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This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector f ∈ R n from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under(More)
This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f ∈ C N and a randomly chosen set of frequencies Ω. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set Ω? A typical result of this paper is as follows. Suppose that f is a(More)
Suppose we wish to recover a vector x 0 ∈ R m (e.g. a digital signal or image) from incomplete and contaminated observations y = Ax 0 + e; A is a n by m matrix with far fewer rows than columns (n m) and e is an error term. Is it possible to recover x 0 accurately based on the data y? To recover x 0 , we consider the solution x to the 1-regularization(More)
—Suppose we are given a vector f in a class F N , 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 in the Euclidean (` 2) metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to(More)
This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number(More)
We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrr odinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for(More)
Let G = (G, +) be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets A + B of finite sets A, B, and related objects such as iterated sumsets kA and difference sets A − B, while the inverse sumset theory of Freiman, Ruzsa, and others characterises those finite sets A for which A + A is small. In(More)
A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi's original combinatorial proof using the Szemerédi regularity lemma and van der(More)
Suppose we wish to transmit a vector f ∈ R n reliably. A frequently discussed approach consists in encoding f with an m by n coding matrix A. Assume now that a fraction of the entries of Af are corrupted in a completely arbitrary fashion. We do not know which entries are affected nor do we know how they are affected. Is it possible to recover f exactly from(More)