
\section{Spectral methods}
\label{sec:spectral}

Spectral methods have emerged as a powerful framework, relying on intrinsic properties of the shapes that are invariant to isometric deformations. Building on this spectral representation, the functional map framework offers a compact and elegant formulation for expressing correspondences between shapes in the spectral domain, reducing the high-dimensional point-wise matching problem to the estimation of a small matrix of coefficients based on frequency decomposition. 
In this section, we first provide a theoretical background on the Laplace–Beltrami operator (LBO) and functional maps, later discussing recent classical refinement methods as well as learning-based methods.  

\subsection{Background}

    










In this subsection, we review the theoretical background for spectral methods. 
\paragraph{Laplace–Beltrami operator.}
The Laplace–Beltrami operator is a generalisation of the Laplace operator to Riemannian manifolds~\cite{guggenheimer2012differential}. 
It is defined as the divergence of the gradient of a function $f$:
\begin{equation}
    \label{eq:laplace}
    \Delta f = \operatorname{div}\left(\nabla f\right).
\end{equation}

\noindent
There exist many different ways to compute the LBO in the discrete setting~\cite{wardetzky2007discrete}. For triangle meshes, the most common discretisation is based on the cotangent Laplacian matrix $L_{\mathcal{M}} \in \mathbb{R}^{n_{\mathcal{M}} \times n_{\mathcal{M}}}$, computed as~\cite{pinkall1993computing}:

\begin{equation}
    \label{eq:lap_mat}
    L_{\mathcal{M}}= A_{\mathcal{M}}^{-1}W_{\mathcal{M}}.
\end{equation}

\noindent
Here $A_{\mathcal{M}}$ is the diagonal matrix of lumped area elements and $W_{\mathcal{M}}$ is the cotangent weight matrix, defined in the following form:
\begin{equation}
    \label{eq:lap_weight}
    W_{ij} = \begin{cases}
\frac{\cot(\alpha_{ij})+\cot(\beta_{ij})}{2} & \text{if } (i,j) \in \mathcal{E} \\
- \sum_{k \in N(i)} W_{ik} & \text{if } i = j \\
0 & \text{otherwise,}
\end{cases}
\end{equation}

\begin{equation}
    \label{eq:area_mat}
    A_{ij} = \begin{cases}
        \sum_{k \in N(i)} \frac{T_{k}}{3} & \text{if } i = j \\
        0 & \text{otherwise.}
    \end{cases}
\end{equation}

\begin{figure}[t]
    \centering
    \includegraphics[width=\linewidth]{figures/03/cotangent_matrix.png}
    \caption{Illustration of the \textbf{notation} used for computing the cotangent weight matrix $W_{ij}$ and the diagonal area matrix $A_{ij}$. The cotangent weight matrix $W_{ij}$ uses the angles opposite the edge between $v_i$ and $v_j$; $T_1$ and $T_2$ denote the areas of the corresponding triangles. The diagonal area matrix $A_{ij}$ assigns each vertex one-third of the total area of its adjacent triangles.}
    \label{fig:cotangent_matrix}
\end{figure}

\noindent
Here $T_k$ is the area of triangle $k$ and $N(i)$ denotes the index of the one-ring neighbour vertices. The notations in~\cref{eq:lap_weight} and~\cref{eq:area_mat} are visualized in~\cref{fig:cotangent_matrix}. 

The eigendecomposition of the LBO plays a crucial role in spectral shape analysis. Let $\Delta_{\mathcal{M}}$ be the LBO on shape $\mathcal{M}$ and
\begin{equation}
    \label{eq:eigendecom}
    \Delta_{\mathcal{M}} \phi_{i} = \lambda_i  \phi_{i},
\end{equation}
where $\phi_{i} \in L^2(\mathcal{M})$ is the $i$-th LBO eigenfunction with corresponding eigenvalue $\lambda_i$.
In the discrete setting, the eigendecomposition can be computed via the generalised eigenvalue problem,
\begin{equation}
    W_{\mathcal{M}} \phi_{i} = \lambda_i A_{\mathcal{M}} \phi_i.
\end{equation}

\noindent Since the LBO is positive semi-definite, its eigenvalues are real and non-negative,
\begin{equation}
    0 = \lambda_0 \leqslant \lambda_1 \leqslant \ldots \lambda_{n_{\mathcal{M}}}.
\end{equation}

\noindent The eigenfunctions of the Laplace–Beltrami operator encode the intrinsic geometry of a surface, their eigenvalues correspond to the frequency, and thus provide a natural basis for representing functions defined on it as well as for frequency filters.
This spectral representation serves as the foundation for functional approaches to shape correspondence.

