# Ghosts in Neural Networks: Existence, Structure and Role of Infinite-Dimensional Null Space

arXiv: https://arxiv.org/abs/2106.04770 · HTML: https://mlsanity.com/paper/2106.04770/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 49 references that are in the map.
- First posted on arXiv 9 Jun 2021.

## Builds on (most cited first, 23 of 23)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (2019)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1709.01953/) (2017)
- [The loss surface of deep and wide neural networks](https://mlsanity.com/paper/1704.08045/) (2017)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Width Provably Matters in Optimization for Deep Linear Neural Networks](https://mlsanity.com/paper/1901.08572/) (2019)
- [Compression based bound for non-compressed network: unified generalization error analysis of large compressible deep neural network](https://mlsanity.com/paper/1909.11274/) (2019)
- [Global Convergence of Adaptive Gradient Methods for An Over-parameterized Neural Network](https://mlsanity.com/paper/1902.07111/) (2019)
- [Universal Approximation Theorem for Equivariant Maps by Group CNNs](https://mlsanity.com/paper/2012.13882/) (2020)

## Built on by (most cited first, 7 of 7)

- [Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis](https://mlsanity.com/paper/2203.01631/) (2022)
- [Universality of Group Convolutional Neural Networks Based on Ridgelet Analysis on Groups](https://mlsanity.com/paper/2205.14819/) (2022)
- [Neural Hilbert Ladders: Multi-Layer Neural Networks in Function Space](https://mlsanity.com/paper/2307.01177/) (2023)
- [Joint Group Invariant Functions on Data-Parameter Domain Induce Universal Neural Networks](https://mlsanity.com/paper/2310.03530/) (2023)
- [A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networks](https://mlsanity.com/paper/2402.15984/) (2024)
- [Deep Ridgelet Transform and Unified Universality Theorem for Deep and Shallow Joint-Group-Equivariant Machines](https://mlsanity.com/paper/2405.13682/) (2024)
- [Approximate Nullspace Augmented Finetuning for Robust Vision Transformers](https://mlsanity.com/paper/2403.10476/) (2024)

## Abstract (arXiv)

We study parameter nonuniqueness in continuous-width depth-two fully connected neural networks. Our main contribution is a direct method for solving the neural-network equation S[γ]=f. Starting from the Fourier expression of the synthesis operator, separation of variables produces a ridgelet particular solution and identifies every homogeneous direction. To isolate the argument, we first prove an abstract reconstruction formula for unitary factorizations, yielding the adjoint, normalized right inverse, and orthogonal solution geometry. We then specialize this formula to neural-network synthesis: for tempered-distribution activations such as ReLU, we equip the activation class As,t with a Hilbert structure, construct compatible coefficient and parameter Hilbert spaces Hs,t and Gs,t, and prove that S:Gs,t→ L2(mathbb Rm) is bounded. The resulting ridgelet expansion exhausts the null space and the complete solution set and identifies the unique minimum-norm parameter distribution. Concrete examples give adjoint ridgelet functions for standard activations. Further developments show that finite-measure null elements admit normalized width-N discretizations with O(N-1/2) output error and characterize how additive parameter perturbations can reveal information encoded in the null space. A Lean 4 blueprint for the main results is available at https://shosonoda.github.io/lean-ridgelet/ .