Triangulating PL functions and the existence of efficient ReLU DNNs (2505.07137v1)

Abstract: We show that every piecewise linear function $f:R^d \to R$ with compact support a polyhedron $P$ has a representation as a sum of so-called `simplex functions'. Such representations arise from degree 1 triangulations of the relative homology class (in $R^{d+1}$) bounded by $P$ and the graph of $f$, and give a short elementary proof of the existence of efficient universal ReLU neural networks that simultaneously compute all such functions $f$ of bounded complexity.

Overview of "Triangulating PL Functions and the Existence of Efficient ReLU DNNs"

In the paper "Triangulating PL functions and the existence of efficient ReLU DNNs," Danny Calegari presents a novel proof concerning the representation of piecewise linear (PL) functions using ReLU neural networks. This paper explores the theoretical aspects of PL functions with compact support, illustrating how they can be formulated as sums of simplex functions, which are fundamentally tied to the geometrical structure of the function’s graph.

Representation of PL Functions

The paper's primary contribution lies in its demonstration that every PL function $f$ from $\mathbb{R}^d$ to $\mathbb{R}$, supported compactly by a polyhedron $P$, can be expressed as a sum of simplex functions arising from degree 1 triangulations involving the relative homology class bounded by $P$ and the graph of $f$. The paper provides a concise, elementary proof of this representation, contributing to the longstanding body of work on efficient architectures of universal ReLU neural networks capable of computing all such functions with bounded complexity.

Simplex Functions and Neural Networks

The key theoretical element introduced is the concept of simplex functions. These functions stem from nondegenerate simplices in $\mathbb{R}^{d+1}$, and their graphs form pyramids with bases that are simple polyhedra in $\mathbb{R}^d$. This structural understanding leads to the construction of a ReLU neural network that efficiently computes these simplex functions using optimal constants. The network’s architecture is fixed and capable of representing a wide range of PL functions given an appropriate parameterization. With respect to depth, width, and size, the neural network is constructed to be more efficient than existing models.

Theoretical Implications and Hyperbolic Geometry

The paper further explores the relationship between hyperbolic geometry and the computational complexity of polyhedral representations, analogizing with the complexity theories in computational settings known from Sleator-Tarjan-Thurston and Cohn-Kenyon-Propp theories. It posits that the algebraic volumes enclosed by totally geodesic immersions in hyperbolic space can be correlated with the lower bounds on the complexity of function representation by simplex functions.

Practical and Future Implications

This work implicates a practical improvement in the construction of neural networks, enhancing efficiency—particularly in scenarios necessitating the representation of complex PL functions. The insights gained from simplicial and hyperbolic geometric perspectives may lead future research toward computational methods that optimize neural network training and function approximation in high-dimensional spaces.

Conclusion

In conclusion, Calegari’s work advances the understanding of how combinatorial and geometric insights can be leveraged to optimize the representation and computational efficiency of PL functions using ReLU neural networks. This mathematical approach enriches theoretical foundations pertinent to both artificial intelligence and computational geometry fields, and paves avenues for further exploration into geometric methods that may optimize computational applications in neural modeling.