Particle Number Conservation and Block Structures in Matrix Product States

Bachmayr, Markus; Götte, Michael; Pfeffer, Max J.

doi:10.48550/arxiv.2104.13483

Cited by 2 publications

(5 citation statements)

References 24 publications

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“…We discuss the problem of function identification from data for tensor train based ansatz spaces and give some insights into when these ansatz spaces can be used efficiently. For this we combine recent results on sample complexity [EST20] and block sparsity of tensor train networks [BGP21] to motivate a novel algorithm for the problem at hand. We then demonstrate the applicability of this algorithm to different problems.…”

Section: Discussionmentioning

confidence: 99%

“…Now that we have seen that it is advantagious to restrict ourselves to the space W d g we need to find a way to do so without loosing the advantages of the tensor train format. In [BGP21] it was rediscovered that if a tensor train is an eigenvector of certain Laplace-like operators it admits a block sparse structure. This means for a tensor train c the components C k have zero blocks.…”

Section: Block Sparse Tensor Trainsmentioning

confidence: 99%

“…Note that we use polynomials since they have been applied successfully in practice, but other function dictionaries can be used as well. Also note that the theory is much more general as shown in [BGP21] and is not restricted to the degree counting operator.…”

Section: Block Sparse Tensor Trainsmentioning

confidence: 99%

“…This representation allows us to parametrize the space of homogeneous polynomials of a given degree in an exact and very efficient manner. This is known to quantum physicists for at least a decade [SPV10] but it was introduced to the mathematics community only recently in [BGP21]. In the language of quantum mechanics one would say that there exists an operator for which the coefficient tensor of any homogeneous polynomial is an eigenvector.…”

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confidence: 99%

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A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression

Götte¹,

Schneider²,

Trunschke³

2021

Preprint

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Low-rank tensors are an established framework for high-dimensional least-squares problems. We propose to extend this framework by including the concept of block-sparsity. In the context of polynomial regression each sparsity pattern corresponds to some subspace of homogeneous multivariate polynomials. This allows us to adapt the ansatz space to align better with known sample complexity results. The resulting method is tested in numerical experiments and demonstrates improved computational resource utilization and sample efficiency.Keywords empirical L 2 approximation • sample efficiency • homogeneous polynomials • sparse tensor networks • alternating least squares IntroductionAn important problem in many applications is the identification of a function from measurements or random samples. For this problem to be well-posed, some prior information about the function has to be assumed and a common requirement is that the function can be approximated in a finite dimensional ansatz space. For the purpose of extracting governing equations the most famous approach in recent years has been SINDy [BPK16]. However, the applicability of SINDy to high-dimensional problems is limited since truly high-dimensional problems require a nonlinear parameterization of the ansatz space. One particular reparametrization that has proven itself in many applications are tensor networks. These allow for a straight-forward extension of SINDy [GKES19] but can also encode additional structure as presented in [GRK + 20]. The compressive capabilities of tensor networks originate from this ability to exploit additional structure like smoothness, locality or self-similarity and have hence been used in solving high-dimensional equations [KK12, KS18, BK20, EPS16]. In the context of optimal control tensor train networks have been utilized for solving the Hamilton-Jacobi-Bellman equation in [DKK21, OSS20], for solving backward stochastic differential equations in [RSN21] and for the calculation of stock options prices in [BEST21, GKS20]. In the context of uncertainty quantification they are used in [ENSW19, ESTW19, ZYO + 15] and in the context of image classification they are used in [KG19, SS16]. A common thread in these publications is the parametrization of a high-dimensional ansatz space by a tensor train network which is then optimized. In most cases this means that the least-squares error of the parametrized function to the data is minimized. There exist many methods to perform this minimization. A well-known algorithm in the mathematics community is the alternating linear scheme (ALS) [Ose11a, HRS12a] , which is related to the famous DMRG method [Whi92] for solving the Schrödinger equation in Quantum Physics. Although, not directly suitable for recovery tasks, it became apparent that DMRG and ALS can be adapted to work in this context. Two of these extensions to the ALS algorithm are the stablilized ALS approximation (SALSA) [GK19] and the block alternating steepest descent for Recovery (bASD) algorithm [ENSW19]. Both adapt the tensor network ...

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Section: Discussionmentioning

confidence: 99%

Section: Block Sparse Tensor Trainsmentioning

confidence: 99%

Section: Block Sparse Tensor Trainsmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression

Götte¹,

Schneider²,

Trunschke³

2021

Preprint

Self Cite

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“…[SPV10]. Intriguingly, bringing this concept to the multi-linear parametrizations of multivariate functions leads to natural restrictions of the function space such as bounding the polynomial degree of the approximation [BGP21,GST21].…”

Section: Introductionmentioning

confidence: 99%

Block sparsity and gauge mediated weight sharing for learning dynamical laws from data

Götte¹,

Fuksa²,

Roth³

et al. 2022

Preprint

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Recent years have witnessed an increased interest in recovering dynamical laws of complex systems in a largely data-driven fashion under meaningful hypotheses. In this work, we propose a method for scalably learning dynamical laws of classical dynamical systems from data. As a novel ingredient, to achieve an efficient scaling with the system size, block sparse tensor trains -instances of tensor networks applied to function dictionaries -are used and the self similarity of the problem is exploited. For the latter, we propose an approach of gauge mediated weight sharing, inspired by notions of machine learning, which significantly improves performance over previous approaches. The practical performance of the method is demonstrated numerically on three one-dimensional systems -the Fermi-Pasta-Ulam-Tsingou system, rotating magnetic dipoles and classical particles interacting via modified Lennard-Jones potentials. We highlight the ability of the method to recover these systems, requiring 1400 samples to recover the 50 particle Fermi-Pasta-Ulam-Tsingou system to residuum of 5 × 10 −7 , 900 samples to recover the 50 particle magnetic dipole chain to residuum of 1.5 × 10 −4 and 7000 samples to recover the Lennard-Jones system of 10 particles to residuum 1.5 × 10 −2 . The robustness against additive Gaussian noise is demonstrated for the magnetic dipole system.Keywords Dynamical laws recovery • machine learning • tensor trains • block sparse tensor trains • tensor networks • gauge mediated weight sharing

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Particle Number Conservation and Block Structures in Matrix Product States

Cited by 2 publications

References 24 publications

A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression

A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression

Block sparsity and gauge mediated weight sharing for learning dynamical laws from data

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