Video Compressive Sensing by Larry Carin ( compressive hyperspectral camera and Compressive video )

Yesterday's series of videos from Richard Hamming are probably sufficient for several Saturday Morning Video sessions. However, this presentation by Larry Carin at USNA just showed up on my radar screen and it is about Compressive hyperspectral camera and Compressive video and presented as an introduction to the subject to an audience that is not made up of specialists.

The supporting presentation slides are Dictionary Learning & Compressive Imaging by Lawrence CarinGuillermo Sapiro and David Brady

Join the CompressiveSensing subreddit or the Google+ Community and post there !
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.

## Friday, April 26, 2013

### It's Friday, It's Hamming's Time: Learning to Learn by Richard Hamming, "You Get What You Measure" / "Error-Correcting Codes"

Forget the MOOCs, it's Friday, it's Hamming Time! Here is a series of videos by Richard Hamming: "Learning to Learn" The Art of Doing Science and Engineering. I featured two at the top but the remaining thirty others are listed below. Which ones did you like ?

Hamming, "You Get What You Measure" (June 1, 1995)

And since we've had some error correcting papers today and yesterday, let's hear it from the horse's mouth:

Hamming, "Error-Correcting Codes" (April 21, 1995)

Here is the rest of the video series "Learning to Learn" The Art of Doing Science and Engineering  by Richard Hamming:

### Correcting Errors in Linear Measurements and Compressed Sensing of Multiple Sources - implementation -

Here is a cross-and-bouquet approach with a greedy algorithm this time. Of note this is the second paper on error correction in two days:

Correcting Errors in Linear Measurements and Compressed Sensing of Multiple Sources by Alexander PetukhovInna Kozlov. The abstract reads:
We present an algorithm for finding sparse solutions of the system of linear equations $\Phi\mathbf{x}=\mathbf{y}$ with rectangular matrices $\Phi$ of size $n\times N$, where $n$ lt N, when measurement vector y is corrupted by a sparse vector of errors e. We call our algorithm the ℓ1-greedy-generous (LGGA) since it combines both greedy and generous strategies in decoding. Main advantage of LGGA over traditional error correcting methods consists in its ability to work efﬁciently directly on linear data measurements. It uses the natural residual redundancy of the measurements and does not require any additional redundant channel encoding. We show how to use this algorithm for encoding-decoding multichannel sources. This algorithm has a signiﬁcant advantage over existing straightforward decoders when the encoded sources have different density/sparsity of the information content. That nice property can be used for very efﬁcient blockwise encoding of the sets of data with a non-uniform distribution of the information. The images are the most typical example of such sources.
The important feature of LGGA is its separation from the encoder. The decoder does not need any additional side information from the encoder except for linear measurements and the knowledge that those measurements created as a linear combination of different sources

The attendant code implementation is here. Thanks Alex

Join the CompressiveSensing subreddit or the Google+ Community and post there !
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.

### Greedy Approach for Subspace Clustering from Corrupted and Incomplete Data - implementation -

Greedy Approach for Subspace Clustering from Corrupted and Incomplete Data by Alexander PetukhovInna Kozlov. The abstract reads:
We describe the Greedy Sparse Subspace Clustering (GSSC) algorithm providing an efficient method for clustering data belonging to a few low-dimensional linear or affine subspaces from incomplete corrupted and noisy data. We provide numerical evidences that, even in the simplest implementation, the greedy approach increases the subspace clustering capability of the existing state-of-the art SSC algorithm significantly.
The attendant code implementation is here. Thanks Alex

Join the CompressiveSensing subreddit or the Google+ Community and post there !
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### Greedy Approach for Low-Rank Matrix Recovery - implementation -

Greedy Approach for Low-Rank Matrix Recovery by Alexander Petukhov, Inna Kozlov. The abstract reads:
We describe the Simple Greedy Matrix Completion Algorithm providing an efficient method for restoration of low-rank matrices from incomplete corrupted entries.
We provide numerical evidences that, even in the simplest implementation, the greedy approach may increase the recovery capability of existing algorithms significantly.

The attendant code implementation is here. Thanks Alex

Join the CompressiveSensing subreddit or the Google+ Community and post there !
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.

