Matrix completion of noisy graph signals via proximal gradient minimization

Abstract

Abstract-This paper takes on the problem of recovering the missing entries of an incomplete matrix, which is known as matrix completion, when the columns of the matrix are signals that lie on a graph and the available observations are noisy. We solve a version of the problem regularized with the Laplacian quadratic form by means of the proximal gradient method, and derive theoretical bounds on the recovery error. Moreover, in order to speed up the convergence of the proximal gradient, we propose an initialization method that utilizes the structural information contained in the Laplacian matrix of the graph.

Generalization Error Bounds for Kernel Matrix Completion and Extrapolation

“…Plugging the latter into the objective of (2) and substituting W = K w B and H = K h C, yields (15) is the objective used by the inductive MC [16]; and therefore, we have shown that inductive MC is a special case of KMC. This leads to the following result.…”

Matrix Completion and Extrapolation via Kernel Regression

“…, where b n,m and c l,m are the entries at (n, m) and (l, m) of the factor matrices B and C from (17), respectively. Therefore, ( 19) can be rewritten as…”

“…One difference between the two loss functions is that (29) does not explicitly limit the rank of the recovered matrix F = unvec(v R ) since it has N L degrees of freedom through γ, while in (17) the rank of F cannot exceed p since B and C are of rank p at most. In fact, the low-rank property is indirectly promoted in (29) through the kernel matrices.…”

“…On the other hand, the regularization terms in ( 17) and ( 29) play a different role in each formulation. The regularization in (17) promotes smoothness on the columns of the estimated factor matrices { Ŵ , Ĥ}; or, in other words, similarity between the rows of { Ŵ , Ĥ} as measured by κ x and κ y . On the contrary, the regularization in (29) promotes smoothness on v, which is tantamount to promoting similarity between the entries of F in accordance with κ z .…”

“…One major hurdle in MC is the computational cost as the matrix size grows. In its formulation as a rank minimization task, MC can be solved via semidefinite programming [1], or proximal gradient minimization [15], [16], [8], [17], which entails a singular value decomposition of the recovered matrix per iteration. Instead, algorithms with lower computational cost are available for the bi-convex formulation based on matrix factorization [4].…”