\paragraph{Functional maps.}
The functional map framework was first introduced by Ovsjanikov et al.~\cite{ovsjanikov_functional_2012}.
Rather than establishing correspondences directly between individual points, it represents them implicitly as linear operators that transfer functions between shapes.
Once the functional map is estimated, point-wise correspondences can be recovered, for example, by transferring delta functions.
This formulation casts the correspondence problem as solving a compact linear system that maps functions to functions, rather than points to points. Given two sets of pointwise descriptor functions $f_{\mathcal{M}} \in \mathbb{R}^{n_{\mathcal{M}} \times d}$ and $f_{\mathcal{N}} \in \mathbb{R}^{n_{\mathcal{N}} \times d}$ and the orthogonal basis sets $\phi_{\mathcal{M}} \in \mathbb{R}^{n_{\mathcal{M}} \times k}$ and $\phi_{\mathcal{N}} \in \mathbb{R}^{n_{\mathcal{N}} \times k}$, which are often chosen as the Laplace-Beltrami eigenfunctions, on each shape, the functional map $C_{\mathcal{MN}} \in \mathbb{R}^{k \times k}$ can be computed as 
\begin{equation}
    C_{\mathcal{MN}} = \arg \min_{C} \|C\phi_{\mathcal{M}}^{\dagger}f_{\mathcal{M}} - \phi_{\mathcal{N}}^{\dagger}f_{\mathcal{N}}\|_F^2,
\end{equation}
where $\phi^{\dagger}_{\bullet} = \phi^{\top}_{\bullet}A_{\bullet}$, which is the pseudo-inverse of $\phi$.

The corresponding point-wise map $\Pi_{\mathcal{NM}}$ can be recovered from the functional map based on the following relationship:
\begin{equation}
    \label{eq:map_relation}
    \Pi_{\mathcal{NM}}\phi_{\mathcal{N}} \approx \phi_{\mathcal{M}}C_{\mathcal{NM}},
\end{equation}
The left-hand side of the equation aligns the LBO eigenfunctions in the spatial domain, while the right-hand side aligns them in the spectral domain.~\cref{fig:functional_map} visualizes the functional map framework. 

\begin{figure*}[t]
    \centering
    \includegraphics[width=.8\linewidth]{figures/03/functional_map_xy_compressed.jpg}
    \caption{Visualisation of the \textbf{functional map framework} as an analogy of Fourier analysis in signal processing. Given two functions $f_{\mathcal{M}}$ and $f_{\mathcal{N}}$ defined on shapes $\mathcal{M}$ and $\mathcal{N}$, respectively, the functional map $C_{\mathcal{MN}}$ transfers $f_{\mathcal{M}}$ from $\mathcal{M}$ to $f_{\mathcal{N}}$ on $\mathcal{N}$,  each function is approximated using the first $k$ LBO eigenfunctions $\phi$ of the corresponding shape, which is similar to Fourier transform that uses sinusoidal functions with different frequencies as basis functions. In this representation, the functional map transfers the coefficients $a$ of the basis functions on $\mathcal{M}$ to the coefficients $b$ on $\mathcal{N}$, thereby encoding the point-wise correspondences $\Pi_{\mathcal{NM}}$ in a compact $k \times k$ matrix.} 
    \label{fig:functional_map}
\end{figure*}


\subsection{Axiomatic methods}
Axiomatic methods assume that noisy or incomplete correspondences are obtained either based on handcrafted features~\cite{sun2009concise,bronstein2010scale,aubry2011wave} or manual annotations and aim to improve the final matching performance by iteratively refining functional maps and point-wise maps as a post-processing step. 
The key idea relies on the relationship between the two representations, as defined in~\cref{eq:map_relation}.

As the simplest axiomatic technique, ICP iteratively converts the functional map to a point-wise map via nearest neighbour search in spectral space~\cite{ovsjanikov_functional_2012}. Follow-up works~\cite{rodola2015point,rodola2017regularized,ezuz2017deblurring} iteratively optimise the functional map and point-wise map by minimising an objective function based on~\cref{eq:map_relation} with additional regularisation terms or constraints encouraging smoothness and bijectivity. 
BCICP~\cite{ren2018continuous} further promotes correspondence continuity and overall coverage by introducing bijective ICP with additional point-wise correspondence smoothing as well as outlier detection and fix. ZoomOut~\cite{melzi_zoomout_2019} gradually increases the spectral resolution of the functional map during the ICP process. 
Despite its simplicity, it achieves better performance than previous, more complex refinement methods. 

