The mathematical representations of data in the Spherical Harmonic (SH)
domain has recently regained increasing interest in the machine learning
community. This technical report gives an in-depth introduction to the
theoretical foundation and practical implementation of SH representations,
summarizing works on rotation invariant and equivariant features, as well as
convolutions and exact correlations of signals on spheres. In extension, these
methods are then generalized from scalar SH representations to Vectorial
Harmonics (VH), providing the same capabilities for 3d vector fields on spheres

在这篇技术报告中，我们深入介绍了球谐（Spherical Harmonic）域中数据的数学表示的理论基础和实际实现，并总结了关于旋转不变性和等变特征、球面信号的卷积和精确相关性的作品。此外，我们还将这些方法从标量球谐表示推广到矢量谐波（Vectorial Harmonics），为球面上的三维矢量场提供了相同的能力。

关于球形调和表示对于标量和向量数据的不变性、等变性、相关性和卷积

On Invariance, Equivariance, Correlation and Convolution of Spherical  Harmonic Representations for Scalar and Vectorial Data

Self-supervised representation learning (SSRL) methods have shown great
success in computer vision. In recent studies, augmentation-based contrastive
learning methods have been proposed for learning representations that are
invariant or equivariant to pre-defined data augmentation operations. However,
invariant or equivariant features favor only specific downstream tasks
depending on the augmentations chosen. They may result in poor performance when
a downstream task requires the counterpart of those features (e.g., when the
task is to recognize hand-written digits while the model learns to be invariant
to in-plane image rotations rendering it incapable of distinguishing "9" from
"6"). This work introduces Contrastive Invariant and Predictive Equivariant
Representation learning (CIPER). CIPER comprises both invariant and equivariant
learning objectives using one shared encoder and two different output heads on
top of the encoder. One output head is a projection head with a
state-of-the-art contrastive objective to encourage invariance to
augmentations. The other is a prediction head estimating the augmentation
parameters, capturing equivariant features. Both heads are discarded after
training and only the encoder is used for downstream tasks. We evaluate our
method on static image tasks and time-augmented image datasets. Our results
show that CIPER outperforms a baseline contrastive method on various tasks,
especially when the downstream task requires the encoding of
augmentation-related information.

本研究通过 Contrastive Invariant and Predictive Equivariant Representation learning (CIPER) 方法，利用共享编码器和两个不同的输出头，旨在同时通过不变性和等变性目标学习表示。CIPER 在静态图像任务和时间增强图像数据集上都表现出较好的性能，特别是当下游任务需要编码与数据增强相关的信息时。

CIPER: 使用对比学习和预测学习结合不变和等变表示

CIPER: Combining Invariant and Equivariant Representations Using Contrastive and Predictive Learning

Features that are equivariant to a larger group of symmetries have been shown
to be more discriminative and powerful in recent studies. However, higher-order
equivariant features often come with an exponentially-growing computational
cost. Furthermore, it remains relatively less explored how rotation-equivariant
features can be leveraged to tackle 3D shape alignment tasks. While many past
approaches have been based on either non-equivariant or invariant descriptors
to align 3D shapes, we argue that such tasks may benefit greatly from an
equivariant framework. In this paper, we propose an effective and practical
SE(3) (3D translation and rotation) equivariant network for point cloud
analysis that addresses both problems. First, we present SE(3) separable point
convolution, a novel framework that breaks down the 6D convolution into two
separable convolutional operators alternatively performed in the 3D Euclidean
and SO(3) spaces. This significantly reduces the computational cost without
compromising the performance. Second, we introduce an attention layer to
effectively harness the expressiveness of the equivariant features. While
jointly trained with the network, the attention layer implicitly derives the
intrinsic local frame in the feature space and generates attention vectors that
can be integrated into different alignment tasks. We evaluate our approach
through extensive studies and visual interpretations. The empirical results
demonstrate that our proposed model outperforms strong baselines in a variety
of benchmarks

本文提出了一个有效的 SE（3）网络，它是针对点云分析领域中 3D 形状对齐任务而设计的，可以利用等变特征来提高性能并解决相对较少探索的旋转等变特征对 3D 形状对齐任务的处理问题。我们采用一个新的框架 SE (3) 可分点卷积来降低计算成本，并在网络中引入一个注意力层来有效地利用等变特征的表达能力。通过广泛的研究和视觉解释，实证结果表明，我们提出的模型在各种基准测试中优于强基线。