Efficient Generalized Spherical CNNs. (arXiv:2010.11661v1 [cs.CV])
<a href="http://arxiv.org/find/cs/1/au:+Cobb_O/0/1/0/all/0/1">Oliver J. Cobb</a>, <a href="http://arxiv.org/find/cs/1/au:+Wallis_C/0/1/0/all/0/1">Christopher G. R. Wallis</a>, <a href="http://arxiv.org/find/cs/1/au:+Mavor_Parker_A/0/1/0/all/0/1">Augustine N. Mavor-Parker</a>, <a href="http://arxiv.org/find/cs/1/au:+Marignier_A/0/1/0/all/0/1">Augustin Marignier</a>, <a href="http://arxiv.org/find/cs/1/au:+Price_M/0/1/0/all/0/1">Matthew A. Price</a>, <a href="http://arxiv.org/find/cs/1/au:+dAvezac_M/0/1/0/all/0/1">Mayeul d&#x27;Avezac</a>, <a href="http://arxiv.org/find/cs/1/au:+McEwen_J/0/1/0/all/0/1">Jason D. McEwen</a>

Many problems across computer vision and the natural sciences require the
analysis of spherical data, for which representations may be learned
efficiently by encoding equivariance to rotational symmetries. We present a
generalized spherical CNN framework that encompasses various existing
approaches and allows them to be leveraged alongside each other. The only
existing non-linear spherical CNN layer that is strictly equivariant has
complexity $mathcal{O}(C^2L^5)$, where $C$ is a measure of representational
capacity and $L$ the spherical harmonic bandlimit. Such a high computational
cost often prohibits the use of strictly equivariant spherical CNNs. We develop
two new strictly equivariant layers with reduced complexity $mathcal{O}(CL^4)$
and $mathcal{O}(CL^3 log L)$, making larger, more expressive models
computationally feasible. Moreover, we adopt efficient sampling theory to
achieve further computational savings. We show that these developments allow
the construction of more expressive hybrid models that achieve state-of-the-art
accuracy and parameter efficiency on spherical benchmark problems.

Many problems across computer vision and the natural sciences require the
analysis of spherical data, for which representations may be learned
efficiently by encoding equivariance to rotational symmetries. We present a
generalized spherical CNN framework that encompasses various existing
approaches and allows them to be leveraged alongside each other. The only
existing non-linear spherical CNN layer that is strictly equivariant has
complexity $mathcal{O}(C^2L^5)$, where $C$ is a measure of representational
capacity and $L$ the spherical harmonic bandlimit. Such a high computational
cost often prohibits the use of strictly equivariant spherical CNNs. We develop
two new strictly equivariant layers with reduced complexity $mathcal{O}(CL^4)$
and $mathcal{O}(CL^3 log L)$, making larger, more expressive models
computationally feasible. Moreover, we adopt efficient sampling theory to
achieve further computational savings. We show that these developments allow
the construction of more expressive hybrid models that achieve state-of-the-art
accuracy and parameter efficiency on spherical benchmark problems.

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