PowerNet: Truncated Matrix Power Series as Quasi-Equivariant Layers

Alex Gabel, Rick Quax, Efstratios Gavves

Keywords: symmetry, deep learning, geometry, Lie theory

Abstract:

Despite being theoretically well-grounded, enforcing strict equivariance in deep learning models has shown to be harmful in some cases. The problem is that most available data does not follow mathematically precise rules, is noisy, and is not strictly group-structured. While soft equivariance approaches attempt to address these issues, they often struggle to maintain group structure and lack strong theoretical guarantees, potentially compromising the benefits of equivariance. Here we introduce the concept of \textit{quasi equivariances}, where group structure is maintained but the associated parameters become distributions, and implement it in the proposed \textit{PowerNet} architecture. Similar to CNNs, PowerNet is constructed by interlacing truncated matrix power series with non-linearities. We show how the base matrix used to define the power series can instill quasi-equivariance in a natural way. Finally, we provide results for augmented MNIST classification and transformation magnitude regression in addition to classification of CIFAR-10.

Primary Area: learning theory

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Submission Number: 4405