## Thursday, April 25, 2013

### Nuit Blanche Reader's Reviews: GlobalSIP, some preprints, abstracts of the Workshop on Sparse Representation of Functions: Analytic and Computational Aspects

If you recall Marco mentioned the GlobalSIP 2013 Symposium on: Low-Dimensional Models and Optimization in Signal Processing. Pierre just twitted about the GlobalSIP Graph Signal Processing workshop, while Waheed twitted about New Sensing and Statistical Inference Methods clearly GlobalSIP will be the place to be in December. Submission deadline for all these workshops is June 1, 2013.

Alex Petukhov sent me the following a week ago:

Dear Igor,

I'd like to present a few sparse recovery related preprints (joint with I.Kozlov)
for Nuit Blanche. One common idea behind the algorithms in all 3 preprints is  a greedy envelope over the existing "basic" algorithm. Generally, this idea is extendable on many other algorithms.

The first preprint is about error resilient compressed sensing. The second one is about low-rank matrix completion from corrupted and  incomplete samples. The third one is about subspace clustering from corrupted, incomplete, and noisy data.

The links to the preprints are here:

Best regards,
Alex Petukhov
Thanks Alex, I will feature those papers tomorrow!

I contacted Gitta Kutyniok and Marc Pfetsch about the availability of the book of abstract for the Workshop on Sparse Representation of Functions: Analytic and Computational Aspects that took place in Berlin, on December 10-14th last year. As it stands it is here. However, the pdf is not searchable and therefore invisible to the search engines. Gitta and Marc kindly sent me a searchable version (which will be posted on the site soon). In the meantime, here are the abstracts:

Abstracts
Communications Inspired Linear Discriminant Analysis
Robert Calderbank
Duke University, USA
We study the problem of supervised linear dimensionality reduction, taking an informationtheoretic viewpoint. The linear projection matrix is designed by maximizing the mutual information between the projected signal and the class label. By harnessing a recent theoretical result on the gradient of mutual information, the above optimization problem can be solved directly using gradient descent, without requiring simpliﬁcation of the objective function. Theoretical analysis and empirical comparison are made between the proposed method and two closely related methods, and comparisons are also made with a method in which Renyi entropy is used to deﬁne the mutual information (in this case the gradient may be computed simply, under a special parameter setting). Relative to these alternative approaches, the proposed method achieves promising results on real datasets.
Learning non-parametric basis independent models from point queries via low-rank methods
Volkan Cevher
EPFL Lausanne, Switzerland
We consider the problem of learning multi-ridge functions of the form f(x) = g(Ax) from point evaluations of f. We assume that the function f is deﬁned on an 2-ball in Rd , g is twice continuously diﬀerentiable almost everywhere, and A 2 Rk d is a rank k matrix, where k is much greater than d. We propose a randomized, polynomial-complexity sampling scheme for estimating such functions. Our theoretical developments leverage recent techniques from low rank matrix recovery, which enables us to derive a polynomial time estimator of the function f along with uniform approximation guarantees. We prove that our scheme can also be applied for learning functions of the form: f(x) = Pk i=1 gi(a Ti x), provided f satisﬁes certain smoothness
conditions in a neighborhood around the origin. We also characterize the noise robustness of the scheme. Finally, we present numerical examples to illustrate the theoretical bounds in action.
Tensor Sparsity of Solutions to High Dimensional PDEs
Wolfgang Dahmen
RWTH Aachen, Germany
Functions of a possibly very large number of variables arise for instance as solutions to parametric families of PDEs or of PDEs with a large number diﬀerentiable variables such as the Schrodinger equation in quantum chemistry or Fokker-Planck equations modeling dilute polymers. In this talk we discuss the (near) tensor sparsity of solutions to certain high dimensional diﬀusion equations resulting from operator splitting applied to Fokker-Planck equations. We report on results obtained jointly with L. Grasedyck, E. Süli, and partly with R. DeVore concerning the question: given (nearly) tensor sparse data for such equations, are the solutions (nearly) tensor sparse and if so, how can this be quantiﬁed?
TBA
Ingrid Daubechies
Duke University, USA
Sparse Stabilization and Optimal Control of the Cucker and Smale System
Massimo Fornasier
TU München, Germany
From a mathematical point of view self-organization can be described as steady patterns, to which certain dynamical systems modeling social dynamics tend autonomously to be attracted. In this talk we explore situations beyond self-organization, in particular how to externally control such dynamical systems in order to eventually enforce pattern formation also in those situations where this wished phenomenon does not result from spontaneous and autonomous convergence. Our focus is on dynamical systems of Cucker-Smale type, modeling consensus emergence, and we question the existence of stabilization and optimal control strategies which require the minimal amount of external intervention for nevertheless inducing consensus in a group of interacting agents. In mathematical terms, our main result realizes the connection between certain variational problems involving 1-norm terms and optimally sparse controls. Our ﬁndings can also be informally stated in terms of the general principle for which a policy maker should always consider more favorable to intervene with stronger actions on the fewest possible instantaneous optimal leaders than try to control more agents with minor strength, in order to achieve group consensus.
Sparse Representation in Hierarchical Tensor Formats
Lars Grasedyck
RWTH Aachen, Germany
The (numerical) linear algebra related to tensor computations plays an immensely powerful role in theoretical and applied sciences: applied mathematics, biology, chemistry, information sciences, physics and many other areas. In this talk we consider high-order or high-dimensional tensors, respectively functions, of the form
A 2 R n n; f : [0;1]d ! R as they appear in multivariate (discrete or continuous) approximation problems. These require special techniques in order to allow a representation and approximation for high dimension d (cf. the curse of dimension). A typical example is the canonical polyadic (CP) format Ai1;:::;id =Xrj=1Yd =1a ;j(i ); f(x1; : : : ; xd) = Xrj=1Yd =1f ;j(x ) with discrete, respectively continuous, functions a ;j : f1; : : : ; ng ! R; f ;j : [0;1] ! R that require O(d n r) degrees of freedom or the representation of dr one-dimensional functions f ;j. However, this simple format comes along with several numerical obstructions, because it is highly non-linear.
In the last four years some new hierarchical formats (namely Tensor Train and Hierarchical Tucker) have been developed and analysed. These include as a subset all tensors of CP format (rank parameter r ﬁxed), but in addition they allow for some nice linear algebra approaches, e.g. a hierarchical SVD in complexity O(d n r 2 +d r4). The hierarchical SVD is similar to the SVD for matrices in the sense that it allows us to reliably compute a quasi-best approximation in the hierarchical format. We will outline the basic linear algebra behind the new approaches as well as open questions and possible applications. In particular, we will consider a standard two- or three-dimensional

