Hybrid Discrete-Continuous Geometric Deep Learning
No existing spherical convolutional neural network (CNN) framework is both computationally scalable and rotationally equivariant. Continuous approaches capture rotational equivariance but are often prohibitively computationally demanding. Discrete approaches offer more favorable computational performance but at the cost of equivariance. We develop a hybrid discrete-continuous (DISCO) group convolution that is simultaneously equivariant and computationally scalable to high-resolution. This approach achieves state-of-the-art (SOTA) performance on many benchmark dense prediction tasks. (Further details can be found in our ICLR paper on Scalable and Equivariant Spherical CNNs by DISCO Convolutions.)
Geometric Deep Learning on groups has many applications, such as analysing observations over the Earth and panoramic 360° photos and videos, to name just a few. However, current approaches suffer a dichotomy: they either exhibit good equivariance properties or good computationally scalability; but not both simultaneously.
Dichotomy: discrete vs continuous approaches
The key goals of Geometric Deep Learning techniques on groups is to encode equivariance to various group transformations (which typically translates to very good performance), while also being highly computationally scalable.
As discussed in our previous TDS article, focusing on the group setting of homogenous spaces with global symmetries, geometric deep learning on groups can be broadly classified into discrete and continuous approaches. Continuous approaches offer equivariance but with a large computational cost. Discrete approaches, on the other hand, are typically relatively computationally efficient but sacrifice equivariance.
Breaking the dichotomy: discrete-continuous (DISCO) approach
At Copernic AI we have recently developed techniques that break this dichotomy (recently published in ICLR [1]). That is, we have developed geometric deep learning techniques on groups that provide excellent equivariance properties, while also being highly computationally efficient so that they can be effectively scaled to huge, high-resolution datasets.
The key to breaking the discrete versus continuous dichotomy is to take a hybrid approach, where some parts of the representation are discretized, to facilitate efficient computation, while other parts are left continuous to facilitate equivariance. Due to its hybrid nature (as illustrated in the diagram below) we name this approach DISCO, for DIScrete-COntinous.
While the DISCO approach is general, we focus on the sphere as the archetypical example of the group setting of homogenous spaces with global symmetries.
Discrete-continuous (DISCO) group convolutions
The DISCO approach is based on convolutional layers, where the DISCO group convolution follows by a careful hybrid representation of the standard group convolution. Some components of the representation are left continuous, to facilitate accurate rotational equivariance, while other components are discretized, to yield scalable computation.
The DISCO group convolution of a signal (i.e. data, feature map) f defined over the group, with a filter
