Matrix Multiplication Tensor Research Articles

On the structure tensor of [formula omitted

The structure tensor of sln, denoted Tsln, is the tensor arising from the Lie bracket bilinear operation on the set of traceless n×n matrices over C. This tensor is intimately related to the well studied matrix multiplication tensor. Studying the structure tensor of sln may provide further insight into the complexity of matrix multiplication and the “hay in a haystack” problem of finding explicit sequences tensors with high rank or border rank. We aim to find new bounds on the rank and border rank of this structure tensor in the case of sl3 and sl4. We additionally provide bounds in the case of the lie algebras so4 and so5. The lower bounds on the border ranks were obtained via various recent techniques, namely Koszul flattenings, border substitution, and border apolarity. Upper bounds on the rank of Tsl3 are obtained via numerical methods that allowed us to find an explicit rank decomposition.

Halos and undecidability of tensor stable positive maps

A map is tensor stable positive (tsp) if is positive for all n, and essential tsp if it is not completely positive or completely co-positive. Are there essential tsp maps? Here we prove that there exist essential tsp maps on the hypercomplex numbers. It follows that there exist bound entangled states with a negative partial transpose (NPT) on the hypercomplex, that is, there exists NPT bound entanglement in the halo of quantum states. We also prove that tensor stable positivity on the matrix multiplication tensor is undecidable, and conjecture that tensor stable positivity is undecidable. Proving this conjecture would imply existence of essential tsp maps, and hence of NPT bound entangled states.

Equivalent polyadic decompositions of matrix multiplication tensors

Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de Groote, who showed that for the multiplication of 2 ×2 matrices with 7 active multiplications, all algorithms are essentially equivalent to Strassen’s algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices (e.g., 2 ×3 by 3 ×2 or 3 ×3 by 3 ×3 matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors.

Apolarity, border rank, and multigraded Hilbert scheme

We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We also define a border rank version of the variety of sums of powers and analyze its usefulness in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some interesting tensors, such as the matrix multiplication tensor. We work in a general setting, where the base variety is not necessarily a Segre or Veronese variety, but an arbitrary smooth toric projective variety. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.

Border Rank Nonadditivity for Higher Order Tensors

Whereas matrix rank is additive under direct sum, in 1981 Sch\"onhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive for higher order tensors has remained open. In this work, we settle this problem by providing analogs of Sch\"onhage's construction for tensors of order four and higher. Sch\"onhage's work was motivated by the study of the computational complexity of matrix multiplication; we discuss implications of our results for the asymptotic rank of higher order generalizations of the matrix multiplication tensor.

The Computational Complexity of Plethysm Coefficients

In two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity.In this paper, we show that deciding positivity of plethysm coefficients is NP-hard and that computing plethysm coefficients is #P-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.

Grothendieck constant is norm of Strassen matrix multiplication tensor

We show that two important quantities from two disparate areas of complexity theory --- Strassen's exponent of matrix multiplication $\omega$ and Grothendieck's constant $K_G$ --- are intimately related. They are different measures of size for the same underlying object --- the matrix multiplication tensor, i.e., the $3$-tensor or bilinear operator $\mu_{l,m,n} : \mathbb{F}^{l \times m} \times \mathbb{F}^{m \times n} \to \mathbb{F}^{l \times n}$, $(A,B) \mapsto AB$ defined by matrix-matrix product over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$. It is well-known that Strassen's exponent of matrix multiplication is the greatest lower bound on (the log of) a tensor rank of $\mu_{l,m,n}$. We will show that Grothendieck's constant is the least upper bound on a tensor norm of $\mu_{l,m,n}$, taken over all $l, m, n \in \mathbb{N}$. Aside from relating the two celebrated quantities, this insight allows us to rewrite Grothendieck's inequality as a norm inequality \[ \lVert\mu_{l,m,n}\rVert_{1,2,\infty} =\max_{X,Y,M\neq0}\frac{|\operatorname{tr}(XMY)|}{\lVert X\rVert_{1,2}\lVert Y\rVert_{2,\infty}\lVert M\rVert_{\infty,1}}\le K_G. \] We prove that Grothendieck's inequality is unique: If we generalize the $(1,2,\infty)$-norm to arbitrary $p,q, r \in [1, \infty]$, \[ \lVert\mu_{l,m,n}\rVert_{p,q,r}=\max_{X,Y,M\neq0}\frac{|\operatorname{tr}(XMY)|}{\|X\|_{p,q}\|Y\|_{q,r}\|M\|_{r,p}}, \] then $(p,q,r )=(1,2,\infty)$ is, up to cyclic permutations, the only choice for which $\lVert\mu_{l,m,n}\rVert_{p,q,r}$ is uniformly bounded by a constant independent of $l,m,n$.