Sparse dictionary learning in the presence of noise & outliers
Rémi Gribonval
Projet METISS, IRISA, Rennes, France
A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various ﬁelds ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. Considering a probabilistic model of sparse signals, we show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations. Joint work with Rodolphe Jenatton & Francis Bach
Breaking the Coherence Barrier: Semi-random Sampling in Compressed Sensing
Anders Hansen
University of Cambridge, United Kingdom
It is well known that compressed sensing relies on two crucial assumptions: incoherence and sparsity. Although there are examples where one has both of these features, there are surprisingly many cases in applications where these criteria are not satisﬁed. In particular, Fourier sampling and wavelet or polynomial reconstruction (both essential in Magnetic Resonance Imaging (MRI)). To overcome this obstacle we introduce a new theory where the assumptions of incoherence and sparsity are replaced by two new features: asymptotic incoherence and asymptotic sparsity. These two ideas combined with diﬀerent semi-random sampling schemes allow for dramatic subsampling in cases that previously were limited by the lack of incoherence/sparsity.
Moreover, we demonstrate that asymptotic incoherence and asymptotic sparsity + semi-random sampling yield a rather intriguing eﬀect: higher resolution allows for substantially better subsampling than with lower resolution. In particular, this new technique is excellent for high resolution problems.
Low-rank tensor completion by Riemannian optimization
Daniel Kressner
EPFL Lausanne
The aim of tensor completion is to ﬁll in missing entries of a partially known tensor under a low-rank constraint. We propose a new algorithm that performs Riemannian optimization techniques on the manifolds of tensors of ﬁxed multilinear rank. More speciﬁcally, a variant of the nonlinear conjugate gradient method is developed. Particular attention is paid to the eﬃcient computation of the diﬀerent ingredients, including the gradient, retraction and vector transport. Examples with synthetic data demonstrate good recovery even if the vast majority of the entries is unknown. Finally, we illustrate the use of the developed algorithm for the approximation of functions with singularities.Joint work with Michael Steinlechner (EPFL) and Bart Vanderecken (U Princeton)
Clustered Sparsity and Separation of Cartoon and Texture
Gitta Kutyniok
TU Berlin, Germany
Natural images are typically a composition of cartoon and texture structures. A medical image might, for instance, show a mixture of gray matter and the skull cap. One common task is to separate such an image into two single images, one containing the cartoon part and the other containing the texture part. Using compressed sensing techniques, numerous inspiring empirical results have already been obtained.
In this paper we provide the ﬁrst thorough theoretical study of the separation of a combination of cartoon and texture structures in a model situation. The methodology we consider expands the image in a combined dictionary consisting of a shearlet tight frame and a Gabor tight frame and minimizes the 1 norm on the analysis side. Sparse approximation properties then force the cartoon components into the shearlet coeﬃcients and the texture components into the Gabor coeﬃcients, thereby separating the image. Utilizing the fact that the coeﬃcients are clustered geometrically and endowing a Gabor system with a scale, we prove that at suﬃciently ﬁne scales
arbitrarily precise separation is possible.
Sparse dual frames
Jakob Lemvig
TU Danmark, Danmark
Frames are generalizations of bases which lead to redundant signal expansions, and they play an important role in many applications, e.g., in the theory of nonuniform sampling, wireless communications, and Sigma-Delta quantization. Sparsity of frame vectors (in some ﬁxed orthonormal basis) is a new paradigm in frame theory that among other things allows for simple representation of the frame vectors and fast analysis and reconstruction procedures. Recently, a general construction method for sparse tight frames was considered in [Casazza, Heinecke, Krahmer, Kutyniok, Optimally sparse frames, IEEE Trans. IT]. In this talk, we study sparse dual frames of a given (not necessarily tight nor sparse) frame. We characterize the optimal sparsity level in the set of all duals and present numerical algorithms for ﬁnding sparse dual with desirable properties.
Joint work with Felix Krahmer and Gitta Kutyniok

