Efficient Generalized Spherical CNNs (2010.11661v3)

Abstract: 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.

• The paper proposes novel spherical CNN layers that reduce complexity from O(C²L⁵) to O(CL⁴) or O(CL³ log L) while maintaining rotational equivariance.
• It leverages a channel-wise structure and constrained convolution steps to optimize parameter efficiency and enable more expressive models.
• The efficiency gains are validated in tasks like spherical MNIST classification, QM7 atomization energy regression, and 3D object retrieval.

Efficient Generalized Spherical CNNs

The paper "Efficient Generalized Spherical CNNs" addresses critical computational challenges in the domain of spherical convolutional neural networks (CNNs), which are pivotal across several disciplines where data inherently lies on spherical manifolds. Examples include 360-degree virtual reality applications, cosmic microwave background studies in cosmology, and molecular chemistry. Traditional CNNs, designed for Euclidean spaces, fail to capture the unique geometric peculiarities of spherical domains, necessitating dedicated methodologies like spherical CNNs.

The paper builds upon existing spherical CNN frameworks, offering a generalized method that subsumes several prominent approaches. The primary impediment to deploying strictly equivariant spherical CNNs has been prohibitive computational complexity, quantified as $\mathcal{O}(C^2L^5)$, where $C$ symbolizes representational capacity and $L$ signifies the spherical harmonic bandlimit. To address this, the paper introduces two novel spherical CNN layers that significantly reduce complexity to either $\mathcal{O}(CL^4)$ or $\mathcal{O}(CL^3 \log L)$, enabling larger and more expressive models to be practically feasible.

The researchers achieved these advancements through a strategic focus on three key areas:

1. Channel-Wise Structure: The introduction of a multi-channel framework allows the efficient utilization of representational resources, aligning more closely with standard CNN practices, thereby reducing computation from previously quadratic dependence to linear relative to representational capacity.
2. Constrained Generalized Convolutions: By segmenting the generalized convolution process into distinct steps—projecting across channels, within each channel, and finally across the channel space—parameter efficiency and computational savings are significantly enhanced.
3. Optimized Degree Mixing Sets: Using minimum spanning tree (MST) principles, they identify essential subsets of degrees to conduct convolutions, thus reducing the spatial complexity from $\mathcal{O}(L^5)$ to $\mathcal{O}(L^4)$ or, optionally, to $\mathcal{O}(L^3\log L)$ with a reduced variant.

Innovatively, this methodology preserves strict rotational equivariance, which is vital for retaining the geometric symmetry inherent in spherical data. Efficiency enhancements extend to adoption of sampling theorems that halve the Nyquist rate compared to older approaches.

The implication of this paper is twofold. Practically, it equips researchers and engineers with tools to deploy spherical CNNs in computationally restrictive environments while achieving state-of-the-art performance, as validated in applications like spherical MNIST classification, QM7 atomization energy regression, and 3D object retrieval tasks. Theoretically, it facilitates further explorations into hybrid models that weave together diverse spherical CNN approaches within a unified framework, potentiating advancements across varied scientific and technological fields.

Looking forward, this research lays groundwork for even more scalable spherical learning paradigms, including potentially extending scattering transforms to spherical contexts. This is pivotal as the integration and analysis of spherical data saturate more fields beyond traditional boundaries, underscoring the transformative potential of efficient spherical CNNs in AI and deep learning landscapes.