# Learning Sub-Patterns in Piecewise Continuous Functions

arXiv: https://arxiv.org/abs/2010.15571 · HTML: https://mlsanity.com/paper/2010.15571/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2020.

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

- [Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer](https://mlsanity.com/paper/2201.13094/) (2022)
- [Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?](https://mlsanity.com/paper/2204.11231/) (2022)
- [The Responsibility Problem in Neural Networks with Unordered Targets](https://mlsanity.com/paper/2304.09499/) (2023)
- [Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts](https://mlsanity.com/paper/2402.03460/) (2024)

## Abstract (arXiv)

Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters; however, this implies that the neural network's activation function must exhibit a degree of continuity which limits the neural network model's uniform approximation capacity to continuous functions. This paper focuses on the case where the discontinuities arise from distinct sub-patterns, each defined on different parts of the input space. We propose a new discontinuous deep neural network model trainable via a decoupled two-step procedure that avoids passing gradient updates through the network's only and strategically placed, discontinuous unit. We provide approximation guarantees for our architecture in the space of bounded continuous functions and universal approximation guarantees in the space of piecewise continuous functions which we introduced herein. We present a novel semi-supervised two-step training procedure for our discontinuous deep learning model, tailored to its structure, and we provide theoretical support for its effectiveness. The performance of our model and trained with the propose procedure is evaluated experimentally on both real-world financial datasets and synthetic datasets.