Tight Conic Relaxations for Rank-one Doubly Nonnegative Matrix Completion

What the math gives to ML

The paper gives constructive tightness conditions for recovering a nonnegative rank-one matrix from sparse entries, expressed through the bipartite sparsity graph, local ratios, and cumulative differences around cycles. The transferable asset is a cycle-consistency certificate: multiplicative ratios along a sparse graph determine whether missing entries can be completed by a single pair of nonnegative factors. This can be used to initialize or compress nonnegative attention, router, or embedding matrices by replacing an expensive dense parameterization with graph-observed entries plus rank-one completion. The most practical first test is a cycle-certified low-rank parameterization for attention or MoE routing, with the paper's ratio bounds used as a stability diagnostic and adaptive regularizer rather than solving a full SDP during every training step.