Results 1  10
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30
Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
, 2004
"... Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear m ..."
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Cited by 1513 (20 self)
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Suppose we are given a vector f in RN. How many linear measurements do we need to make about f to be able to recover f to within precision ɛ in the Euclidean (ℓ2) metric? Or more exactly, suppose we are interested in a class F of such objects— discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy ɛ? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal f ∈ F decay like a powerlaw (or if the coefficient sequence of f in a fixed basis decays like a powerlaw), then it is possible to reconstruct f to within very high accuracy from a small number of random measurements. typical result is as follows: we rearrange the entries of f (or its coefficients in a fixed basis) in decreasing order of magnitude f  (1) ≥ f  (2) ≥... ≥ f  (N), and define the weakℓp ball as the class F of those elements whose entries obey the power decay law f  (n) ≤ C · n −1/p. We take measurements 〈f, Xk〉, k = 1,..., K, where the Xk are Ndimensional Gaussian
A Probabilistic and RIPless Theory of Compressed Sensing
, 2010
"... This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models — e.g. Gaussian, frequency measurements — discussed in the literature, ..."
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Cited by 95 (3 self)
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This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models — e.g. Gaussian, frequency measurements — discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) — they make use of a much weaker notion — or a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.
Reconstruction and subgaussian operators in Asymptotic Geometric Analysis
 FUNCT. ANAL
"... We present a randomized method to approximate any vector v from some set T ⊂ R n. The data one is given is the set T, vectors (Xi) k i=1 of R n and k scalar products (〈Xi, v〉) k i=1, where (Xi) k i=1 are i.i.d. isotropic subgaussian random vectors in R n, and k ≪ n. We show that with high probabilit ..."
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Cited by 73 (13 self)
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We present a randomized method to approximate any vector v from some set T ⊂ R n. The data one is given is the set T, vectors (Xi) k i=1 of R n and k scalar products (〈Xi, v〉) k i=1, where (Xi) k i=1 are i.i.d. isotropic subgaussian random vectors in R n, and k ≪ n. We show that with high probability, any y ∈ T for which (〈Xi, y〉) k i=1 is close to the data vector (〈Xi, v〉) k i=1 will be a good approximation of v, and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}valued vectors with i.i.d, symmetric entries, yields new information on the geometry of faces of random {−1, 1}polytope; we show that a kdimensional random {−1, 1}polytope with n vertices is mneighborly for very large m ≤ ck / log(c ′ n/k). The proofs are � based on new estimates on the behavior of the empirical process supf∈F �k−1 �k i=1 f 2 (Xi) − Ef 2 � when F is a subset of the L2 sphere. The estimates are given in terms of the γ2 functional with respect to the ψ2 metric on F, and hold both in exponential probability and in expectation.
Euclidean embeddings in spaces of finite volume ratio via random matrices
 J. Reine Angew. Math
, 2005
"... Let (R N, � · �) be the space R N equipped with a norm � · � whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N × n matrix with N> n, whose entries are independent random variables satisfying some moment assumptions. We show that with hig ..."
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Cited by 16 (9 self)
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Let (R N, � · �) be the space R N equipped with a norm � · � whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N × n matrix with N> n, whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the ndimensional Euclidean space (R n,  · ) onto its image in (R N, � · �): there exist α, β> 0 such that for all x ∈ R n, α √ Nx  ≤ �Γx � ≤ β √ N x. This solves a conjecture of Schechtman on random embeddings of ℓ n 2 into ℓN 1. 1
Empirical processes with a bounded ψ1 diameter
, 2009
"... We study the empirical process supf∈F N −1 ∑N i=1 f 2 (Xi) − Ef 2 , where F is a class of meanzero functions on a probability space (Ω, µ) and (Xi) N i=1 are selected independently according to µ. We present a sharp bound on this supremum that depends on the ψ1 diameter of the class F (rather th ..."
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Cited by 16 (3 self)
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We study the empirical process supf∈F N −1 ∑N i=1 f 2 (Xi) − Ef 2 , where F is a class of meanzero functions on a probability space (Ω, µ) and (Xi) N i=1 are selected independently according to µ. We present a sharp bound on this supremum that depends on the ψ1 diameter of the class F (rather than on the ψ2 one) and on the complexity parameter γ2(F, ψ2). In addition, we present optimal bounds on the random diameters sup f∈F max I=m ( ∑ i∈I f 2 (Xi)) 1/2 using the same parameters. As applications, we extend several well known results in Asymptotic Geometric Analysis to any isotropic, logconcave ensemble on R n.
On weakly bounded empirical processes
 Math. Ann
, 2008
"... Let F be a class of functions on a probability space (Ω, µ) and let X1,..., Xk be independent random variables distributed according to µ. We establish high probability tail estimates of the form supf∈F {i: f(Xi)  ≥ t} using a natural parameter associated with F. We use this result to analyze we ..."
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Cited by 12 (6 self)
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Let F be a class of functions on a probability space (Ω, µ) and let X1,..., Xk be independent random variables distributed according to µ. We establish high probability tail estimates of the form supf∈F {i: f(Xi)  ≥ t} using a natural parameter associated with F. We use this result to analyze weakly bounded empirical processes indexed by F and processes of the form Zf = ∣k−1 ∑k i=1 fp(Xi) − Ef  p ∣ for p> 1. We also present some geometric applications of this approach, based on properties of the random operator Γ = k−1/2 ∑k i=1
Covering numbers and “low M∗estimate” for quasiconvex bodies
, 1996
"... This article gives estimates on covering numbers and diameters of random proportional sections and projections of symmetric quasiconvex bodies in R n. These results were known for the convex case and played an essential role in development of the theory. Because duality relations can not be appl ..."
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Cited by 11 (6 self)
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This article gives estimates on covering numbers and diameters of random proportional sections and projections of symmetric quasiconvex bodies in R n. These results were known for the convex case and played an essential role in development of the theory. Because duality relations can not be applied in the quasiconvex setting, new ingredients were introduced that give new understanding for the convex case as well.
Diameters of sections and coverings of convex bodies
 Journal Functional Analysis
"... We study the diameters of sections of convex bodies in RN determined by a random N × n matrix Γ, either as kernels of Γ ∗ or as images of Γ. Entries of Γ are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random vari ..."
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Cited by 7 (2 self)
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We study the diameters of sections of convex bodies in RN determined by a random N × n matrix Γ, either as kernels of Γ ∗ or as images of Γ. Entries of Γ are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random variables. We show that if a symmetric convex body K in RN has one well bounded kcodimensional section, then for any m> ck random sections of K of codimension m are also well bounded, where c ≥ 1 is an absolute constant. It is noteworthy that in the Gaussian case, when Γ determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c = 1. 0
A Geometric Lemma and Duality of Entropy Numbers
"... 1 Introduction We shall study in this note the following conjecture, to which we shall refer as the "Geometric Lemma"; we state it first in a somewhat imprecise form. Let n; N be positive integers with k: = log N o / n. If S ae IRn is a finite set whose cardinality doesn't exceed N an ..."
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Cited by 5 (3 self)
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1 Introduction We shall study in this note the following conjecture, to which we shall refer as the "Geometric Lemma"; we state it first in a somewhat imprecise form. Let n; N be positive integers with k: = log N o / n. If S ae IRn is a finite set whose cardinality doesn't exceed N and such that its convex hull K: = conv S admits an equally small Euclidean 1net (i.e., K can be covered by no more than N translates of the unit Euclidean ball D), then 12 D 6ae K.