Learning invariant representations has been the longstanding approach to
self-supervised learning. However, recently progress has been made in
preserving equivariant properties in representations, yet do so with highly
prescribed architectures. In this work, we propose an invariant-equivariant
self-supervised architecture that employs Capsule Networks (CapsNets) which
have been shown to capture equivariance with respect to novel viewpoints. We
demonstrate that the use of CapsNets in equivariant self-supervised
architectures achieves improved downstream performance on equivariant tasks
with higher efficiency and fewer network parameters. To accommodate the
architectural changes of CapsNets, we introduce a new objective function based
on entropy minimisation. This approach, which we name CapsIE (Capsule Invariant
Equivariant Network), achieves state-of-the-art performance across all
invariant and equivariant downstream tasks on the 3DIEBench dataset, while
outperforming supervised baselines. Our results demonstrate the ability of
CapsNets to learn complex and generalised representations for large-scale,
multi-task datasets compared to previous CapsNet benchmarks. Code is available
at this https URL

通过使用 Capsule Networks (CapsNets) 构建不变 - 等变自监督架构，提出了 CapsIE (Capsule Invariant Equivariant Network) 的目标函数，以提高等变任务的下游性能，具有更高的效率和更少的网络参数。在 3DIEBench 数据集上，CapsNets 能学习出复杂且普遍化的表示，相比之前的 CapsNet 基准数据，达到最先进的性能，超越了监督模型的性能表现。

胶囊网络投影器是等变和不变学习器

Capsule Network Projectors are Equivariant and Invariant Learners

We develop a novel deep learning architecture for naturally complex-valued
data, which is often subject to complex scaling ambiguity. We treat each sample
as a field in the space of complex numbers. With the polar form of a
complex-valued number, the general group that acts in this space is the product
of planar rotation and non-zero scaling. This perspective allows us to develop
not only a novel convolution operator using weighted Fr\'echet mean (wFM) on a
Riemannian manifold, but also a novel fully connected layer operator using the
distance to the wFM, with natural equivariant properties to non-zero scaling
and planar rotation for the former and invariance properties for the latter.
Compared to the baseline approach of learning real-valued neural network
models on the two-channel real-valued representation of complex-valued data,
our method achieves surreal performance on two publicly available
complex-valued datasets: MSTAR on SAR images and RadioML on radio frequency
signals. On MSTAR, at 8% of the baseline model size and with fewer than 45,000
parameters, our model improves the target classification accuracy from 94% to
98% on this highly imbalanced dataset. On RadioML, our model achieves
comparable RF modulation classification accuracy at 10% of the baseline model
size.

本文提出一种新的深度学习架构用于自然复数数据，并利用极径形式将每个样本视为复数空间中的一个领域，设计了一种基于 Riemannian 流形上加权 Frechet 平均的卷积算子和基于距离到 wFM 的全连接层算子，具有非零缩放和平面旋转下的自然等变性及后者的不变性，相较于基于实数神经网络的两通道实数表示法，我们的方法在 MSTAR 和 RadioML 复数数据集上均获得了出色的表现。