# On Functional Dimension and Persistent Pseudodimension

arXiv: https://arxiv.org/abs/2410.17191 · HTML: https://mlsanity.com/paper/2410.17191/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 22 Oct 2024.

## Builds on (most cited first, 15 of 15)

- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Reverse-Engineering Deep ReLU Networks](https://mlsanity.com/paper/1910.00744/) (2019)
- [The Modern Mathematics of Deep Learning](https://mlsanity.com/paper/2105.04026/) (2021)
- [Hidden symmetries of ReLU networks](https://mlsanity.com/paper/2306.06179/) (2023)
- [On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks](https://mlsanity.com/paper/2008.09052/) (2020)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [Functional dimension of feedforward ReLU neural networks](https://mlsanity.com/paper/2209.04036/) (2022)
- [An Embedding of ReLU Networks and an Analysis of their Identifiability](https://mlsanity.com/paper/2107.09370/) (2021)
- [Local Identifiability of Deep ReLU Neural Networks: the Theory](https://mlsanity.com/paper/2206.07424/) (2022)
- [Geometry-induced Regularization in Deep ReLU Neural Networks](https://mlsanity.com/paper/2402.08269/) (2024)

## Built on by (most cited first, 1 of 1)

- [Regularization Implies balancedness in the deep linear network](https://mlsanity.com/paper/2511.01137/) (2025)

## Abstract (arXiv)

For any fixed feedforward ReLU neural network architecture, it is well-known that many different parameter settings can determine the same function. It is less well-known that the degree of this redundancy is inhomogeneous across parameter space. In this work, we discuss two locally applicable complexity measures for ReLU network classes and what we know about the relationship between them: (1) the local functional dimension [14, 18], and (2) a local version of VC dimension that we call persistent pseudodimension. The former is easy to compute on finite batches of points; the latter should give local bounds on the generalization gap, which would inform an understanding of the mechanics of the double descent phenomenon [7].