Computational Solver Comparison for Basis Pursuit
Marc Pfetsch
The problem of ﬁnding a minimum 1-norm solution to an underdetermined linear system (basis pursuit) is undoubtedly one of the central problems in compressed sensing. Many specialized solution approaches have been proposed and (matlab) implementations are available. This talks presents an extensive numerical comparison of seven such solvers and discusses the benchmarking methodology. Moreover, we propose a heuristic optimality check (HOC) as a general tool for l_1-minimization, which often allows for early termination by “guessing” a primal-dual optimal pair based on an approximate support.
Joint work with Dirk Lorenz and Andreas Tillmann
Scalable Frames
Friedrich Philipp
TU Berlin, Germany
Tight frames can be characterized as those frames which possess optimal numerical stability properties. Here, we consider the question of modifying a general frame to generate a tight frame by simply scaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal matrix. A frame is called scalable, if such a diagonal matrix exists. We derive various characterizations of scalable frames and provide a geometrical interpretation of scalability in terms of conical surfaces. Finally, it is shown that – loosely speaking – the set of scalable frames is thin in the set of frames.

Function interpolation via weighted 1-minimization
Holger Rauhut
University of Bonn, Germany
We consider the problem of interpolating a function from sample values. Building on insights from compressive sensing, we study weighted 1-minimization as recovery/interpolation method. Assuming that the function to be recovered has a sparse expansion in terms of the Fourier system or more general orthonormal systems including orthogonal polynomials, compressive sensing predicts that the function can be recovered from few samples at randomly chosen locations via (unweighted) 1-minimization. In practice, however, strict sparsity occurs rarely and has to be replaced with approximate sparsity. While also this setup is already well-understood, we take
a step further and take into consideration that functions often possess a certain smoothness. Hence, low-order Fourier coeﬃcients are more likely to appear in the best s-term approximation than high-order Fourier coeﬃcients. Taking this observation into account, we use weighted lp norms with p 1 on the Fourier coeﬃcients in order to model the functions to be reconstructed. The corresponding natural recovery method turns out to be weighted l1-minimization. We will present theoretical results and promising numerical experiments. This approach is able to overcome certain limitations in the context of recovering spherical harmonic expansions. If time permits also connections to numerically solving parametric (stochastic) partial diﬀerential equations will be discussed.
Joint work with Rachel Ward
High accuracy ﬁnite frame quantization using Sigma-Delta schemes
Rayan Saab
Duke University, USA
In this talk, we address the digitization of oversampled signals in the ﬁnite-dimensional setting. In particular, we show that by quantizing the N-dimensional frame coeﬃcients of signals in Rd using Sigma-Delta quantization schemes, it is possible to achieve root- exponential accuracy in the oversampling rate := N=d (even when one bit per measurement is used). These are currently the best known error rates in this context. In particular, such error rates holds for an “optimal” family of frames (the Sobolev self-dual frames), as well as to the well-known Harmonic frames, and surprisingly to random frames from appropriate distributions. We also discuss connections and applications to quantization of compressed sensing measurements as well as open problems.
Joint work with F. Krahmer, R. Ward, and O. Yilmaz
BPdual: Another Solver for BP, BPDN, NNLS, and More
Michael Saunders
Stanford University, USA
Many imaging and compressed sensing applications seek sparse solutions to under-determined least-squares problems. The basis pursuit (BP) approach minimizes the 1-norm of the solution, and BP denoising (BPDN) balances it against the least-squares ﬁt. The duals of these problems are conventional linear and quadratic programs. We introduce the following parameterization of the BPDN problem and its dual:
min x; y kxk1 +12 kyk22 Ax + y = b;
maxybTy 12 kyk22e ATy e;
where e is a vectors of 1s. Exploring the eﬀectiveness of active-set methods for solving the BPDN dual led us to the generalized problems BP : min x; y c(x) Tx + 12 kyk22
Ax + y = b; QP : max y bTy  12 kyk22l ATy u;
where
cj(x) = (j if xj 0,uj if xj gt 0. Our algorithm for solving QP uniﬁes several existing algorithms and is applicable to large-scale examples.
Joint work with Michael Friedlander

