# Universal Approximation Theorem for Input-Connected Multilayer Perceptrons

arXiv: https://arxiv.org/abs/2601.14026 · HTML: https://mlsanity.com/paper/2601.14026/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 20 Jan 2026.

## Builds on (most cited first, 9 of 9)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Deep, Skinny Neural Networks are not Universal Approximators](https://mlsanity.com/paper/1810.00393/) (2018)
- [Approximation capability of two hidden layer feedforward neural networks with fixed weights](https://mlsanity.com/paper/2101.09181/) (2021)
- [On the approximation by single hidden layer feedforward neural networks with fixed weights](https://mlsanity.com/paper/1708.06219/) (2017)
- [A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function](https://mlsanity.com/paper/1601.00013/) (2015)

## Abstract (arXiv)

We present the Input-Connected Multilayer Perceptron (IC-MLP), a feedforward neural network architecture in which each hidden neuron receives, in addition to the outputs of the preceding layer, a direct affine connection from the raw input. We first study this architecture in the univariate setting and give an explicit and systematic description of IC-MLPs with an arbitrary finite number of hidden layers, including iterated formulas for the network functions. In this setting, we prove a universal approximation theorem showing that deep IC-MLPs can approximate any continuous function on a closed interval of the real line if and only if the activation function is nonlinear. We then extend the analysis to vector-valued inputs and establish a corresponding universal approximation theorem for continuous functions on compact subsets of ℝn.