# Null Space Properties of Neural Networks with Applications to Image Steganography

arXiv: https://arxiv.org/abs/2401.10262 · HTML: https://mlsanity.com/paper/2401.10262/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 1 Jan 2024.

## Builds on (most cited first, 13 of 13)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Survey of Hallucination in Natural Language Generation](https://mlsanity.com/paper/2202.03629/) (2022)
- [Adversarial Examples Are Not Bugs, They Are Features](https://mlsanity.com/paper/1905.02175/) (2019)
- [On Faithfulness and Factuality in Abstractive Summarization](https://mlsanity.com/paper/2005.00661/) (2020)
- [Universal adversarial perturbations](https://mlsanity.com/paper/1610.08401/) (2016)
- [One pixel attack for fooling deep neural networks](https://mlsanity.com/paper/1710.08864/) (2017)
- [Threat of Adversarial Attacks on Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/1801.00553/) (2018)
- [Adversarial Examples Make Strong Poisons](https://mlsanity.com/paper/2106.10807/) (2021)
- [Outlier Detection through Null Space Analysis of Neural Networks](https://mlsanity.com/paper/2007.01263/) (2020)
- [Quantifying Overfitting: Evaluating Neural Network Performance through Analysis of Null Space](https://mlsanity.com/paper/2305.19424/) (2023)
- [Hiding data inside images using orthogonal moments](https://mlsanity.com/paper/1910.07383/) (2019)

## Built on by (most cited first, 2 of 2)

- [Make it SING: Analyzing Semantic Invariants in Classifiers](https://mlsanity.com/paper/2603.14610/) (2026)
- [What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry](https://mlsanity.com/paper/2607.13046/) (2026)

## Abstract (arXiv)

This paper explores the null space properties of neural networks. We extend the null space definition from linear to nonlinear maps and discuss the presence of a null space in neural networks. The null space of a given neural network can tell us the part of the input data that makes no contribution to the final prediction so that we can use it to trick the neural network. This reveals an inherent weakness in neural networks that can be exploited. One application described here leads to a method of image steganography. Through experiments on image datasets such as MNIST, we show that we can use null space components to force the neural network to choose a selected hidden image class, even though the overall image can be made to look like a completely different image. We conclude by showing comparisons between what a human viewer would see, and the part of the image that the neural network is actually using to make predictions and, hence, show that what the neural network ``sees'' is completely different than what we would expect.