v2004.06.19 - Convex Optimization

4.16. SELF SIMILARITY... 201A(l) = 1 2n∑d i,i−l (586)which is a sum of some diagonal of the EDM D. To select the ij th entry ofmatrix ∆ ,i=1√dij ∆ = e T i∆e j = e T j∆e i = tr(e j e T i ∆) = 〈e i e T j , ∆〉 (587)where here,D ∆ = ∆ ◦ ∆ (588)and where ◦ denotes the Hadamard (entry-wise) product. Each scaled entry√2dij = √ 2 tr(e j e T i ∆) = 1 √2tr ( (e i e T j + e j e T i )∆ ) (589)is a coefficient of orthogonal projection of ∆ (§E.6.4) on the range of a vectorized(§2.1.1) member of the orthonormal basis for the vector space S M 0 :E ij = √ 1 ( )ei e T j + e j e T i , 1 ≤ i < j ≤ M (53)2The self-similarity function is therefore equivalent to the Parseval relation,[43] [109] [110] [111] giving a total energy of projection:A(l) =n∑tr(e i e T i−lD) = tri=1i > ln∑e i e T i−l D (590)i=1i > l