# High-Dimensional Uncertainty Quantification via Tensor Regression with Rank Determination and Adaptive Sampling

arXiv: https://arxiv.org/abs/2103.17236 · HTML: https://mlsanity.com/paper/2103.17236/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 62 references that are in the map.
- First posted on arXiv 31 Mar 2021.

## Builds on (most cited first, 16 of 16)

- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Tensor Regression Networks](https://mlsanity.com/paper/1707.08308/) (2017)
- [Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark](https://mlsanity.com/paper/2002.01290/) (2020)
- [Bayesian Tensor Regression](https://mlsanity.com/paper/1509.06490/) (2015)
- [Low-Rank Matrix and Tensor Completion via Adaptive Sampling](https://mlsanity.com/paper/1304.4672/) (2013)
- [Tensor Regression Meets Gaussian Processes](https://mlsanity.com/paper/1710.11345/) (2017)
- [A least-squares method for sparse low rank approximation of multivariate functions](https://mlsanity.com/paper/1305.0030/) (2013)
- [An Efficient Multi-fidelity Bayesian Optimization Approach for Analog Circuit Synthesis](https://mlsanity.com/paper/1912.00392/) (2019)
- [Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition](https://mlsanity.com/paper/1407.3023/) (2014)
- [Active Sampling for Accelerated MRI with Low-Rank Tensors](https://mlsanity.com/paper/2012.12496/) (2020)
- [Stochastic Testing Simulator for Integrated Circuits and MEMS: Hierarchical and Sparse Techniques](https://mlsanity.com/paper/1409.4822/) (2014)
- [Stochastic Testing Method for Transistor-Level Uncertainty Quantification Based on Generalized Polynomial Chaos](https://mlsanity.com/paper/1409.4831/) (2014)
- [A Big-Data Approach to Handle Process Variations: Uncertainty Quantification by Tensor Recovery](https://mlsanity.com/paper/1603.06119/) (2016)
- [Stochastic Collocation with Non-Gaussian Correlated Process Variations: Theory, Algorithms and Applications](https://mlsanity.com/paper/1808.09720/) (2018)
- [Prediction of multi-dimensional spatial variation data via Bayesian tensor completion](https://mlsanity.com/paper/1901.00578/) (2019)
- [Efficient Uncertainty Modeling for System Design via Mixed Integer Programming](https://mlsanity.com/paper/1907.05700/) (2019)

## Built on by (most cited first, 1 of 1)

- [Tensor Shape Search for Optimum Data Compression](https://mlsanity.com/paper/2205.10651/) (2022)

## Abstract (arXiv)

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 ℓq/ ℓ2 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 two-stage 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.