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On PDE Characterization of Smooth Hierarchical Functions Computed by Neural Networks.

Khashayar Filom¹, Roozbeh Farhoodi², Konrad Paul Kording³

¹Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A. filom@umich.edu.

Neural Computation

|October 28, 2021

Summary

Deep neural networks can compute complex functions. This study shows that the functions computed by hierarchical neural networks satisfy specific partial differential equations (PDEs) based on network topology.

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Deep Neural Networks for Image-Based Dietary Assessment

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Published on: March 13, 2021

Area of Science:

Computational theory
Artificial intelligence
Mathematical analysis

Background:

Neural networks approximate functions, but understanding their expressive power is key.
Hierarchical feedforward networks compose simpler functions to compute complex ones.

Purpose of the Study:

To investigate the expressivity of hierarchical feedforward neural networks.
To identify algebraic constraints on functions computed by these networks.

Main Methods:

Studying functions implemented by composing simpler functions in two specific regimes.
Analyzing functions that are compositions of functions with fewer inputs or nonlinear univariate functions applied to linear multivariate functions.
Deriving partial differential equations (PDEs) based on network topology.

Main Results:

Established that computed functions satisfy nontrivial algebraic partial differential equations (PDEs).
These PDEs depend solely on the network's topology and involve only partial derivatives.
Conjectured that these PDE constraints, with additional conditions, characterize representable functions.

Conclusions:

The study provides a step towards an algebraic description of function spaces in neural networks.
Derived PDEs offer potential new tools for constructing neural networks.
The findings are verified for various architectures, including neuroscientifically relevant tree structures.