A Sparse Low-Rank Biclique Decomposition for Graphs

Low-rank plus sparse decompositions, such as robust PCA, typically model the low-rank component as dense and the sparse component as corruption. This viewpoint is poorly suited to graph adjacency matrices, where sparsity is structural and dense low-rank factors destroy the combinatorial and computational properties of the graph. We propose signed biclique decomposition (SBD), a sparse low-rank decomposition for integer-weighted graphs that represents the structured component as a sum of sparse signed binary rank-one factors, together with a sparse signed residual. The resulting representation remains discrete, interpretable, sparse, and compatible with standard linear algebra. We formulate fixed-rank and greedy SBD objectives and derive a gain-based greedy algorithm with theoretical guarantees. Empirically, SBD recovers block structure in noisy graphs, while its factorized representation enables efficient sparse matrix-matrix multiplication (SpMM). On real graphs with strong shared-neighborhood structure, SBD achieves compression factors above $10$ and SpMM speedups up to $6\times$ over cuSPARSE, suggesting a practical alternative to dense low-rank decompositions for graphs.