Machine generated contents note: 1. Introduction to compressed sensing Mark A. Davenport, Marco F. Duarte, Yonina C. Eldar and Gitta Kutyniok; 2. Second generation sparse modeling: structured and collaborative signal analysis Alexey Castrodad, Ignacio Ramirez, Guillermo Sapiro, Pablo Sprechmann and Guoshen Yu; 3. Xampling: compressed sensing of analog signals Moshe Mishali and Yonina C. Eldar; 4. Sampling at the rate of innovation: theory and applications Jose Antonia Uriguen, Yonina C. Eldar, Pier Luigi Dragotta and Zvika Ben-Haim; 5. Introduction to the non-asymptotic analysis of random matrices Roman Vershynin; 6. Adaptive sensing for sparse recovery Jarvis Haupt and Robert Nowak; 7. Fundamental thresholds in compressed sensing: a high-dimensional geometry approach Weiyu Xu and Babak Hassibi; 8. Greedy algorithms for compressed sensing Thomas Blumensath, Michael E. Davies and Gabriel Rilling; 9. Graphical models concepts in compressed sensing Andrea Montanari; 10. Finding needles in compressed haystacks Robert Calderbank, Sina Jafarpour and Jeremy Kent; 11. Data separation by sparse representations Gitta Kutyniok; 12. Face recognition by sparse representation Arvind Ganesh, Andrew Wagner, Zihan Zhou, Allen Y. Yang, Yi Ma and John Wright.

Compressed sensing : theory and applications

In this paper the sequence Banach space ψ (Z) is defined for a class of convex functions ψ , and properties of the Kand Jinterpolation spaces (E0,E1)θ ,ψ,K and (E0,E1)θ ,ψ,J for a Banach couple E = (E0,E1) and θ ∈ (0,1) are studied. Mathematics subject classification (2000): 46B70.

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On ψ- interpolation spaces

CONTENTSIntroduction Chapter I. Existence and uniqueness of generalized solutions of the Cauchy problem and of initial boundary-value problems ??1.1. Theorems of existence and non-existence ??1.2. Uniqueness theorems Chapter II. Estimates and differential properties of generalized solutions ??2.1. The character of the dependence on the data ??2.2. Regularity and non-regularity Chapter III. Behaviour of generalized solutions for small values of time ??3.1. Finite rate of propagation of perturbations ??3.2. Inertia ??3.3. Shrinking of the size of the support Chapter IV. Behaviour of generalized solutions for large values of time ??4.1. Stabilization in finite time ??4.2. Localization of perturbations ??4.3. Asymptotic formulae References

http://iopscience.iop.org/article/10.1070/RM1987v042n02ABEH001309/pdf

Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations

Consider an $m \times N$ matrix $\Phi$ with the restricted isometry property of order k and level $\delta$; that is, the norm of any k-sparse vector in $\mathbb{R}^N$ is preserved to within a multiplicative factor of $1 \pm \delta$ under application of $\Phi$. We show that by randomizing the column signs of such a matrix $\Phi$, the resulting map with high probability embeds any fixed set of $p = O(e^k)$ points in $\mathbb{R}^N$ into $\mathbb{R}^m$ without distorting the norm of any point in the set by more than a factor of $1 \pm 4 \delta$. Consequently, matrices with the restricted isometry property and with randomized column signs provide optimal Johnson–Lindenstrauss embeddings up to logarithmic factors in N. In particular, our results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices; for partial Fourier and partial Hadamard matrices, we improve the recent bound $m \gtrsim \delta^{-4} \log(p) \log^4(N)$ given by Ailon and Liberty to $m...

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New and Improved Johnson–Lindenstrauss Embeddings via the Restricted Isometry Property

Linear Inverse Problems.- Large-Scale Inverse Problems in Imaging.- Regularization Methods for Ill-Posed Problems.- Distance Measures and Applications to Multi-Modal Variational Imaging.- Energy Minimization Methods.- Compressive Sensing.- Duality and Convex Programming.- EM Algorithms.- Iterative Solution Methods.- Level Set Methods for Structural Inversion and Image Reconstructions.- Expansion Methods.- Sampling Methods.- Inverse Scattering.- Electrical Impedance Tomography.- Synthetic Aperture Radar Imaging.- Tomography.- Optical Imaging.- Photoacoustic and Thermoacoustic Tomography: Image Formation Principles.- Mathematics of Photoacoustic and Thermoacoustic Tomography.- Wave Phenomena.- Statistical Methods in Imaging.- Supervised Learning by Support Vector Machines.- Total Variation in Imaging.- Numerical Methods and Applications in Total Variation Image Restoration.- Mumford and Shah Model and its Applications in Total Variation Image Restoration.- Local Smoothing Neighbourhood Filters.- Neighbourhood Filters and the Recovery of 3D Information.- Splines and Multiresolution Analysis.- Gabor Analysis for Imaging.- Shaper Spaces.- Variational Methods in Shape Analysis.- Manifold Intrinsic Similarity.- Image Segmentation with Shape Priors: Explicit Versus Implicit Representations.- Starlet Transform in Astronomical Data Processing.- Differential Methods for Multi-Dimensional Visual Data Analysis.- Wave fronts in Imaging, Quinto.- Ultrasound Tomography, Natterer.- Optical Flow, Schnoerr.- Morphology, Petros.- Maragos.- PDEs, Weickert. - Registration, Modersitzki. - Discrete Geometry in Imaging, Bobenko, Pottmann.-Visualization, Hege.- Fast Marching and Level Sets, Osher.- Couple Physics Imaging, Arridge.- Imaging in Random Media, Borcea.- Conformal Methods, Gu.- Texture, Peyre.- Graph Cuts, Darbon.- Imaging in Physics with Fourier Transform (i.e. Phase Retrieval e.g Dark field imaging), J. R. Fienup.- Electron Microscopy, Oktem Ozan.- Mathematical Imaging OCT (this is also FFT based), Mark E. Brezinski.- Spect, PET, Faukas, Louis.

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Handbook of mathematical methods in imaging

We give a new explicit construction of n×N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ϵ>0, large N, and any n satisfying N1−ϵ≤n≤N, we construct RIP matrices of order k≥n1/2+ϵ and constant δ=n−ϵ. This overcomes the natural barrier k=O(n1/2) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose kth moments are uniformly small for 1≤k≤N (Turan's power sum problem), which improves upon known explicit constructions when (logN)1+o(1)≤n≤(logN)4+o(1). This latter construction produces elementary explicit examples of n×N matrices that satisfy the RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (logN)1+o(1)≤n≤(logN)5/2+o(1).

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Explicit constructions of RIP matrices and related problems

Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best n-term approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases.

The Thresholding Greedy Algorithm, Greedy Bases, and Duality

We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then X has an almost greedy basis. We show that c0 is the only L∞ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized.

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Denka Kutzarova

Papers

Explicit constructions of RIP matrices and related problems

The Thresholding Greedy Algorithm, Greedy Bases, and Duality

On the existence of almost greedy bases in Banach spaces

k-β and k-nearly uniformly convex Banach spaces

Modified mixed Tsirelson spaces