# What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry

arXiv: https://arxiv.org/abs/2607.13046 · HTML: https://mlsanity.com/paper/2607.13046/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 39 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 19 of 19)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Deep Neural Network Fingerprinting by Conferrable Adversarial Examples](https://mlsanity.com/paper/1912.00888/) (2019)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Federated Learning: Opportunities and Challenges](https://mlsanity.com/paper/2101.05428/) (2021)
- [IPGuard: Protecting Intellectual Property of Deep Neural Networks via Fingerprinting the Classification Boundary](https://mlsanity.com/paper/1910.12903/) (2019)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Fingerprinting Deep Neural Networks Globally via Universal Adversarial Perturbations](https://mlsanity.com/paper/2202.08602/) (2022)
- [Outlier Detection through Null Space Analysis of Neural Networks](https://mlsanity.com/paper/2007.01263/) (2020)
- [Null Space Properties of Neural Networks with Applications to Image Steganography](https://mlsanity.com/paper/2401.10262/) (2024)

## Abstract (arXiv)

We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation π of a Lie group G on a space V and a learned function fcolon V → ℝ, we define two objects measuring the symmetry invisible to f. The null fiber at a point x ∈ V is the set NG(f,x) = g ∈ G : f(π(g-1) · x) = f(x) of group elements whose inverse action on x is undetectable by f. When NG(f,x) is independent of x, it coincides with the stabilizer StabG(f), the largest subgroup of G under which f is invariant. For smooth maps to ℝ, the preimage theorem guarantees that null fibers have dimension at least dim G - 1 at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of f. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under SO(3) and spherical image classification under the M"obius group PSL(2, mathbbC). The framework applies uniformly to classical neural networks and variational quantum circuits.