Querying a Matrix through Matrix-Vector Products

We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 ⃛ q and observes Mv . Here i can be randomized as well chosen adaptively a function of i-1 Motivated by applications sketching in distributed computation, linear algebra, streaming models, connections areas such communication complexity property testing, we initiate study number queries needed solve various fundamental problems. problems three broad categories, including statistics problems, graph For example, required approximate rank, trace, maximum eigenvalue, norms M; compute AND/OR/Parity each column or row M, decide whether there are identical columns rows is symmetric, diagonal, unitary; defined connected triangle-free. also show separations for that allowed obtain only querying on right, versus query both left right. depending underlying field product occurs in. form (bipartite adjacency signed edge-vertex incidence matrix) represent graph. Surprisingly, very few works discuss this model, believe thorough investigation model would beneficial different application areas.