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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 13:25:52 +0000Error correction via linear programming
https://resolver.caltech.edu/CaltechAUTHORS:20110609-075242535
Authors: {'items': [{'id': 'Candes-E', 'name': {'family': 'Candes', 'given': 'Emmanuel'}}, {'id': 'Rudelson-M', 'name': {'family': 'Rudelson', 'given': 'Mark'}}, {'id': 'Tao-T', 'name': {'family': 'Tao', 'given': 'Terence'}}, {'id': 'Vershynin-R', 'name': {'family': 'Vershynin', 'given': 'Roman'}}]}
Year: 2005
DOI: 10.1109/SFCS.2005.5464411
Suppose we wish to transmit a vector f Є 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 the corrupted m-dimensional vector y? This paper proves that under suitable conditions on the coding matrix A, the input f is the unique solution to the ℓ_1 -minimization problem (‖x‖ℓ_1: = ∑_i |xi|) min ‖y − Ag‖ℓ_1 g^∈Rn provided that the fraction of corrupted entries is not too large, i.e. does not exceed some strictly positive constant ρ ∗ (numerical values for ρ ^∗ are given). In other words, f can be recovered exactly by solving a simple convex optimization problem; in fact, a linear program. We report on numerical experiments suggesting that ℓ_1-minimization is amazingly effective; f is recovered exactly even in situations where a very significant fraction of the output is corrupted.
In the case when the measurement matrix A is Gaussian,
the problem is equivalent to that of counting lowdimensional
facets of a convex polytope, and in particular
of a random section of the unit cube. In this case we can
strengthen the results somewhat by using a geometric functional
analysis approach.https://authors.library.caltech.edu/records/38b5r-q6w74