A Rank 18 Waring Decomposition of sM 〈3〉 with 432 Symmetries

The recent discovery that the exponent of matrix multiplication is determined by the rank of the symmetrized matrix multiplication tensor has invigorated interest in better understanding symmetrized matrix multiplication. Author present an explicit rank 18 Waring decomposition of and describe its symmetry group.

The geometry of rank decompositions of matrix multiplication II: 3 × 3 matrices

This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank 23 decompositions for the 3×3 matrix multiplication tensor M〈3〉. All our decompositions have symmetry groups that include the standard cyclic permutation of factors but otherwise exhibit a range of behavior. One of them has 11 cubes as summands and admits an unexpected symmetry group of order 12.We establish basic information regarding symmetry groups of decompositions and outline two approaches for finding new rank decompositions of M〈n〉 for larger n.

Polynomials and the exponent of matrix multiplication

The exponent of matrix multiplication is the smallest constant ω such that two n × n matrices may be multiplied by performing O ( n ω + ε ) arithmetic operations for every ε > 0 . Determining the constant ω is a central question in both computer science and mathematics. Strassen [Linear Algebra Appl. 52/53 (1983) 645–685] showed that ω is also governed by the tensor rank of the matrix multiplication tensor. We define certain symmetric tensors, that is, cubic polynomials, and our main result is that their symmetric rank also grows with the same exponent ω, so that ω can be computed in the symmetric setting, where it may be easier to determine. In particular, we study the symmetrized matrix multiplication tensor s M ⟨ n ⟩ defined on an n × n matrix A by s M ⟨ n ⟩ ( A ) = trace ( A 3 ) . The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent ω.

Tensor surgery and tensor rank

We introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices and edges. We show that tensor surgery is capable of preserving the low rank structure of an initial tensor decomposition and thus allows to prove nontrivial upper bounds on tensor rank, border rank and asymptotic rank of the final tensors. We illustrate our method with a number of examples. Tensor surgery on the triangle graph, which corresponds to the matrix multiplication tensor, leads to nontrivial rank upper bounds for all odd cycle graphs, which correspond to the tensors of iterated matrix multiplication. In the asymptotic setting we obtain upper bounds in terms of the matrix multiplication exponent $\omega$ and the rectangular matrix multiplication parameter $\alpha$. These bounds are optimal if $\omega$ equals two. We also give examples that illustrate that tensor surgery on general graphs might involve the absorption of virtual hyperedges and we provide an example of tensor surgery on a hypergraph. Besides its relevance in algebraic complexity theory, our work has applications in quantum information theory and communication complexity.

The border support rank of two-by-two matrix multiplication is seven

We show that the border support rank of the tensor corresponding to two-by-two matrix multiplication is seven over the complex numbers. We do this by constructing two polynomials that vanish on all complex tensors with format four-by-four-by-four and border rank at most six, but that do not vanish simultaneously on any tensor with the same support as the two-by-two matrix multiplication tensor. This extends the work of Hauenstein, Ikenmeyer, and Landsberg. We also give two proofs that the support rank of the two-by-two matrix multiplication tensor is seven over any field: one proof using a result of De Groote saying that the decomposition of this tensor is unique up to sandwiching, and another proof via the substitution method. These results answer a question asked by Cohn and Umans. Studying the border support rank of the matrix multiplication tensor is relevant for the design of matrix multiplication algorithms, because upper bounds on the border support rank of the matrix multiplication tensor lead to upper bounds on the computational complexity of matrix multiplication, via a construction of Cohn and Umans. Moreover, support rank has applications in quantum communication complexity.