RHM~\cite{ezuz2019reversible} prompts reversible harmonic maps, thereby resulting in correspondences with lower conformal distortion. FSF~\cite{pai2021fast} theoretically analyses the relationship between functional maps and point-wise maps and proposes a point-wise map recovery method based on spectral alignment of adjoint maps. Later, DiscreteOpt~\cite{ren2021discrete} introduces the \emph{proper} functional map space and optimises the functional maps in the proper functional map space, leading to superior performance in the context of accuracy and smoothness. The \emph{proper} functional map space is the set of functional maps that arise from pointwise correspondences:
\begin{equation}\label{eq:fmap-to-pmap}
    \left\{C_{\mathcal{MN}}\; |\; \exists \Pi_{\mathcal{NM}}, \;\mathrm{s.t.}\;C_{\mathcal{MN}} = \phi_{\mathcal{N}}^{\dagger}\Pi_{\mathcal{NM}}\phi_{\mathcal{M}}\right\}.
\end{equation}
SmoothFM~\cite{magnet2022smooth} extends DiscreteOpt by introducing spatial smoothness based on minimising the Dirichlet energy, similar to RHM~\cite{ezuz2019reversible}.

Instead of using LBO eigenfunctions for functional map computation, several works~\cite{panine2022non,donati_complex_2022,hartwig_elastic_2023} propose to use different eigenfunctions to improve the matching performance in terms of non-isometry and orientation awareness.
Another line of works encourages cycle-consistency~\cite{huang2014functional,bernard2019hippi,huang2020consistent} in the case of processing a collection of shapes.

The study of axiomatic methods establishes a solid theoretical analysis of the relationship between functional maps and point-wise maps. However, in real-world applications, the axiomatic methods often fail due to the violation of the underlying assumptions (e.g., near-isometric deformation). Therefore, it is interesting to combine axiomatic methods with learning-based approaches to improve the matching accuracy. Previous learning-based approaches~\cite{halimi2019unsupervised,roufosse2019unsupervised} typically use them as a post-processing step to improve the matching accuracy. Recent learning-based methods~\cite{attaiki_understanding_2023,cao_unsupervised_2023,magnet_memory-scalable_2024} are inspired by the insights from those axiomatic methods~\cite{ren2021discrete,melzi_zoomout_2019}. It is interesting to see that future research in this area integrates and extends existing axiomatic methods within end-to-end learning-based frameworks, combining strong theoretical foundations with data-driven learning to further improve matching accuracy.


        
        
        
    


\begin{table}[t]
\centering
\small
\begin{tabular}{lcccccc}
\toprule
\textbf{Method} & \textbf{Cat.} & \textbf{MS} & \textbf{NI} & \textbf{P} & \textbf{TN} \\
\midrule
ZoomOut~\cite{melzi_zoomout_2019} & \textbf{\textcolor{blue}{A}} & \xmark & \cmark & \cmark & \xmark \\
ConsistentZoomOut~\cite{huang2020consistent} & \textbf{\textcolor{blue}{A}} & \cmark & \cmark & \cmark & \xmark \\
RHM~\cite{ezuz2019reversible} & \textbf{\textcolor{blue}{A}} & \xmark & \cmark & \xmark & \xmark \\
ComplexFMaps~\cite{donati_complex_2022} & \textbf{\textcolor{blue}{A}} & \xmark & \cmark & \xmark & \xmark \\
DiscreteOpt~\cite{ren2021discrete} & \textbf{\textcolor{blue}{A}} & \xmark & \cmark & \xmark & \xmark \\
SmoothFM~\cite{magnet2022smooth} & \textbf{\textcolor{blue}{A}} & \xmark & \cmark & \xmark & \xmark \\
\midrule
GeomFMaps~\cite{donati2020deep} & \textbf{\textcolor{orange!90!black}{S}} & \xmark & \xmark & \xmark & \xmark \\
DPFM~\cite{attaiki2021dpfm} & \textbf{\textcolor{orange!90!black}{S}} & \xmark & \xmark & \cmark & \xmark \\
SRFeat~\cite{li_srfeat_2022} & \textbf{\textcolor{orange!90!black}{S}} & \xmark & \cmark & \xmark & \xmark \\
EchoMatch~\cite{xie_echomatch_2025} & \textbf{\textcolor{orange!90!black}{S}} & \xmark & \xmark & \cmark & \xmark \\
DenoisFM~\cite{zhuravlev_denoising_2025} & \textbf{\textcolor{orange!90!black}{S}} & \xmark & \cmark & \xmark & \xmark \\
FRIDU~\cite{rimon_fridu_2025} & \textbf{\textcolor{orange!90!black}{S}} & \xmark & \xmark & \xmark & \xmark \\
DiffuMatch~\cite{pierson2025diffumatch} & \textbf{\textcolor{orange!90!black}{S}} & \xmark & \cmark & \xmark & \xmark \\
\midrule
UnsupFMNet~\cite{halimi2019unsupervised} & \textbf{\textcolor{green!50!black}{U}} & \xmark & \xmark & \xmark & \xmark \\
SURFMNet~\cite{roufosse2019unsupervised} & \textbf{\textcolor{green!50!black}{U}} & \xmark & \xmark & \xmark & \xmark \\
DPFM-unsup~\cite{attaiki2021dpfm} & \textbf{\textcolor{green!50!black}{U}} & \xmark & \xmark & \cmark & \xmark \\
UDMSM~\cite{cao2022unsupervised} & \textbf{\textcolor{green!50!black}{U}} & \cmark & \xmark & \xmark & \xmark \\
AttentiveFMaps~\cite{li_learning_2022} & \textbf{\textcolor{green!50!black}{U}} & \xmark & \cmark & \xmark & \xmark \\
ULRSSM~\cite{cao_unsupervised_2023} & \textbf{\textcolor{green!50!black}{U}} & \xmark & \cmark & \cmark & \cmark \\
G-msm~\cite{eisenberger2023g} & \textbf{\textcolor{green!50!black}{U}} & \cmark & \cmark & \xmark & \cmark \\
HybridFMaps~\cite{bastian2024hybrid} & \textbf{\textcolor{green!50!black}{U}} & \xmark & \cmark & \xmark & \cmark \\
DiffZoomOut~\cite{magnet_memory-scalable_2024} & \textbf{\textcolor{green!50!black}{U}} & \xmark & \cmark & \xmark & \cmark \\
\bottomrule
\end{tabular}

