Asymptotic tensor rank of graph tensors: beyond matrix multiplication

Christandl, Matthias; Vrana, Péter; Zuiddam, Jeroen

doi:10.1007/s00037-018-0172-8

Cited by 22 publications

(28 citation statements)

References 25 publications

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“…The following theorem is our main technical result on which the rest of the paper rests. We note that for the tensor Kronecker product the statement is well-known in the context of algebraic complexity theory [6,7,16,17,8].…”

Section: Degeneration and Restrictionmentioning

confidence: 83%

“…In the language of graph tensors [8], Proposition 18 says that tensor rank is not multiplicative under taking disjoint unions of graphs.…”

Section: Tensor Rank Is Not Multiplicative Under the Tensor Productmentioning

confidence: 99%

“…(This graphical notation is borrowed from [8].) Each tensor has rank, border rank and asymptotic rank equal to 2, since they are essentially identity matrices.…”

Section: Introductionmentioning

confidence: 99%

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Tensor rank is not multiplicative under the tensor product

Christandl

Jensen

Zuiddam

2018

Linear Algebra and its Applications

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The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an -tensor. The tensor product of s and t is a (k + )-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specifically, if a tensor t has border rank strictly smaller than its rank, then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power. The "tensor Kronecker product" from algebraic complexity theory is related to our tensor product but different, namely it multiplies two k-tensors to get a k-tensor. Nonmultiplicativity of the tensor Kronecker product has been known since the work of Strassen.It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, lower bounds on border rank obtained from general- M. Christandl et al. / Linear Algebra and its Applications 543 (2018) [125][126][127][128][129][130][131][132][133][134][135][136][137][138][139] ized flattenings (including Young flattenings) multiply under the tensor product.

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Section: Degeneration and Restrictionmentioning

confidence: 83%

“…In the language of graph tensors [8], Proposition 18 says that tensor rank is not multiplicative under taking disjoint unions of graphs.…”

Section: Tensor Rank Is Not Multiplicative Under the Tensor Productmentioning

confidence: 99%

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Tensor rank is not multiplicative under the tensor product

Christandl

Jensen

Zuiddam

2018

Linear Algebra and its Applications

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“…, 0), etc. It is shown in [VC15] and [CVZ16], and independently in [HX15], that for all k ≥ 3 we have Q(D (1,k−1) ) = Q(Φ (1,k−1) ) = 2 h(k −1 ) , where h(p) denotes the binary entropy function. In [CVZ16] it is shown that Q(D (2,2) ) = Q(Φ (2,2) ) = 2 which in [AVZ] is extended to, for all even k ≥ 4, Q(D (k/2,k/2) ) = Q(Φ (k/2,k/2) ) = 2.…”

Section: Tight Tensorsmentioning

confidence: 99%

“…Namely, statement (29) becomes false when instead we let Φ ⊆ I 1 × · · · × I k with k ≥ 4 and we let the right-hand side of the equation be max P ∈P(Φ) min i 2 H(Pi) , see [CVZ16, Example 1.1.38]. In[CVZ16] the construction of Theorem 4.4 is extended to obtain a lower bound for Q(Φ) when k ≥ 4. This lower bound is not known to be tight in general.…”

mentioning

confidence: 99%

Universal points in the asymptotic spectrum of tensors

Christandl

Vrana

Zuiddam

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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The asymptotic restriction problem for tensors is to decide, given tensors s and t, whether the nth tensor power of s can be obtained from the (n+o(n))th tensor power of t by applying linear maps to the tensor legs (this we call restriction), when n goes to infinity. In this context, Volker Strassen, striving to understand the complexity of matrix multiplication, introduced in 1986 the asymptotic spectrum of tensors. Essentially, the asymptotic restriction problem for a family of tensors X , closed under direct sum and tensor product, reduces to finding all maps from X to the nonnegative reals that are monotone under restriction, normalised on diagonal tensors, additive under direct sum and multiplicative under tensor product, which Strassen named spectral points. Spectral points are by definition an upper bound on asymptotic subrank and a lower bound on asymptotic rank. Strassen created the support functionals, which are spectral points for oblique tensors, a strict subfamily of all tensors.Universal spectral points are spectral points for the family of all tensors. The construction of nontrivial universal spectral points has been an open problem for more than thirty years. We construct for the first time a family of nontrivial universal spectral points over the complex numbers, using the theory of quantum entropy and covariants: the quantum functionals.In the process we connect the asymptotic spectrum of all tensors to the quantum marginal problem and to the entanglement polytope. In entanglement theory, our results amount to the first construction of additive entanglement monotones for the class of stochastic local operations and classical communication.To demonstrate the asymptotic spectrum, we reprove (in hindsight) recent results on the cap set problem by reducing this problem to computing the lowest point in the asymptotic spectrum of the reduced polynomial multiplication tensor, a prime example of Strassen. A better understanding of our universal spectral points construction may lead to further progress on related combinatorial questions. We additionally show that the quantum functionals characterise asymptotic slice rank for complex tensors.

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Supervised machine learning to estimate instabilities in chaotic systems: Estimation of local Lyapunov exponents

Ayers

Lau

Amezcua

et al. 2023

Quart J Royal Meteoro Soc

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In chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real‐time knowledge about the error growth could enable strategies to adjust the modelling and forecasting infrastructure on the fly to increase accuracy and/or reduce computation time. For example, one could change the ensemble size, the distribution and type of target observations, and so forth. Local Lyapunov exponents are known indicators of the rate at which very small prediction errors grow over a finite time interval. However, their computation is very expensive: it requires maintaining and evolving a tangent linear model, orthogonalisation algorithms and storing large matrices. In this feasibility study, we investigate the accuracy of supervised machine learning in estimating the current local Lyapunov exponents, from input of current and recent time steps of the system trajectory, as an alternative to the classical method. Thus machine learning is not used here to emulate a physical model or some of its components, but “nonintrusively” as a complementary tool. We test four popular supervised learning algorithms: regression trees, multilayer perceptrons, convolutional neural networks, and long short‐term memory networks. Experiments are conducted on two low‐dimensional chaotic systems of ordinary differential equations, the Rössler and Lorenz 63 models. We find that on average the machine learning algorithms predict the stable local Lyapunov exponent accurately, the unstable exponent reasonably accurately, and the neutral exponent only somewhat accurately. We show that greater prediction accuracy is associated with local homogeneity of the local Lyapunov exponents on the system attractor. Importantly, the situations in which (forecast) errors grow fastest are not necessarily the same as those in which it is more difficult to predict local Lyapunov exponents with machine learning.

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Asymptotic tensor rank of graph tensors: beyond matrix multiplication

Cited by 22 publications

References 25 publications

Tensor rank is not multiplicative under the tensor product

Tensor rank is not multiplicative under the tensor product

Universal points in the asymptotic spectrum of tensors

Supervised machine learning to estimate instabilities in chaotic systems: Estimation of local Lyapunov exponents

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