Neural models learn data representations that lie on low-dimensional
manifolds, yet modeling the relation between these representational spaces is
an ongoing challenge. By integrating spectral geometry principles into neural
modeling, we show that this problem can be better addressed in the functional
domain, mitigating complexity, while enhancing interpretability and
performances on downstream tasks. To this end, we introduce a multi-purpose
framework to the representation learning community, which allows to: (i)
compare different spaces in an interpretable way and measure their intrinsic
similarity; (ii) find correspondences between them, both in unsupervised and
weakly supervised settings, and (iii) to effectively transfer representations
between distinct spaces. We validate our framework on various applications,
ranging from stitching to retrieval tasks, demonstrating that latent functional
maps can serve as a swiss-army knife for representation alignment.

将光谱几何原理融入神经建模中，提出一个多功能框架，通过在功能域中比较不同空间，测量其内在相似性，找到它们之间的对应关系，并有效地在不同空间之间转移表示，从而在各种应用中验证了我们的框架，展示了潜在的功能映射可以作为表示对齐的多功能工具。

潜在的功能映射

Latent. Functional Map

Learning representations that capture rich semantic relationships and
accommodate propositional calculus poses a significant challenge. Existing
approaches are either contrastive, lacking theoretical guarantees, or fall
short in effectively representing the partial orders inherent to rich
visual-semantic hierarchies. In this paper, we propose a novel approach for
learning visual representations that not only conform to a specified semantic
structure but also facilitate probabilistic propositional reasoning. Our
approach is based on a new nuclear norm-based loss. We show that its minimum
encodes the spectral geometry of the semantics in a subspace lattice, where
logical propositions can be represented by projection operators.

我们提出了一种基于核范数损失的新方法，用于学习既符合指定语义结构又有助于概率命题推理的视觉表征，最小化该方法编码了语义的谱几何在子空间格点上，可以用投影算子表示逻辑命题。

学习视觉语义子空间表示进行命题推理

Learning Visual-Semantic Subspace Representations for Propositional  Reasoning

The question whether one can recover the shape of a geometric object from its
Laplacian spectrum ('hear the shape of the drum') is a classical problem in
spectral geometry with a broad range of implications and applications. While
theoretically the answer to this question is negative (there exist examples of
iso-spectral but non-isometric manifolds), little is known about the practical
possibility of using the spectrum for shape reconstruction and optimization. In
this paper, we introduce a numerical procedure called isospectralization,
consisting of deforming one shape to make its Laplacian spectrum match that of
another. We implement the isospectralization procedure using modern
differentiable programming techniques and exemplify its applications in some of
the classical and notoriously hard problems in geometry processing, computer
vision, and graphics such as shape reconstruction, pose and style transfer, and
dense deformable correspondence.

本文提出了一种名为 “同谱化” 的数值方法，利用现代可微编程技术来应用于几何处理、计算机视觉和图形学等领域中的一些古老而艰难的问题。

等谱化，或者如何从听觉中获取形状，样式与对应关系

Isospectralization, or how to hear shape, style, and correspondence

To unify general relativity and quantum theory is hard in part because they
are formulated in two very different mathematical languages, differential
geometry and functional analysis. A natural candidate for bridging this
language gap, at least in the case of the euclidean signature, is the
discipline of spectral geometry. It aims at describing curved manifolds in
terms of the spectra of their canonical differential operators. As an immediate
benefit, this would offer a clean gauge-independent identification of the
metric's degrees of freedom in terms of invariants that should be ready to
quantize. However, spectral geometry is itself hard and has been plagued by
ambiguities. Here, we regularize and break up spectral geometry into small
finite-dimensional and therefore manageable steps. We constructively
demonstrate that this strategy works at least in two dimensions. We can now
calculate the shapes of 2-dimensional objects from their vibrational spectra.

本研究旨在通过谱几何学的方法，描述弯曲流形的谱，以便于识别量子化的指标，进而计算振动谱，以构造二维对象的形状。