This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper we: establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix multiplication decompositions, give a geometric proof of the theorem of [3] establishing the symmetry group of Strassen’s algorithm, and present two particularly nice s...

We review two similar concepts of hierarchical rank of tensors (which extend the matrix rank to higher order tensors): the TT-rank and the H-rank (hierarchical or H-Tucker rank). Based on this notion of rank, one can define a data-sparse representation of tensors involving O(dnk + dk 3) data for order d tensors with mode sizes n and rank k. Simple examples underline the differences and similari...

Recall that in Lecture 13, a randomized algorithm was described for computing a low rank approximation to the eigendecomposition of a matrix A. A drawback to this method is that the matrix A must be accessed multiple times (twice), which may not be possible in streaming models where A cannot be stored in memory [1]. For the streaming model, we require a single pass algorithm, where A is accesse...

Let C be a set of multisets that is closed under taking submultisets (i.e. when X ∈ C and Y ⊆ X, we have Y ∈ C); then C is a (simplicial) multicomplex. If every member of C is a set, C is a (simplicial) complex. We denote by Fi the number of multisets of cardinality i in C. When the multiset of largest cardinality has r elements, the F-vector of C is (F0, F1, . . . , Fr). A complex or multicomp...

The positive semidefinite rank of a nonnegative (m×n)-matrix S is the minimum number q such that there exist positive semidefinite (q × q)-matrices A1, . . . , Am, B1, . . . , Bn such that S(k, l) = trA∗kBl. The most important lower bound technique on nonnegative rank only uses the zero/nonzero pattern of the matrix. We characterize the power of lower bounds on positive semidefinite rank based ...

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all Q-rank one lattices in semisimple groups of R-rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion