Border Rank Research Articles

Border rank of powers of ternary quadratic forms

We determine the border rank of each power of any quadratic form in three variables. Since the problem for rank 1 and rank 2 quadratic forms can be reduced to determining the rank of powers of binary forms, we primarily focus on non-degenerate quadratic forms. We begin by considering the quadratic form qn=x12+…+xn2 in an arbitrary number n of variables. We determine the apolar ideal of any power qns, proving that it corresponds to the homogeneous ideal generated by the harmonic polynomials of degree s+1. Using this result, we select a specific ideal contained in the apolar ideal for each power of a quadratic form in three variables, which, without loss of generality, we assume to be the form q3. After verifying certain properties, we utilize the recent technique of border apolarity to establish that the border rank of any power q3s is equal to the rank of its middle catalecticant matrix, namely (s+1)(s+2)/2.

Rank of a tensor and quantum entanglement

The rank of a tensor is analysed in the context of quantum entanglement. A pure quantum state v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for the characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state v is called entangled, if it cannot be written in the product form, v ≠ v 1 ⊗ v 2 ⊗ ⋯ ⊗ v d , what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed.

Concise tensors of minimal border rank

We determine defining equations for the set of concise tensors of minimal border rank in $${\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m$$ when $$m=5$$ and the set of concise minimal border rank $$1_*$$ -generic tensors when $$m=5,6$$ . We solve the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case $$m=5$$ . Our proofs utilize two recent developments: the 111-equations defined by Buczyńska–Buczyński and results of Jelisiejew–Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland’s normal form for 1-degenerate tensors satisfying Strassen’s equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $${\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5$$ .

Bounds on complexity of matrix multiplication away from Coppersmith–Winograd tensors

We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, and monomial algebras. We analyze them using Strassen's laser method and obtain an upper bound 2.431 on ω. We also explain how in certain monomial cases using the laser method directly is less profitable than first degenerating. Our results form possible paths in the search for valuable tensors for the laser method away from Coppersmith-Winograd tensors.

Multigraded apolarity

AbstractWe generalize methods to compute various kinds of rank to the case of a toric variety X embedded into projective space using a very ample line bundle . We find an upper bound on the cactus rank. We use this to compute rank, border rank, and cactus rank of monomials in when X is , the Hirzebruch surface , or the weighted projective plane .

Hermitian K-theory via oriented Gorenstein algebras

AbstractWe show that the hermitian K-theory space of a commutative ringRcan be identified, up to𝐀1{\mathbf{A}^{1}}-homotopy, with the group completion of the groupoid of oriented finite GorensteinR-algebras, i.e., finite locally freeR-algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith–Winograd tensor.

Maximum relative distance between real rank-two and rank-one tensors

It is shown that the relative distance in Frobenius norm of a real symmetric order-d tensor of rank-two to its best rank-one approximation is upper bounded by sqrt{1-(1-1/d)^{d-1}}. This is achieved by determining the minimal possible ratio between spectral and Frobenius norm for symmetric tensors of border rank two, which equals left( 1-{1}/{d}right) ^{(d-1)/{2}}. These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case.

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.

Schur apolarity

Inspired by the classic apolarity theory of symmetric tensors, the aim of this paper is to introduce the Schur apolarity theory, i.e. an apolarity for any irreducible representation of the special linear group SL(V). This allows to describe decompositions of structured tensors whose elementary elements are tensors that represent flags of the vector space V. The main result is the Schur apolarity lemma which is the analogous of the apolarity lemma of symmetric apolarity theory. Eventually we study the rank tensors of low border rank related to specific varieties giving rise also to simple algorithms.

Irreversibility of structure tensors of modules

Determining the matrix multiplication exponent $$\omega $$ is one of the greatest open problems in theoretical computer science. We show that it is impossible to prove $$\omega = 2$$ by starting with structure tensors of modules of fixed degree and using arbitrary restrictions. It implies that the same is impossible by starting with $$1_A$$ -generic non-diagonal tensors of fixed size with minimal border rank. This generalizes the work of Bläser and Lysikov (Slice rank of block tensors and irreversibility of structure tensors of algebras 170, 2020). Our methods come from both commutative algebra and complexity theory.

Guarantees for Existence of a Best Canonical Polyadic Approximation of a Noisy Low-Rank Tensor

The canonical polyadic decomposition (CPD) of a low-rank tensor plays a major role in data analysis and signal processing by allowing for unique recovery of underlying factors. However, it is well known that the low-rank CPD approximation problem is ill-posed. That is, a tensor may fail to have a best rank $R$ CPD approximation when $R>1$. This article gives deterministic bounds for the existence of best low-rank tensor approximations over ${\mathbb{K}}={\mathbb{R}}$ or ${\mathbb{K}}={\mathbb{C}}$. More precisely, given a tensor ${\mathcal T} \in {\mathbb{K}}^{I \times I \times I}$ of rank $R \leq I$, we compute the radius of a Frobenius norm ball centered at ${\mathcal T}$ in which best ${\mathbb{K}}$-rank $R$ approximations are guaranteed to exist. In addition we show that every ${\mathbb{K}}$-rank $R$ tensor inside of this ball has a unique canonical polyadic decomposition. This neighborhood may be interpreted as a neighborhood of “mathematical truth" in which CPD approximation and computation are well-posed. In pursuit of these bounds, we describe low-rank tensor decomposition as a “joint generalized eigenvalue" problem. Using this framework, we show that, under mild assumptions, a low-rank tensor which has rank strictly greater than border rank is defective in the sense of algebraic and geometric multiplicities for joint generalized eigenvalues. Bounds for existence of best low-rank approximations are then obtained by establishing perturbation theoretic results for the joint generalized eigenvalue problem. In this way we establish a connection between existence of best low-rank approximations and the tensor spectral norm. In addition we solve a “tensor Procrustes problem" which examines orthogonal compressions for pairs of tensors. The main results of the article are illustrated by a variety of numerical experiments.

Rank and border rank of Kronecker powers of tensors and Strassen's laser method

We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor T_{cw,q} is the square of its border rank for q > 2 and that the border rank of its Kronecker cube is the cube of its border rank for q > 4. This answers questions raised implicitly by Coppersmith & Winograd (1990, §11)and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range.In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, T_{skewcw,q}. For q = 2, the Kronecker square of this tensor coincides with the 3times 3 determinant polynomial, det_{3} in mathbb{C}^{9} otimes mathbb{C}^{9} otimes mathbb{C}^{9}, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two.We determine new upper bounds for the (Waring) rank and the (Waring) border rank of det_3, exhibiting a strict submultiplicative behaviour for T_{skewcw,2} which is promising for the laser method.We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in mathbb{C}^{3} otimes mathbb{C}^{3} otimes mathbb{C}^{3}.

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.

Open -ranks with respect to Segre and Veronese varieties

Let be an integral and non-degenerate variety. Recall (A. Bialynicki-Birula, A. Schinzel, J. Jelisiejew and others) that for any the open rank is the minimal positive integer such that for each closed set there is a set with and , where denotes the linear span. For an arbitrary we give an upper bound for in terms of the upper bound for when is a point in the maximal proper secant variety of and a similar result using only points with submaximal border rank. We study when is a Segre variety (points with -rank and ) and when is a Veronese variety (points with -rank or with border rank ).

On the tensor rank of the 3 x 3 permanent and determinant

The tensor rank and border rank of the $3 \times 3$ determinant tensor are known to be $5$ if the characteristic is not two. In characteristic two, the existing proofs of both the upper and lower bounds fail. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well.

Minimal divergence for border rank-2 tensor approximation

ABSTRACT A tensor is the sum of at least elementary tensors. In addition, a ‘border rank’ is defined: holds if r is the minimum integer such that is a limit of rank-r tensors. Usually, the set of rank-r tensors is not closed, i.e. tensors with may exist. It is easy to see that in such a case the representation of rank-r tensors contains diverging elementary tensors as approaches In a first part, we recall results about the uniform strength of the divergence in the case of general nonclosed tensor formats (restricted to finite dimensions). The second part discusses the r-term format for infinite-dimensional tensor spaces. It is shown that the general situation is very similar to the behaviour of finite-dimensional model spaces. The third part contains the main result: it is proved that in the case of the divergence strength is , i.e. if and the parameters of increase at least proportionally to

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.

Systems of Polynomial Equations, Higher-Order Tensor Decompositions, and Multidimensional Harmonic Retrieval: A Unifying Framework. Part II: The Block Term Decomposition

In Part I we proposed a multilinear algebra framework to solve 0-dimensional systems of polynomial equations with simple roots. We extend this framework to incorporate multiple roots: a block term decomposition (BTD) of the null space of the Macaulay matrix reveals the dual (sub)space of a disjoint root in each term. The BTD is the joint triangularization of multiplication tables and a three-way generalization of the Jordan canonical form in the matrix case, intimately related to the border rank of a tensor. We hint at and illustrate flexible numerical optimization-based algorithms.

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.

Vanishing Hessian, wild forms and their border VSP

Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree dge 3 as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczyńska and Buczyński, we study the border varieties of sums of powers underline{{mathrm {VSP}}} of these forms in the corresponding multigraded Hilbert schemes.