# Nonlinear System Identification via Tensor Completion

arXiv: https://arxiv.org/abs/1906.05746 · HTML: https://mlsanity.com/paper/1906.05746/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2019.

## Built on by (most cited first, 4 of 4)

- [Implicit Regularization in Hierarchical Tensor Factorization and Deep Convolutional Neural Networks](https://mlsanity.com/paper/2201.11729/) (2022)
- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [TTML: tensor trains for general supervised machine learning](https://mlsanity.com/paper/2203.04352/) (2022)
- [Complex-valued Adaptive System Identification via Low-Rank Tensor Decomposition](https://mlsanity.com/paper/2306.16428/) (2023)

## Abstract (arXiv)

Function approximation from input and output data pairs constitutes a fundamental problem in supervised learning. Deep neural networks are currently the most popular method for learning to mimic the input-output relationship of a general nonlinear system, as they have proven to be very effective in approximating complex highly nonlinear functions. In this work, we show that identifying a general nonlinear function y = f(x1,ldots,xN) from input-output examples can be formulated as a tensor completion problem and under certain conditions provably correct nonlinear system identification is possible. Specifically, we model the interactions between the N input variables and the scalar output of a system by a single N-way tensor, and setup a weighted low-rank tensor completion problem with smoothness regularization which we tackle using a block coordinate descent algorithm. We extend our method to the multi-output setting and the case of partially observed data, which cannot be readily handled by neural networks. Finally, we demonstrate the effectiveness of the approach using several regression tasks including some standard benchmarks and a challenging student grade prediction task.