How Neighborly Can a Centrally Symmetric Polytope Be?
Discrete & Computational Geometry
Reproducible Research in Computational Harmonic Analysis
Computing in Science and Engineering
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
Dense error correction via l1-minimization
IEEE Transactions on Information Theory
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In "Counting faces of randomly projected polytopes when the projection radically lowers dimension" the authors proved an asymptotic sampling theorem for sparse signals, showing that n random measurements permit to reconstruct an N-vector having k nonzeros provided n 2 ċ k ċ log(N/n)(1+o(1)) reconstruction uses l1 minimization. They also proved an asymptotic rate theorem, showing existence of real error-correcting codes for messages of length N which can correct all possible k-element error patterns using just n generalized checksum bits, where n 2e ċ k ċ log(N/n)(1+o(1)) decoding uses l1 minimization. Both results require an asymptotic framework, with N growing large. For applications, on the other hand, we are concerned with specific triples k, n, N. We exhibit triples (k, n, N) for which Compressed Sensing Matrices and Real Error-Correcting Codes surely exist and can be obtained with high probability by random sampling. These derive from exponential bounds on the probability of drawing 'bad' matrices. The bounds give conditions effective at finite-N, and converging to the known sharp asymptotic conditions for large N. Compared to other finite-N bounds known to us, they are much stronger, and much more explicit. Our bounds derive from asymptotics in "Counting faces of randomly projected polytopes when the projection radically lowers dimension" counting the expected number of k-dimensional faces of the randomly projected simplex TN-1 and cross-polytope CN. We develop here finite-N bounds on the expected discrepancy between the number of k-faces of the projected polytope AQ and its generator Q, for Q = TN-1 and CN. Our bounds also imply existence of interesting geometric objects. Thus, we exhibit triples (k, n, N) for which polytopes with 2N vertices can be centrally k-neighborly.