Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces

Compressed sensing

This report demonstrates theoretically and empirically that a greedy algorithm called
Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension
d given O(mln d) random linear measurements of that signal. This is a massive improvement
over previous results, which require O(m2) measurements. The new results for OMP are comparable
with recent results for another approach called Basis Pursuit (BP). In some settings, the
OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal
recovery problems.

/pdf/signal-recovery-from-random-measurements-via-orthogonal-2uks7d8txo.pdf

Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case

The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

/pdf/the-probabilistic-method-5g63k0itke.pdf

The Probabilistic Method

Suppose we are given a vector f in a class FsubeRopfN , e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision epsi in the Euclidean (lscr2) metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the nth largest entry of the vector |f| (or of its coefficients in a fixed basis) obeys |f|(n)lesRmiddotn-1p/, where R>0 and p>0. Suppose that we take measurements yk=langf# ,Xkrang,k=1,...,K, where the Xk are N-dimensional Gaussian vectors with independent standard normal entries. Then for each f obeying the decay estimate above for some 0<p<1 and with overwhelming probability, our reconstruction ft, defined as the solution to the constraints yk=langf# ,Xkrang with minimal lscr1 norm, obeys parf-f#parlscr2lesCp middotRmiddot(K/logN)-r, r=1/p-1/2. There is a sense in which this result is optimal; it is generally impossible to obtain a higher accuracy from any set of K measurements whatsoever. The methodology extends to various other random measurement ensembles; for example, we show that similar results hold if one observes a few randomly sampled Fourier coefficients of f. In fact, the results are quite general and require only two hypotheses on the measurement ensemble which are detailed

/pdf/near-optimal-signal-recovery-from-random-projections-2mmu9nt75f.pdf

Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Introduction 1. Notation and preliminary background 2. Gaussian variables. K-convexity 3. Ellipsoids 4. Dvoretzky's theorem 5. Entropy, approximation numbers, and Gaussian processes 6. Volume ratio 7. Milman's ellipsoids 8. Another proof of the QS theorem 9. Volume numbers 10. Weak cotype 2 11. Weak type 2 12. Weak Hilbert spaces 13. Some examples: the Tsirelson spaces 14. Reflexivity of weak Hilbert spaces 15. Fredholm determinants Final remarks Bibliography Index.

The volume of convex bodies and Banach space geometry

The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C*-algebras. The third (and shorter) part of the book describes applications to non self-adjoint operator algebras, and similarity problems. In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of 'length' of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find that this book has much to offer.

/pdf/introduction-to-operator-space-theory-382yool2d5.pdf

Introduction to Operator Space Theory

Absolutely summing operators and basic applications Factorization through a Hilbert space Type and cotype. Kwapien's theorem The ""abstract"" version of Grothendieck's theorem Grothendieck's theorem Banach spaces satisfying Grothendieck's theorem Applications of the volume ratio method Banach lattices $C^*$-algebras Counterexamples to Grothendieck's conjecture.

Factorization of Linear Operators and Geometry of Banach Spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ
 x(ɛ)/ɛ
 r whenɛ → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(ɛ)/ɛ
 q → ∞ whenɛ → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.

Gilles Pisier

Papers

The volume of convex bodies and Banach space geometry

Introduction to Operator Space Theory

Factorization of Linear Operators and Geometry of Banach Spaces

Martingales with values in uniformly convex spaces

Probabilistic methods in the geometry of Banach spaces