Optimization methods for frame conditioning and application to graph Laplacian scaling

for all x∈H . When A = B the frame is said to be tight and if in addition, A = B = 1 it is termed a Parseval frame. When F = { fi}i=1 is a frame, we shall abuse notations and denote by F again, the n×M matrix whose ith column is fi, and where n is the dimension of H . Using this notation, the frame operator is the n×n matrix S=FF∗ where F∗ is the adjoint of F . It is a folklore to note that F is a frame if and only if S is a positive definite operator and the optimal lower frame bound, A, coincides with the lowest eigenvalue of S while the optimal upper frame bound, B, equals the largest eigenvalue of S. We refer to [?, ?, ?] for more details on frame theory. It is apparent that tight frames are optimal frames in the sense that the condition number of their frame operator is 1. We recall that, the condition number of a matrix A, denoted κ(A), is defined as the ratio of the largest singular value and the smallest singular value of A, i.e., κ(A) = σmax(A)/σmin(A). By analogy, for a frame in a Hilbert space { fi}i=1 ⊆H with optimal frame bounds A and B, we define the condition number of the frame to be the condition number of its associated frame operator κ({ fi}) := κ(S) = B/A. In particular, if a frame is Parseval then its condition number equals 1. In fact, a frame is tight if and only if its condition number is 1. Scalable frames were precisely introduced to turn a non optimal (non-tight) frame into an optimal one, by just rescaling the length of each frame vector. More precisely,