Rank adaptive tensor recovery based model reduction for partial differential equations with high-dimensional random inputs

Tang, Kejun; Liao, Qifeng

doi:10.1016/j.jcp.2020.109326

Cited by 5 publications

(3 citation statements)

References 38 publications

(131 reference statements)

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“…In this work, we focus on the low rank tensor train representation to construct the random projection f . Tensor decompositions are widely used for data compression [5,[19][20][21][22][23][24]. The Tensor Train (TT) decomposition, also called Matrix Product States (MPS) [25][26][27][28], gives the following benefits-low rank TT-formats can provide compact representations of projection matrices and efficient basic linear algebra operations of matrix-by-vector products [29].…”

Section: Introductionmentioning

confidence: 99%

Tensor Train Random Projection

Feng¹,

Tang²,

He³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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This work proposes a Tensor Train Random Projection (TTRP) method for dimension reduction, where pairwise distances can be approximately preserved. Our TTRP is systematically constructed through a Tensor Train (TT) representation with TT-ranks equal to one. Based on the tensor train format, this random projection method can speed up the dimension reduction procedure for high-dimensional datasets and requires fewer storage costs with little loss in accuracy, compared with existing methods. We provide a theoretical analysis of the bias and the variance of TTRP, which shows that this approach is an expected isometric projection with bounded variance, and we show that the scaling Rademacher variable is an optimal choice for generating the corresponding TT-cores. Detailed numerical experiments with synthetic datasets and the MNIST dataset are conducted to demonstrate the efficiency of TTRP.

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Section: Introductionmentioning

confidence: 99%

Tensor Train Random Projection

Feng¹,

Tang²,

He³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

Self Cite

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“…Multiway data or multidimensional arrays, represented by tensors, are ubiquitous in real-world applications, such as recommendation systems [1,2], computer version [3], medical imaging [4] and uncertainty quantification [5].…”

Section: Introductionmentioning

confidence: 99%

Streaming probabilistic tensor train decomposition

Huang¹,

Yu²,

Liao³

2023

Preprint

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The Bayesian streaming tensor decomposition method is a novel method to discover the low-rank approximation of streaming data. However, when the streaming data comes from a high-order tensor, tensor structures of existing Bayesian streaming tensor decomposition algorithms may not be suitable in terms of representation and computation power. In this paper, we present a new Bayesian streaming tensor decomposition method based on tensor train (TT) decomposition. Especially, TT decomposition renders an efficient approach to represent high-order tensors. By exploiting the streaming variational inference (SVI) framework and TT decomposition, we can estimate the latent structure of high-order incomplete noisy streaming tensors. The experiments in synthetic and real-world data show the accuracy of our algorithm compared to the state-of-the-art Bayesian streaming tensor decomposition approaches.

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“…However, there is a fundamental question: how can we determine the tensor rank and the associated model complexity? Because it is hard to exactly determine a tensor rank a-priori [30], existing methods often use a tensor rank pre-specified by the user or use a greedy method to update the tensor rank until convergence [24,31,32]. These methods often offers inaccurate rank estimation and are complicated in computation.…”

Section: Introductionmentioning

confidence: 99%

High-Dimensional Uncertainty Quantification via Active and Rank-Adaptive Tensor Regression

Zhang

2020

2020 IEEE 29th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS)

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Fabrication process variations can significantly influence the performance and yield of nano-scale electronic and photonic circuits. Stochastic spectral methods have achieved great success in quantifying the impact of process variations, but they suffer from the curse of dimensionality. Recently, low-rank tensor methods have been developed to mitigate this issue, but two fundamental challenges remain open: how to automatically determine the tensor rank and how to adaptively pick the informative simulation samples. This paper proposes a novel tensor regression method to address these two challenges. We use a q / 2 group-sparsity regularization to determine the tensor rank. The resulting optimization problem can be efficiently solved via an alternating minimization solver. We also propose a twostage adaptive sampling method to reduce the simulation cost. Our method considers both exploration and exploitation via the estimated Voronoi cell volume and nonlinearity measurement respectively. The proposed model is verified with synthetic and some realistic circuit benchmarks, on which our method can well capture the uncertainty caused by 19 to 100 random variables with only 100 to 600 simulation samples.

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Rank adaptive tensor recovery based model reduction for partial differential equations with high-dimensional random inputs

Cited by 5 publications

References 38 publications

Tensor Train Random Projection

Tensor Train Random Projection

Streaming probabilistic tensor train decomposition

High-Dimensional Uncertainty Quantification via Active and Rank-Adaptive Tensor Regression

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