Deterministic Completion of Rectangular Matrices Using Asymmetric Ramanujan Graphs: Exact and Stable Recovery

New and explicit constructions of unbalanced Ramanujan bipartite graphs

“…7.1, a significant motivation for the study of explicit constructions of Ramanujan graphs comes from the matrix completion problem. This theme was explored in detail in [8] where it was shown that it is possible to guarantee exact completion of an unknown low-rank matrix, if the sampling set corresponds to the edge set of a Ramanujan bigraph. While that set of results is interesting in itself, it has left open the question of just how Ramanujan bigraphs are to be constructed.…”

“…Then it is shown that, if the number of samples | | is sufficiently large, then with high probabilitŷ X does indeed equal X . A significant departure is made in a recent paper by a subset of the present authors [8], where the sample set is chosen in a deterministic fashion, as the edge set of a Ramanujan bigraph. In [8] a sufficient condition is derived for the solution of (12) to equal X .…”

“…While [8] gives the first deterministic solution to the matrix completion problem, it does not address the issue of "missing measurements." In any real-life application of matrix completion, it is a fact that a few elements of the matrix X are "missing" and thus cannot be measured.…”

“…Explicit constructions of such graphs have multifaceted applications in areas such as computer science, coding theory, and compressed sensing. In particular, in [8] the first and third authors show that Ramanujan graphs can provide the first deterministic solution to the so-called matrix completion problem. Prior to the publication of [8], the matrix completion problem had only a probabilistic solution.…”

“…In particular, in [8] the first and third authors show that Ramanujan graphs can provide the first deterministic solution to the so-called matrix completion problem. Prior to the publication of [8], the matrix completion problem had only a probabilistic solution. The explicit construction of Ramanujan graphs has been classically well studied.…”