# Nonlinear System Identification via Tensor Completion

## TL;DR

This paper introduces a novel tensor completion approach for nonlinear system identification, offering an alternative to neural networks that handles multi-output and partial data scenarios effectively.

## Contribution

The paper formulates nonlinear system identification as a tensor completion problem and extends it to multi-output and partially observed data cases, demonstrating its effectiveness.

## Key findings

- Effective in standard regression benchmarks
- Handles multi-output systems and partial data
- Outperforms neural networks in certain scenarios

## Abstract

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 $y = f(x_1,\ldots,x_N)$ 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.