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arxiv_ml 90% Match Research Paper Theoretical computer scientists,Machine learning theorists,Researchers in neural network analysis,AI researchers focused on interpretability 1 week ago

The Computational Complexity of Counting Linear Regions in ReLU Neural Networks

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📄 Abstract

Abstract: An established measure of the expressive power of a given ReLU neural network is the number of linear regions into which it partitions the input space. There exist many different, non-equivalent definitions of what a linear region actually is. We systematically assess which papers use which definitions and discuss how they relate to each other. We then analyze the computational complexity of counting the number of such regions for the various definitions. Generally, this turns out to be an intractable problem. We prove NP- and #P-hardness results already for networks with one hidden layer and strong hardness of approximation results for two or more hidden layers. Finally, on the algorithmic side, we demonstrate that counting linear regions can at least be achieved in polynomial space for some common definitions.
Authors (3)
Moritz Stargalla
Christoph Hertrich
Daniel Reichman
Submitted
May 22, 2025
arXiv Category
cs.CC
arXiv PDF

Key Contributions

This paper systematically clarifies different definitions of linear regions in ReLU networks and analyzes their computational complexity. It proves NP- and #P-hardness for counting these regions even for single-hidden-layer networks, and strong hardness of approximation results for deeper networks, while showing polynomial space algorithms exist for some definitions.

Business Value

Provides fundamental theoretical understanding of neural network complexity, which can inform the design of more efficient and interpretable models, and guide research into approximation algorithms.

Paper Metadata

Innovation Type

Theoretical/Analytical

Deployment Feasibility

High (theoretical work), informs future practical developments.

Limitations Addressed

The lack of clear definitions and the unknown computational complexity of counting linear regions, a key measure of neural network expressivity.

Performance Gains

N/A (theoretical results)

Technical Tags

ReLU Neural NetworksLinear RegionsComputational ComplexityExpressive PowerNP-hardness#P-hardnessApproximation HardnessPolynomial SpaceInput Space PartitioningModel Theory

Research Topics

Theoretical Computer ScienceMachine Learning TheoryNeural Network AnalysisComputational ComplexityAI Interpretability

Methods & Architectures

Complexity analysisProof of NP-hardnessProof of #P-hardnessProof of hardness of approximationPolynomial space algorithms ReLU Neural Networks (single and multi-hidden layer)

Applications & Tasks

Theoretical Machine Learning Neural Network Analysis AI Model Understanding Determining the computational complexity of counting linear regionsClarifying definitions of linear regionsEstablishing hardness results for counting linear regions Analyzing neural network expressivityUnderstanding model behaviorTheoretical analysis of neural networks

Related Fields

Computer ScienceMathematicsMachine LearningComplexity Theory

Keywords

ReLU networksLinear regionsComputational complexityExpressive powerNP-hardness#P-hardnessApproximation algorithmsNeural network theoryInput space partitioningModel interpretabilityDeep learning theoryComplexity theory

Academic Context

#Theoretical Computer Science#Machine Learning Theory#Neural Network Analysis#Computational Complexity#AI Interpretability

Commercial Potential

Potential Products
Theoretical frameworks for model analysisTools for complexity assessment of NNs
Target Industries
Technology (AI Research)Academia
Use Case Examples
Understanding the theoretical limits of simple neural networksGuiding the development of more efficient network architecturesFormal analysis of model decision boundaries
Competitive Edge
Establishes fundamental theoretical limits and complexities related to a key aspect of neural network expressivity.
Market Opportunity
N/A (theoretical)
Revenue Models
N/A (theoretical)

Resource Requirements

Compute Needs
Low (theoretical analysis)
Data Requirements
N/A (theoretical)
Scalability
Focuses on the inherent complexity of the problem, not algorithmic scalability.

Production Readiness

Maturity Level

Theoretical Foundation

Time to Market

N/A (theoretical)

Patent Potential

Low

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