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Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
TLDR
It is shown how one can reconstruct a piecewise constant object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f. Expand
Decoding by linear programming
  • E. Candès, T. Tao
  • Computer Science, Mathematics
  • IEEE Transactions on Information Theory
  • 15 February 2005
TLDR
F can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program) and numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. Expand
Stable signal recovery from incomplete and inaccurate measurements
Suppose we wish to recover a vector x_0 Є R^m (e.g., a digital signal or image) from incomplete and contaminated observations y = Ax_0 + e; A is an n by m matrix with far fewer rows than columns (n «Expand
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
  • E. Candès, T. Tao
  • Mathematics, Computer Science
  • IEEE Transactions on Information Theory
  • 25 October 2004
TLDR
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. Expand
The Power of Convex Relaxation: Near-Optimal Matrix Completion
  • E. Candès, T. Tao
  • Mathematics, Computer Science
  • IEEE Transactions on Information Theory
  • 9 March 2009
TLDR
This paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). Expand
Nonlinear dispersive equations : local and global analysis
Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations WaveExpand
Topics in Random Matrix Theory
  • T. Tao
  • Mathematics
  • 21 March 2012
The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the fieldExpand
The primes contain arbitrarily long arithmetic progressions
We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers ofExpand
Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$
The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown toExpand
On the multilinear restriction and Kakeya conjectures
We prove d-linear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families ofExpand
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