On Comon’s and Strassen’s Conjectures

Abstract

Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.

Veronese varieties and Hilbert functions

“…For example, in the study of Waring rank, that is the tensor rank for symmetric tensors, in the paper [CCG12]. Another example is the study of Strassen's Conjecture, a crucial conjecture in complexity theory, now proved to be false in general [Shi19], but still open in the relevant case of symmetric tensors, see [CCC + 18] and [CMM18]. As a last example, we also mention the study of the identifiability of tensors, which plays a crucial role in Algebraic Statistic, see [ABC18], [AC20] and [AGMO18].…”

On rank decomposition and semi-symmetric rank decomposition of semi-symmetric tensors

“…Also, in [28], Comon's conjecture was proved for the case of rank of the symmetric tensor is not greater than its order. Casarotti et al investigated the additional aspects of Comon's conjecture [7].…”

Counterexamples to Strassen’s direct sum conjecture

“…This approach has led to interesting generalizations and relaxations of Conjecture 1, to further sufficient conditions for tensors to satisfy it, and, therefore, to many new classes of tensors for which the conjecture holds. Let us mention the paper [14] on the so-called e-computable tensors, the work [13] devoted to symmetric tensors, the paper [45] dealing with the cactus rank and catalecticant bound, the work [27] proving Conjecture 1 for tensors whose ranks can be computed by a particular adaptation of the so-called substitution method, the paper [15] studying the spaces of feasible rank decompositions in context of the direct sum conjecture, the work [12] containing further reformulations and generalizations of Conjecture 1 in terms of secant varieties, the monograph [25] containing a detailed discussion of this conjecture and its consequences for algebraic geometry, and a recent survey paper [16] on the topic. Different rank-decomposition problems do also take an important place in linear algebra and combinatorics.…”