Solution-recovery in l1-norm for non-square linear systems - CiteSeerX
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Solution-recovery in l1-norm for non-square linear systems - CiteSeerX
two âsolution-recoveryâ problems have been the focus of a number of recent works. .... was corrupted twice (i.e., h1 and h2 did not have non-zero entries at the ...
Solution-recovery in `1-norm for non-square linear systems: deterministic conditions and open questions∗ Yin Zhang Technical Report TR05-06 Department of Computational and Applied Mathematics Rice University, Houston, Texas, 77005, U.S.A. June, 2005, Revised August, 2005
Abstract Consider an over-determined linear system AT x ≈ b and an under-determined linear
system By = c. Given b = AT xˆ + h, under what conditions xˆ will minimize the residual AT x − b in `1 -norm (i.e., khk1 = minx kAT x − bk1 )? On the other hand, given c = Bh,
under what conditions h will be the minimum `1 -norm solution of By = c? These
two “solution-recovery” problems have been the focus of a number of recent works. Moreover, these two problems are equivalent under appropriate conditions on the data sets (A, b) and (B, c). In this paper, we give deterministic conditions for these solutionrecoverary problems and raise a few open questuions. Some of the results in this paper are already known or partially known, but our derivations are different and thus may provide different perspectives.
1
Introduction
Let us consider the `1 -norm approximation of an over-determined linear system: (O1) :
