Non-Euclidean data is frequently encountered across different fields, yet
there is limited literature that addresses the fundamental challenge of
training neural networks with manifold representations as outputs. We introduce
the trick named Deep Extrinsic Manifold Representation (DEMR) for visual tasks
in this context. DEMR incorporates extrinsic manifold embedding into deep
neural networks, which helps generate manifold representations. The DEMR
approach does not directly optimize the complex geodesic loss. Instead, it
focuses on optimizing the computation graph within the embedded Euclidean
space, allowing for adaptability to various architectural requirements. We
provide empirical evidence supporting the proposed concept on two types of
manifolds, $SE(3)$ and its associated quotient manifolds. This evidence offers
theoretical assurances regarding feasibility, asymptotic properties, and
generalization capability. The experimental results show that DEMR effectively
adapts to point cloud alignment, producing outputs in $ SE(3) $, as well as in
illumination subspace learning with outputs on the Grassmann manifold.

通过将外在流形表征嵌入到深度神经网络中，Deep Extrinsic Manifold Representation (DEMR) 方法用于视觉任务，不直接优化复杂的测地损失，而是侧重于优化嵌入欧氏空间内的计算图，以适应不同的架构要求；在两种流形类型 ——$SE (3)$ 及其相关商流形上，提供了理论上的可行性、渐近性质和泛化能力的实验证据。DEMR 在点云对齐和光照子空间学习方面表现出良好的适应性。

深度外在流形表示的视觉任务

Deep Extrinsic Manifold Representation for Vision Tasks

Recently, the equivariance of models with respect to a group action has
become an important topic of research in machine learning. However, imbuing an
architecture with a specific group equivariance imposes a strong prior on the
types of data transformations that the model expects to see. While
strictly-equivariant models enforce symmetries, real-world data does not always
conform to such strict equivariances, be it due to noise in the data or
underlying physical laws that encode only approximate or partial symmetries. In
such cases, the prior of strict equivariance can actually prove too strong and
cause models to underperform on real-world data. Therefore, in this work we
study a closely related topic, that of almost equivariance. We provide a
definition of almost equivariance that differs from those extant in the current
literature and give a practical method for encoding almost equivariance in
models by appealing to the Lie algebra of a Lie group. Specifically, we define
Lie algebra convolutions and demonstrate that they offer several benefits over
Lie group convolutions, including being well-defined for non-compact groups.
From there, we pivot to the realm of theory and demonstrate connections between
the notions of equivariance and isometry and those of almost equivariance and
almost isometry, respectively. We prove two existence theorems, one showing the
existence of almost isometries within bounded distance of isometries of a
general manifold, and another showing the converse for Hilbert spaces. We then
extend these theorems to prove the existence of almost equivariant manifold
embeddings within bounded distance of fully equivariant embedding functions,
subject to certain constraints on the group action and the function class.
Finally, we demonstrate the validity of our approach by benchmarking against
datasets in fully equivariant and almost equivariant settings.

在机器学习领域，最近，模型对于群作用的等变性已经成为重要的研究主题。本研究通过利用 Lie 群的 Lie 代数，提出了一种新颖的几乎等变性定义，并给出了一种将几乎等变性编码到模型中的实用方法。进一步，该研究还证明了等变性与等距性的关联以及几乎等变性与几乎等距性的关联，并在一定约束条件下，证明了几乎等变性流形嵌入函数与完全等变性嵌入函数之间的存在性。最后，通过在完全等变性和几乎等变性设置下的数据集上进行基准测试，证明了该方法的有效性。

通过李代数卷积实现准等变性

Almost Equivariance via Lie Algebra Convolutions

Zero-shot learning (ZSL) enables solving a task without the need to see its
examples. In this paper, we propose two ZSL frameworks that learn to synthesize
parameters for novel unseen classes. First, we propose to cast the problem of
ZSL as learning manifold embeddings from graphs composed of object classes,
leading to a flexible approach that synthesizes "classifiers" for the unseen
classes. Then, we define an auxiliary task of synthesizing "exemplars" for the
unseen classes to be used as an automatic denoising mechanism for any existing
ZSL approaches or as an effective ZSL model by itself. On five visual
recognition benchmark datasets, we demonstrate the superior performances of our
proposed frameworks in various scenarios of both conventional and generalized
ZSL. Finally, we provide valuable insights through a series of empirical
analyses, among which are a comparison of semantic representations on the full
ImageNet benchmark as well as a comparison of metrics used in generalized ZSL.
Our code and data are publicly available at
this https URL

本文提出了两个使用零样本学习（ZSL）系统，该系统使用流形嵌入和自动生成视觉分类器来解决具有未知类的任务，并且在五个类视觉识别基准数据集上表现出卓越的性能。