\caption{
Overview of \textbf{spectral} shape correspondence methods.
Each method is categorized by approach (\textbf{Cat.}): 
axiomatic (\textbf{\textcolor{blue}{A}}), 
supervised (\textbf{\textcolor{orange!90!black}{S}}), 
unsupervised (\textbf{\textcolor{green!50!black}{U}}).
The remaining columns indicate supported properties: 
multi-shape consistency (\textbf{MS}), 
non-isometric deformations (\textbf{NI}), 
partiality (\textbf{P}), 
topological noise (\textbf{TN}).
}
\label{tab:spectral_methods}
\end{table}


\subsection{Supervised methods}
Instead of relying on handcrafted features~\cite{sun2009concise,bronstein2010scale,aubry2011wave,salti2014shot} for finding correspondences, learning-based methods attempt to use neural networks to learn the feature descriptors directly from training data. FMNet~\cite{litany2017deep} was among the first supervised methods to learn a non-linear transformation of SHOT descriptors~\cite{salti2014shot} based on ground-truth correspondences.
Instead of directly obtaining soft correspondences based on feature similarity, learning-based methods first compute a functional map based on extracted features on each shape $f_{\mathcal{M}} \in \mathbb{R}^{n_{\mathcal{M}} \times c}, f_{\mathcal{N}} \in \mathbb{R}^{n_{\mathcal{N}} \times c}$ by solving the linear least square problem:
\begin{equation}
    \label{eq:fmap}  C_{\mathcal{MN}}=\mathrm{argmin}_{C}~ \Energy_{\mathrm{data}}\left(C\right)+\lambda \Energy_{\mathrm{reg}}\left(C\right).
\end{equation}
Here, minimizing $\Energy_{\mathrm{data}}=\left\|C\phi_{\mathcal{M}}^{\dagger}f_{\mathcal{M}}-\phi_{\mathcal{N}}^{\dagger}f_{\mathcal{N}}\right\|^{2}_{F}$ enforces descriptor preservation, while minimizing the regularization term $\Energy_{\mathrm{reg}}$ imposes some form of structural properties~\cite{ovsjanikov_functional_2012}. After obtaining the functional map, the soft correspondences can be computed as:
\begin{equation}
    \label{eq:soft_corr}
    \Pi_{\mathcal{NM}}^{\mathrm{soft}} = \operatorname{softmax}\left(\phi_{\mathcal{N}}C_{\mathcal{MN}}\phi_{\mathcal{M}}^{\dagger}\right).
\end{equation}
The loss is measured as the geodesic distance error between the predicted soft correspondences and the ground truth. 
Instead of using geodesic distance, which is computationally expensive, GeomFMaps~\cite{donati2020deep} directly compute the loss in the spectral domain based on the difference between the predicted functional map and the ground-truth functional map. 
DPFM~\cite{attaiki2021dpfm} extends this idea to partial shape matching and, for the first time, enables partial-to-partial shape matching by introducing an additional overlap prediction module. 
Afterwards, SRFeat~\cite{li_srfeat_2022} introduces additional contrastive loss to improve the local discriminative power of learned features and to overcome the limitation that functional maps mainly focus on low-frequency areas. 
EchoMatch~\cite{xie_echomatch_2025} improves the partial-to-partial matching performance by introducing the concept of correspondence reflection. Correspondence reflection indicates that the valid correspondences for partial shapes should be bijective. 
Another line of research focuses on learning semantically meaningful basis functions to replace the standard LBO eigenfunctions~\cite{marin2020correspondence,huang_multiway_2022} and thereby improve matching performance. 
Recently, NAM~\cite{vigano2025nam} extended the concept of functional maps to non-linear neural adjoint maps based on multi-layer perceptions. 