Tensor completion and tensor recovery in recent tensor formats
Reinhold Schneider
TU Berlin, Germany
Hierarchical Tucker tensor format (Hackbusch) and a particular case Tensor Trains (TT) (Tyrtyshnikov) have been introduced recently oﬀering stable and robust approximation by a low order cost, and generalizing the well established Tucker format.

Branch & Cut for L0-Minimization
Andreas Tillmann
While there are many heuristics for ﬁnding a sparse solution to an underdetermined linear equation system, hardly any exact solution methods are known beside the trivial total enumeration procedure. Since ﬁnding the sparsest solution, i.e., minimizing the 0-quasinorm of  a vector x satisfying Ax = b, is NP-hard, there is little hope of devising a polynomial-time solution algorithm. In this talk, we discuss work in progress on a novel Branch & Cut scheme for 0-minimization.
Random Sampling in Numerical Linear Algebra
Sivan Toledo
University Tel Aviv, Israel
The talk will focus on random sampling in numerical linear algebra. Traditionally, algorithms in numerical linear algebra strive for accuracy (small residual errors), which straightforward random-sampling methods do not always deliver. The talk will explain how the accuracy of randomly-sampled approximations can be improved using residual-correction methods, such as LSQR. The talk will describe cases where we do have an appropriate residual correction method and cases where we do not have such methods. We will discuss both mixing-based (random projection) methods, which are appropriate for dense problems, and leverage-scorebased methods, which are appropriate for sparse problems.
Hammersley and Fibonacci QMC beat Sparse Grids
Tino Ullrich
Hausdorﬀ-Center for Mathematics/Institute for Numerical Simulation, Bonn, Germany
We constructively prove new asymptotically optimal error bounds for numerical integration in bivariate periodic Besov spaces with dominating mixed smoothness S r
p;qB(T2 ) where 1 p; q 1 and r gt 1=p. Our ﬁrst result uses Quasi-Monte Carlo integration on Fibonacci lattice rules and improves on the so far best known upper bound achieved by using cubature formula taking function values from a Sparse Grid. It is well known that there is no proper counterpart for Fibonacci lattice rules in higher dimensions. To this end, our second result is based on Hammersley (or Van der Corput) type point grids. Instead of exploiting a Hlawka-Zaremba type discrepancy duality, which is limited to 1=p lt r 1, we extend Hinrichs’ recent results to larger orders r, namely 1=p lt r lt 2. This direct approach is strongly conjectured to have a
proper counterpart for higher orders r and, in addition, for functions on the d-torus Td
. Last, but not least, we prove that any cubature rule based an a sparse grid in d dimensions has a signiﬁcantly worse error order than the previously described methods. These results are a ﬁrst step to understand the problem of optimal recovery of functions from a discrete set of function values from a completely new direction.
Nuclear norm optimization in system identiﬁcation
Lieven Vandenberghe
UCLA, USA
The nuclear norm (sum of singular values) of a matrix plays an important role in extensions of 1-norm techniques for sparse optimization to optimization problems involving matrix rank minimization. It has been used successfully for applications in machine learning, signal and image processing, and statistics. Formulations based on the nuclear norm penalty are also increasingly used in dynamical system identiﬁcation, as an alternative to the singular value decomposition commonly used for low-rank matrix approximations in subspace identiﬁcation algorithms. The nuclear norm approach is attractive for several reasons. It preserves linear structure (such as block Hankel structure) in the low-rank matrix approximation, additional constraints or regularization terms are easily included in the formulation, and missing measurement data can be
handled in a straightforward manner. In the talk we will discuss several system identiﬁcation algorithms based on the nuclear norm penalty, as well as ﬁrst-order algorithms for solving the resulting convex optimization problems.
