\section{Other Approaches}
\label{sec:machine_learning_approaches}
Several methods cannot easily be assigned to one of the aforementioned paradigms. We present these approaches and their core ideas in the following. 

\subsubsection{Regionwise Matchings}\
Apart from a few exceptions, most methods establish correspondences on a point-to-point basis.
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\add{For shapes belonging to the same semantic class but exhibiting substantial geometric variability, it is often more appropriate to ask for correspondences between regions rather than specific points. In this spirit, \cite{Ganapathi-Subramanian16-stableRegionCorres} exploit the observation that, although the absolute values of descriptor functions may differ for non-isometric shapes, their relative ordering remains comparable. For example, while curvature values at corresponding locations may not be identical across shapes, regions of high curvature tend to correspond. Based on this insight, they define similarity in terms of rank: Points are considered similar if they share the same or similar ranks \wrt multiple descriptor functions. Based on this similarity measure they match corresponding regions.}
%

\cite{kleiman2019ex-robustStructure} explore the idea of finding the correspondence between coarse regions of the surfaces, making the approach less sensitive against local isometric distortion. First, the shapes are jointly decomposed based on a descriptor. As a consequence, the method inherits the properties of the descriptor, so ideally the descriptor is invariant under near-isometric deformations and robust to noise. The decomposition is followed by a graph matching step, which allows to extract the correspondence between regions. 

Zero-Shot 3D Shape Correspondence \cite{abdelreheem2023lb-zeroShot} approaches correspondence from a purely semantic perspective. It derives a sparse, region-level alignment using pretrained vision and language models. First, the class of each input shape is predicted from multiple rendered views. Language models then infer typical parts for both shapes and establish relations between them. Finally, a segmentation model partitions the surfaces according to the provided semantic labels.

\subsubsection{Learned Embeddings}\
% point embeddings
Limp \cite{cosmo2020limp} is a standard VAE with PointNet \cite{qi2017pointnet} as its encoder. Unlike deformation-based methods, it employs an intrinsic loss that enforces distance consistency under interpolation: For a convex combination of latent codes, the loss requires that the geodesic distances on the decoded shape match the corresponding convex combination of the input distances. Similarly, \cite{yang2020mappingInACycle} use a point-based network to learn pointwise features. These features will be used to extract a vertex-to-vertex map. The learning process is regularized by enforcing cycle-consistency between the maps extracted between the source shape, a rigidly transformed source shape, and the target. 

Effectively, \cite{attaiki2022in-ncp} propose a \emph{denoising} strategy for maps of a shape collection, called Neural Correspondence Prior. The approach has two stages. In the fist stage, any unsupervised method can be used to predict correspondences of possibly poor quality containing outliers. In the second stage, a neural network is is trained with these poor correspondences with a point-to-point loss. Based on this neural network, feature embeddings are learned from which the correspondences can be extracted through nearest-neighbor search or a functional map pipeline. 

% latent embedding
Deep Coupled Embedding \cite{zeng2024in-coe} learns a high-dimensional embedding for points of point clouds. They combine a variant of DiffusionNet \cite{sharp2022fm-diffusionnet} with cross-attention. The network is trained with an unsupervised loss promoting mutual consistency between the two embedded point clouds. The vertex-to-vertex map is finally retrieved by a nearest-neighbor search of the coupled embeddings. Similarly, for partial shapes, \cite{bracha2024dataset-pfaust} use DiffusionNet \cite{sharp2022fm-diffusionnet} with shared parameters for both input shapes with a cosine similarity loss to compute an embedding.

\subsubsection{Convolutional Operators}\
\add{To enable neural networks to process surface meshes both effectively and efficiently, a number of intrinsic mesh convolution operators have been proposed. Among them, SpiralNet++~\cite{Gong2019-SpiralNet++} introduces a formulation that explicitly serializes the local mesh neighborhood of each vertex into a fixed-length spiral ordering. The resulting ordered feature vector is processed by a fully connected layer, allowing the network to learn discriminative local features, which in turn can be used to successfully predict vertex-to-vertex maps.}

\add{FanNet~\cite{Sucu2025-FanNet} follows a related serialization-based paradigm, but organizes the local neighborhood around a vertex as a fan-shaped structure composed of multiple spokes. Starting from a central vertex, spokes are grown by selecting successive edges such that each spoke remains as straight as possible in the intrinsic geometry of the surface. To improve robustness \wrt mesh resolution and irregular connectivity, FanNet samples representative vertices along each spoke at prescribed distances from the center vertex. The resulting structured neighborhood encoding captures local geometric context while maintaining computational efficiency, making it well suited for dense correspondence estimation.}


