Counterexamples to Strassen’s direct sum conjecture

The multiplicative complexity of systems of bilinear forms (and, in particular, the famous question of fast matrix multiplication) is an important area of research in modern theory of computation. One of the foundational papers on the topic is Strassen’s work [20], which contains an O(n 7/ ln ) algorithm for the multiplication of two n×n matrices. In his subsequent paper [21] published in 1973, Strassen asked whether the multiplicative complexity of the union of two bilinear systems depending on different variables is equal to the sum of the multiplicative complexities of both systems. A stronger version of this problem was proposed in the 1981 paper [10] by Feig and Winograd, who asked whether any optimal algorithm that computes such a pair of bilinear systems must compute each system separately. These questions became known as the direct sum conjecture and strong direct sum conjecture, respectively, and they were attracting a notable amount of attention during the four decades. As Feig and Winograd wrote, ‘either a proof of, or a counterexample to, the direct sum conjecture will be a major step forward in our understanding of complexity of systems of bilinear forms.’ The modern formulation of this conjecture is based on a natural representation of a bilinear system as a three-dimensional tensor, that is, an array of elements T (i|j|k) taken from a field F , where the triples (i, j, k) run over the Cartesian product of finite indexing sets I, J,K. A tensor T is called decomposable if T = a⊗b⊗c (which should be read as T (i|j|k) = aibjck), for some vectors a ∈ FI , b ∈ FJ , c ∈ FK . The rank of a tensor T , or the multiplicative complexity of the corresponding bilinear system, is the smallest r for which T can be written as a sum of r decomposable tensors with entries in F . We denote this quantity by rankF T , and we note that the rank of a tensor may change if one allows to take the entries of decomposable tensors as above from an extension of F , see [3]. Taking the union of two bilinear systems depending on disjoint sets of variables corresponds to the direct sum operation on tensors. More precisely, if T and T ′ are tensors with disjoint indexing sets I, I , J, J ,K,K , then we can define the direct sum T⊕T ′ as a tensor with indexing sets I ∪ I , J ∪ J , K ∪ K ′ such that the (I|J |K) block equals T and (I ′|J ′|K ) block equals T , and all entries outside of these blocks are zero. In other words, direct sums of tensors are a multidimensional analogue of block-diagonal matrices; a basic result of linear algebra says that the ranks of such matrices are equal to the sums of the ranks of their diagonal blocks. Strassen’s direct sum conjecture is a three-dimensional analogue of this statement.