The Geometry of Rank Decompositions of Matrix Multiplication I: 2 × 2 Matrices

ABSTRACTThis 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 Burichenko establishing the symmetry group of Strassen's algorithm, and present two particularly nice subfamilies in the Strassen family of decompositions.

Entanglement Distillation from Greenberger–Horne–Zeilinger Shares

We study the problem of converting a product of Greenberger-Horne-Zeilinger (GHZ) states shared by subsets of several parties in an arbitrary way into GHZ states shared by every party. Our result is that if SLOCC transformations are allowed, then the best asymptotic rate is the minimum of bipartite log-ranks of the initial state. This generalizes a result by Strassen on the asymptotic subrank of the matrix multiplication tensor.

Fundamental invariants of orbit closures

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental SL-invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure. For the power sums we can write down this fundamental invariant explicitly in most cases. Our constructions generalize the two Aronhold invariants on ternary cubics. For the other objects we identify the invariant function conditional on intriguing combinatorial problems much like the well-known Alon–Tarsi conjecture on Latin squares. We provide computer calculations in small dimensions for these cases. As a main tool for our analysis, we determine the stabilizers, and we establish the polystability of all the mentioned forms and tensors (including the generic ones).

On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication

ABSTRACTWe make an in-depth study of the known border rank (i.e., approximate) algorithms for the matrix multiplication tensor encoding the multiplication of an n × 2 matrix by a 2 × 2 matrix.

Nontriviality of equations and explicit tensors in [formula omitted] of border rank at least [formula omitted

For even (resp. odd) m, I show the Young-flattening equations for border rank of tensors in Cm⊗Cm⊗Cm of [7] are nontrivial up to border rank 2m−3 (resp. 2m−5) by writing down explicit tensors on which the equations do not vanish. Thus these tensors have border rank at least 2m−2 (resp. 2m−4). The result implies that there are nontrivial equations for border rank 2n2−n that vanish on the matrix multiplication tensor for n×n matrices. I also study the border rank of the tensors of [1] and the equations of [4]. I show the tensors T2k∈Ck⊗C2k⊗C2k of [1], despite having rank equal to 2k+1−1, have border rank equal to 2k. I show the equations for border rank of [4] are trivial in the case of border rank 2m−1 and determine their precise non-vanishing on the matrix multiplication tensor.

Tensor Rank: Matching Polynomials and Schur Rings

We study the polynomial equations vanishing on tensors of a given rank. By means of polarization we reduce them to elements $$A$$ A of the group algebra $${\mathbb {Q}}[S_n\times S_n]$$ Q [ S n × S n ] and describe explicit linear equations on the coefficients of $$A$$ A to vanish on tensors of a given rank. Further, we reduce the study to the Schur ring over the group $$S_n\times S_n$$ S n × S n that arises from the diagonal conjugacy action of $$S_n$$ S n . More closely, we consider elements of $${\mathbb {Q}}[S_n\times S_n]$$ Q [ S n × S n ] vanishing on tensors of rank $$n-1$$ n - 1 and describe them in terms of triples of Young diagrams, their irreducible characters, and nonvanishing of their Kronecker coefficients. Also, we construct a family of elements in $${\mathbb {Q}}[S_n\times S_n]$$ Q [ S n × S n ] vanishing on tensors of rank $$n-1$$ n - 1 and illustrate our approach by a sharp lower bound on the border rank of an explicitly produced tensor. Finally, we apply this construction to prove a lower bound $$5n^2/4$$ 5 n 2 / 4 on the border rank of the matrix multiplication tensor (being, of course, weaker than the best known one $$(2-\epsilon )\cdot n^2$$ ( 2 - ∈ ) · n 2 , due to Landsberg, Ottaviani).