Algorithms and complexity for nonnegative matrix factorization
Steve Vavasis
Nonnegative matrix factorization (NMF) is a tool for approximating a large nonnegative matrix with much smaller nonnegative sparse factors. Its power stems from its ability to automatically ﬁnd features in the large matrix. It is widely used in machine learning applications including classifying text, ﬁnding features in images, interpreting the results of bioarray experiments, and even analysis of musical compositions. Recently several results have emerged concerning the theoretical complexity of NMF: it is NP-hard in general, but some interesting special cases are solvable in polynomial time. Convex relaxation in particular has proven to be a powerful solution method. Guaranteed polynomial-time algorithms and corresponding complexity bounds are a very promising line of attack in understanding NMF.
Parts of this talk represent joint work with N. Gillis, X. V. Doan and K.-C. Toh.
Learning Functions of Few Arbitrary Linear Parameters in High Dimensions
Jan Vybiral
TU Berlin, Germany
We study the uniform approximation of functions of many variables with the following inner structure. We assume, that f(x) = g(Ax), where x 2 Rd , A is a k d matrix and g is a (smooth) function on Rk . Both g and A are unknown and their recovery is a part of the problem. Under certain smoothness and variation assumptions on the function g, and an arbitrary choice of the matrix A, we present a sampling choice of the points drawn at random for each function approximation and algorithms for computing the approximating function. Due to the arbitrariness of A, the choice of the sampling points will be according to suitable random distributions and our results hold with overwhelming probability. Our approach uses tools taken from the compressed sensing framework, recent Chernoﬀ bounds for sums of positive-semideﬁnite matrices, and classical stability bounds for invariant subspaces of singular value decompositions.
Symbology-based Algorithms for Robust Bar Code Recovery
Rachel Ward
University of Texas, USA
UPC bar codes can be characterized as sparse representations with respect to a certain symbology basis. We exploit this low-dimensional structure and introduce a greedy bar code reconstruction algorithm which can recover UPC bar codes from very noisy measurements and inaccurate parameter information. Extensions to general bar codes, radio-frequency identiﬁcation, and text denoising will be discussed
Gitta tells me that the searchable version will be on the site shortly. Thank you Gitta and Marc !

Image Credit: NASA/JPL-Caltech
This image was taken by Rear Hazcam: Right B (RHAZ_RIGHT_B) onboard NASA's Mars rover Curiosity on Sol 235 (2013-04-04 10:54:53 UTC).
Full Resolution

### Robust error correction for real-valued signals via message-passing decoding and spatial coupling

So we now have some use of the magic seeded matrices for error correcting codes:

We revisit the error correction scheme of real-valued signals when the codeword is corrupted by gross errors on a fraction of entries and a small noise on all the entries. Combining the recent developments of approximate message passing and the spatially-coupled measurement matrix in compressed sensing we show that the error correction and its robustness towards noise can be enhanced considerably. We discuss the performance in the large signal limit using previous results on state evolution, as well as for finite size signals through numerical simulations. Even for relatively small sizes, the approach proposed here outperforms convex-relaxation-based decoders.
The ASPICS solver with the attendant seeded matrices is located at: http://aspics.krzakala.org/

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## Wednesday, April 24, 2013

### Compressive Sensing: What Is It Good For ?

I just gave a talk at Supelec today. The whole presentation can be downloaded here [version 2 is here]. It was a good crowd. Thank you Francesca, Elsa and Pierre for the invitation.

PS: Yes, the title of this talk comes from Seinfeld's "The Marine Biologist".

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