Optimization Methods for Frame Conditioning and Application to Graph Laplacian Scaling

Abstract

A frame is scalable if each of its vectors can be rescaled in such a way that the resulting set becomes a Parseval frame. In this paper, we consider four different optimization problems for determining if a frame is scalable. We offer some algorithms to solve these problems. We then apply and extend our methods to the problem of reweighing (finite) graph so as to minimize the condition number of the resulting Laplacian.

Tight Frame Graphs Arising as Line Graphs

“…the pump journal of undergraduate research 4 (2021), [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] We see in Figure 2 that the original frame is robust to the loss of any two vectors. On the other hand, the sequence F = {f 1 , f 2 , f 3 , f 3 } is a frame for R 2 that represents the diamond graph but which is robust to only one erasure, since reconstruction would be impossible in the event that both x, f 1 and x, f 2 were lost.…”

“…Now, L(Γ) can be decomposed as L(Γ) = B(Γ)B(Γ) * , where B = B(Γ) is an oriented incidence matrix of Γ. Since the frame operator must be full rank, the authors of [5] create an (n−1)×(n−1) matrix L 0 by restricting B to the (n−1)-dimensional subspace spanned by the orthonormal eigenvectors x 1 , . .…”