v2007.11.26 - Convex Optimization

276 CHAPTER 4. SEMIDEFINITE PROGRAMMINGthe most sparse solution is respectively x = [ 0 0 √ 2 ] T ∈ R 3 (confer (597))and x = [ 0 1 0 ] T ∈ R 3 . Given random data, in Matlab notation,A=randn(m,n), index=round((n−1)∗rand(1)) +1, b=A(:,index)(651)where m and n are selected arbitrarily, the sparsest solution is x=e index ∈ R nfrom the standard basis. Although these sparsest solutions are recoverableby inspection, we seek to discern them instead by convex iteration; namely,by iterating problem sequence (649) (435). From the numerical data given,cardinality ‖x‖ 0 = 1 is expected. Iteration continues until x T y vanishes (towithin some numerical precision); id est, until desired cardinality is achieved.All three examples return a correct cardinality-1 solution to withinmachine precision in few iterations, but are occasionally subject to stall.Stalls are remedied by reinitializing y to a random state. Stalling is not an inevitable behavior. Convex iteration succeeds, for someproblem types, almost all the time:4.5.1.0.2 Example. Signal dropout.Signal dropout is an old problem; well studied from both an industrial andacademic perspective. Essentially dropout means momentary loss or gap ina signal, while passing through some channel, caused by some man-madeor natural phenomenon. The signal lost is assumed completely destroyedsomehow. What remains within the time-gap is system or idle channel noise.The signal could be voice over Internet protocol (VoIP), for example, audiodata from a compact disc (CD) or video data from a digital video disc (DVD),a television transmission over cable or the airwaves, or a typically ravagedcell phone communication, etcetera.Here we consider signal dropout in a discrete-time signal corrupted byadditive white noise assumed uncorrelated to the signal. The linear channelis assumed to introduce no filtering. We create a discretized windowedsignal for this example by positively combining k randomly chosen vectorsfrom a discrete cosine transform (DCT) basis denoted Ψ∈ R n×n . Frequencyincreases, in the Fourier sense, from DC toward Nyquist as column index ofbasis Ψ increases. Otherwise, details of the basis are unimportant exceptfor its orthogonality Ψ T = Ψ −1 . Transmitted signal is denoteds = Ψz ∈ R n (652)