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This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as(More)
This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f/spl isin/R/sup 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 suitable conditions(More)
Suppose we wish to recover a vector x0 ∈ R (e.g. a digital signal or image) from incomplete and contaminated observations y = Ax0 + 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 x0 accurately based on the data y? To recover x0, we consider the solution x to the `1-regularization problem min(More)
Suppose we are given a vector f in a class FsubeRopf<sup>N </sup>, 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 (lscr<sub>2</sub>) metric? This paper shows that if the objects of interest are sparse in a fixed basis or(More)
In many important statistical applications, the number of variables or parameters p is much larger than the number of observations n. Suppose then that we have observations y =Xβ+z, where β ∈R is a parameter vector of interest, X is a data matrix with possibly far fewer rows than columns, n≪ p, and the zi’s are i.i.d. N(0, σ ). Is it possible to estimate β(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 <i>matrix completion</i> problem, and comes up in a great number of applications, including the famous <i>Netflix Prize</i> and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from(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)
We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemerédi’s theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This(More)
First of all, we would like to thank all the discussants for their interest and comments, as well as for their thorough investigation. The comments all underlie the importance and timeliness of the topics discussed in our paper, namely, accurate statistical estimation in high dimensions. We would also like to thank the editors for this opportunity to(More)
Let A be a subset of a finite field F := Z/qZ for some prime q. If |F | < |A| < |F |1−δ for some δ > 0, then we prove the estimate |A+ A|+ |A · A| ≥ c(δ)|A|1+ε for some ε = ε(δ) > 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for(More)