# Topology and Geometry of the Learning Space of ReLU Networks: Connectivity and Singularities

arXiv: https://arxiv.org/abs/2602.00693 · HTML: https://mlsanity.com/paper/2602.00693/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 76 references that are in the map.
- First posted on arXiv 31 Jan 2026.

## Builds on (most cited first, 30 of 49)

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [SparseGPT: Massive Language Models Can Be Accurately Pruned in One-Shot](https://mlsanity.com/paper/2301.00774/) (2023)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Rethinking the Value of Network Pruning](https://mlsanity.com/paper/1810.05270/) (2018)
- [Sparsity in Deep Learning: Pruning and growth for efficient inference and training in neural networks](https://mlsanity.com/paper/2102.00554/) (2021)
- [Pruning neural networks without any data by iteratively conserving synaptic flow](https://mlsanity.com/paper/2006.05467/) (2020)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [DepGraph: Towards Any Structural Pruning](https://mlsanity.com/paper/2301.12900/) (2023)
- [A Survey on Deep Neural Network Pruning-Taxonomy, Comparison, Analysis, and Recommendations](https://mlsanity.com/paper/2308.06767/) (2023)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Drawing Early-Bird Tickets: Towards More Efficient Training of Deep Networks](https://mlsanity.com/paper/1909.11957/) (2019)
- [Dynamic Model Pruning with Feedback](https://mlsanity.com/paper/2006.07253/) (2020)
- [Path-SGD: Path-Normalized Optimization in Deep Neural Networks](https://mlsanity.com/paper/1506.02617/) (2015)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [Winning the Lottery with Continuous Sparsification](https://mlsanity.com/paper/1912.04427/) (2019)
- [Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics](https://mlsanity.com/paper/2012.04728/) (2020)
- [Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Bias](https://mlsanity.com/paper/2110.13905/) (2021)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [Operation-Aware Soft Channel Pruning using Differentiable Masks](https://mlsanity.com/paper/2007.03938/) (2020)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Deep Learning is Singular, and That's Good](https://mlsanity.com/paper/2010.11560/) (2020)
- [Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows](https://mlsanity.com/paper/2307.00144/) (2023)

## Built on by (most cited first, 6 of 6)

- [Most ReLU Networks Admit Identifiable Parameters](https://mlsanity.com/paper/2605.03601/) (2026)
- [The Symmetries of Three-Layer ReLU Networks](https://mlsanity.com/paper/2605.18319/) (2026)
- [From Approximation Rates to Loss-Landscape Barrier Decay in Shallow ReLU Networks](https://mlsanity.com/paper/2602.17596/) (2026)
- [Stratifying Reinforcement Learning with Signal Temporal Logic](https://mlsanity.com/paper/2604.04923/) (2026)
- [A Theory of Saddle Escape in Deep Nonlinear Networks](https://mlsanity.com/paper/2605.01288/) (2026)
- [On Parameter Symmetries and Conservation Laws in Gradient Flow](https://mlsanity.com/paper/2609.34549/) (2026)

## Abstract (arXiv)

Understanding the properties of the parameter space in feed-forward ReLU networks is critical for effectively analyzing and guiding training dynamics. After initialization, training under gradient flow decisively restricts the parameter space to an algebraic variety that emerges from the homogeneous nature of the ReLU activation function. In this study, we examine two key challenges associated with feed-forward ReLU networks built on general directed acyclic graph (DAG) architectures: the (dis)connectedness of the parameter space and the existence of singularities within it. We extend previous results by providing a thorough characterization of connectedness, highlighting the roles of bottleneck nodes and balance conditions associated with specific subsets of the network. Our findings clearly demonstrate that singularities are intricately connected to the topology of the underlying DAG and its induced sub-networks. We discuss the reachability of these singularities and establish a principled connection with differentiable pruning. We validate our theory with simple numerical experiments.