v2010.10.26 - Convex Optimization

4.7. CONSTRAINING RANK OF INDEFINITE MATRICES 387perfect reconstruction? Does that number of samples represent compressionof the original data?We claim that perfect reconstruction of the MIT logo can be reliablyachieved with as few as 700 samples y=[y i ]∈ R 700 from this one-pixelcamera. That number represents only 19% of information obtainable from3726 micromirrors. 4.62 (Figure 114)Our approach to reconstruction is to look for low-rank solution to anunderdetermined system:find XX ∈ R 46×81subject to A vec X = yrankX ≤ 5(865)where vec X is the vectorized (37) unknown image matrix. Each row offat matrix A is one realization of a pseudorandom pattern applied to themicromirrors. Since these patterns are deterministic (known), then the i thsample y i equals A(i, :) vec Y ; id est, y = A vec Y . Perfect reconstructionhere means optimal solution X ⋆ equals scene Y ∈ R 46×81 to within machineprecision.Because variable matrix X is generally not square or positive semidefinite,we constrain its rank by rewriting the problem equivalentlyfindW 1 ∈ R 46×46 , W 2 ∈ R 81×81 , X ∈ R 46×81subject toXA vec[X = y]W1 XrankX T ≤ 5W 2(866)This rank constraint on the composite (block) matrix insures rank X ≤ 5for any choice of dimensionally compatible matrices W 1 and W 2 . But tosolve this problem by convex iteration, we alternate solution of semidefinite4.62 That number (700 samples) is difficult to achieve, as reported in [300,6]. If a minimalbasis for the MIT logo were instead constructed, only five rows or columns worth ofdata (from a 46×81 matrix) are linearly independent. This means a lower bound onachievable compression is about 5×46 = 230 samples plus 81 samples column encoding;which corresponds to 8% of the original information. (Figure 114)