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Depth-Bounds for Neural Networks via the Braid Arrangement

large-language-models › model-architecture

📄 Abstract

Abstract: We contribute towards resolving the open question of how many hidden layers are required in ReLU networks for exactly representing all continuous and piecewise linear functions on $\mathbb{R}^d$. While the question has been resolved in special cases, the best known lower bound in general is still 2. We focus on neural networks that are compatible with certain polyhedral complexes, more precisely with the braid fan. For such neural networks, we prove a non-constant lower bound of $\Omega(\log\log d)$ hidden layers required to exactly represent the maximum of $d$ numbers. Additionally, under our assumption, we provide a combinatorial proof that 3 hidden layers are necessary to compute the maximum of 5 numbers; this had only been verified with an excessive computation so far. Finally, we show that a natural generalization of the best known upper bound to maxout networks is not tight, by demonstrating that a rank-3 maxout layer followed by a rank-2 maxout layer is sufficient to represent the maximum of 7 numbers.

Authors (3)

Moritz Grillo

Christoph Hertrich

Georg Loho

Submitted

February 13, 2025

arXiv Category

cs.LG

arXiv PDF

Key Contributions

This paper establishes non-constant lower bounds on the depth of ReLU networks for representing certain piecewise linear functions, specifically proving $\Omega(\log\log d)$ layers are needed for the max function on $d$ numbers under assumptions related to the braid fan. It also provides a combinatorial proof for the necessity of 3 layers for computing the max of 5 numbers and shows limitations of generalizing upper bounds to maxout networks.

Business Value

Contributes to a fundamental understanding of neural network expressivity, which can guide the design of more efficient and powerful architectures in the future.

Paper Metadata

Innovation Type

Theoretical Analysis & Bounds

Deployment Feasibility

Low (theoretical research, not directly applicable to deployment)

Limitations Addressed

Addresses the open question regarding the minimum depth required for ReLU networks to represent arbitrary piecewise linear functions, improving upon the existing lower bound of 2.

Technical Tags

neural network depthReLU networkspiecewise linear functionslower boundsbraid arrangementmaxout networkscomputational complexityrepresentational capacitycombinatorial proof

Research Topics

Theoretical Deep LearningNeural Network ArchitecturesComputational ComplexityFunction Approximation

Methods & Architectures

Combinatorial AnalysisGeometric Analysis (Braid Arrangement)Proof Techniques ReLU NetworksMaxout Networks

Applications & Tasks

Theoretical Computer Science Machine Learning Theory Determining Minimum Network DepthUnderstanding Representational Power Representing piecewise linear functionsComputing the maximum of numbers

Related Fields

Theoretical Computer ScienceMachine Learning TheoryDeep LearningComputational Geometry

Keywords

neural network depthReLUpiecewise linearlower boundbraid arrangementmaxoutrepresentational capacitycomputational complexityfunction approximationnetwork architecturetheoretical computer sciencecombinatorial proof

Academic Context

#Theoretical Deep Learning#Neural Network Architectures#Computational Complexity#Function Approximation

Commercial Potential

Competitive Edge

Advances the theoretical understanding of neural network depth requirements, providing tighter bounds than previously known.

Market Opportunity

Indirect impact on the vast market for AI/ML models.

Revenue Models

N/A

Resource Requirements

Compute Needs

Minimal (theoretical analysis)

Data Requirements

Not applicable (theoretical study)

Deployment Constraints

Not applicable

Scalability

Focuses on theoretical scalability with input dimension 'd'.

Production Readiness

Maturity Level

Foundational Research

Time to Market

Long-term (impacts future architecture design)

Patent Potential

Very Low (theoretical results)

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