Algebraic Neural Networks: Stability to Deformations

09/03/2020
βˆ™
by   Alejandro Parada-Mayorga, et al.
βˆ™
0
βˆ™

In this work we study the stability of algebraic neural networks (AlgNNs) with commutative algebras which unify CNNs and GNNs under the umbrella of algebraic signal processing. An AlgNN is a stacked layered structure where each layer is conformed by an algebra π’œ, a vector space β„³ and a homomorphism ρ:π’œβ†’End(β„³), where End(β„³) is the set of endomorphims of β„³. Signals in each layer are modeled as elements of β„³ and are processed by elements of End(β„³) defined according to the structure of π’œ via ρ. This framework provides a general scenario that covers several types of neural network architectures where formal convolution operators are being used. We obtain stability conditions regarding to perturbations which are defined as distortions of ρ, reaching general results whose particular cases are consistent with recent findings in the literature for CNNs and GNNs. We consider conditions on the domain of the homomorphisms in the algebra that lead to stable operators. Interestingly, we found that these conditions are related to the uniform boundedness of the FrΓ©chet derivative of a function p:End(β„³)β†’End(β„³) that maps the images of the generators of π’œ on End(β„³) into a power series representation that defines the filtering of elements in β„³. Additionally, our results show that stability is universal to convolutional architectures whose algebraic signal model uses the same algebra.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

βˆ™ 10/22/2020

Stability of Algebraic Neural Networks to Small Perturbations

Algebraic neural networks (AlgNNs) are composed of a cascade of layers e...
βˆ™ 08/23/2021

Convolutional Filtering and Neural Networks with Non Commutative Algebras

In this paper we provide stability results for algebraic neural networks...
βˆ™ 05/29/2018

Classification Stability for Sparse-Modeled Signals

Despite their impressive performance, deep convolutional neural networks...
βˆ™ 04/23/2021

Algebraic combinatory models

We introduce an equationally definable counterpart of the notion of comb...
βˆ™ 04/12/2021

Quantum protocols at presence of non-abelian superselection rules in the framework of algebraic model

In this paper, we study the influence of non-abelian superselection rule...
βˆ™ 02/16/2021

Finite Atomized Semilattices

We show that every finite semilattice can be represented as an atomized ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.