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From Universal Approximation Theorem to Tropical Geometry of Multi-Layer Perceptrons

large-language-models › model-architecture

📄 Abstract

Abstract: We revisit the Universal Approximation Theorem(UAT) through the lens of the tropical geometry of neural networks and introduce a constructive, geometry-aware initialization for sigmoidal multi-layer perceptrons (MLPs). Tropical geometry shows that Rectified Linear Unit (ReLU) networks admit decision functions with a combinatorial structure often described as a tropical rational, namely a difference of tropical polynomials. Focusing on planar binary classification, we design purely sigmoidal MLPs that adhere to the finite-sum format of UAT: a finite linear combination of shifted and scaled sigmoids of affine functions. The resulting models yield decision boundaries that already align with prescribed shapes at initialization and can be refined by standard training if desired. This provides a practical bridge between the tropical perspective and smooth MLPs, enabling interpretable, shape-driven initialization without resorting to ReLU architectures. We focus on the construction and empirical demonstrations in two dimensions; theoretical analysis and higher-dimensional extensions are left for future work.

Authors (2)

Yi-Shan Chu

Yueh-Cheng Kuo

Submitted

October 16, 2025

arXiv Category

stat.ML

arXiv PDF

Key Contributions

Revisits the Universal Approximation Theorem using tropical geometry to introduce a geometry-aware initialization for sigmoidal MLPs. This approach allows for the construction of MLPs whose decision boundaries align with prescribed shapes at initialization, offering an interpretable alternative to ReLU networks without sacrificing approximation power.

Business Value

Provides a deeper theoretical understanding of neural networks, potentially leading to more stable and interpretable models in applications where decision boundary shapes are critical.

Paper Metadata

Innovation Type

theoretical framework and initialization method

Deployment Feasibility

Low for direct deployment, high for informing the design of future ML architectures and training procedures.

Limitations Addressed

Addresses the lack of interpretable and shape-driven initialization methods for sigmoidal MLPs, providing a theoretical bridge between tropical geometry and practical network construction.

Performance Gains

Enables interpretable, shape-driven initialization without resorting to ReLU architectures.

Technical Tags

universal approximation theoremtropical geometrymulti-layer perceptronssigmoidal activationrectified linear unit (relu)decision boundariesinitializationcombinatorial structureinterpretable AI

Research Topics

neural network theorygeometric deep learningnetwork initializationunderstanding deep learninginterpretable ML

Methods & Architectures

Tropical geometry analysisGeometry-aware initializationConstructive proof Multi-layer Perceptrons (MLPs)Sigmoidal MLPsReLU networks

Applications & Tasks

theoretical computer science machine learning theory understanding neural network behaviorimproving network initialization constructing sigmoidal MLPsachieving shape-driven initialization

Related Fields

mathematicsalgebraic geometrycomputational complexitydeep learning theory

Keywords

Universal Approximation Theoremtropical geometrymulti-layer perceptronMLPsigmoidal activationReLUinitializationdecision boundaryinterpretable AIneural network theoryconstructive methodsshape-driven initialization

Academic Context

#neural network theory#geometric deep learning#network initialization#understanding deep learning#interpretable ML

Commercial Potential

Use Case Examples

Designing neural networks with specific classification boundary shapes from the outset.

Competitive Edge

Offers a novel theoretical perspective and practical initialization technique that complements existing methods for MLP design and understanding.

Market Opportunity

N/A (theoretical research)

Revenue Models

N/A (theoretical research)

Resource Requirements

Compute Needs

Minimal, primarily for theoretical analysis and small-scale experiments.

Data Requirements

Not applicable for the core theoretical contribution; may require synthetic data for demonstration.

Deployment Constraints

Primarily theoretical; practical application depends on integration into ML frameworks.

Scalability

The theoretical framework is general, but its application to very large networks might require further investigation.

Production Readiness

Maturity Level

Theoretical Foundation

Time to Market

5+ years

Patent Potential

Low, as it's a theoretical contribution.

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