Despite the appealing matching performance of supervised learning methods, they still face challenges in generalisation. 
Due to the lack of large annotated training data with ground-truth correspondences across different object categories, supervised methods cannot generalise well to unseen object categories and thus limit their practical applications. 
Further research in this area should focus on improving generalisation to unseen object categories and on creating large-scale training datasets~\cite{ehm2025beyond,liu_stable-score_2025}.    

\subsection{Unsupervised methods}
Due to the lack of large, well-annotated training datasets with ground-truth correspondences, many learning-based methods focus on developing unsupervised training strategies. 
Starting from UnsupFMNet~\cite{halimi2019unsupervised}, the method proposes an unsupervised loss to train FMNet~\cite{litany2017deep} based on geodesic distance preservation:
\begin{equation}
    \mathcal{L}_{\mathrm{geo}} = \frac{1}{n_{\mathcal{M}}}\|D_{\mathcal{M}} - \Pi_{\mathcal{MN}} D_{\mathcal{N}} \Pi_{\mathcal{MN}}^{\top} \|_{{F}}^{2}, 
\end{equation}
where $D_{\mathcal{M}} \in \mathbb{R}^{n_{\mathcal{M}} \times n_{\mathcal{M}}}$ and $D_{\mathcal{N}} \in \mathbb{R}^{n_{\mathcal{N}} \times n_{\mathcal{N}}}$ are the geodesic distance matrices for shape $\mathcal{M}$ and shape $\mathcal{N}$, respectively. Similar to~\cite{litany2017deep}, the soft correspondences are obtained based on~\cref{eq:soft_corr}. Later work~\cite{aygun2020unsupervised} replaces geodesic distance matrices with heat kernels to be more efficient and robust. Instead of directly regularising the soft correspondences, follow-up works attempt to regularise the more compact functional maps. SURFMNet~\cite{roufosse2019unsupervised} is one of the first works in this direction. It exploits the structure of the functional map to enable unsupervised learning. Specifically, it uses Laplacian commutativity as the $\Energy_{\mathrm{reg}}$ in~\cref{eq:fmap}, which can be expressed as
\begin{equation}
    \label{eq:l_lap}
    \left\|C_{\mathcal{MN}}\Lambda_{\mathcal{M}} - \Lambda_{\mathcal{N}}C_{\mathcal{MN}}\right\|_{F}^{2},
\end{equation}
where $\Lambda_{\mathcal{M}}$ and $\Lambda_{\mathcal{N}}$ are the diagonal eigenvalue matrix of shape $\mathcal{M}$ and $\mathcal{N}$, respectively. As for the unsupervised loss terms, SURFMNet uses bijectivity and orthogonality regularisation to promote the matching to be bijective and area-preserving~\cite{ovsjanikov_functional_2012}:
\begin{equation}
    \label{eq:l_orth}
    \mathcal{L}_{\mathrm{orth}} = \left\|C_{\mathcal{MN}}^{\top}C_{\mathcal{MN}}-I\right\|^{2}_{F}+\left\|C_{\mathcal{NM}}^{\top}C_{\mathcal{NM}}-I\right\|^{2}_{F},
\end{equation}
\begin{equation}
    \label{eq:l_bij}
    \mathcal{L}_{\mathrm{bij}}=\left\|C_{\mathcal{MN}}C_{\mathcal{NM}}-I\right\|^{2}_{F}+\left\|C_{\mathcal{NM}}C_{\mathcal{MN}}-I\right\|^{2}_{F}.
\end{equation}
Later work~\cite{sharma2020weakly} systematically analyses each component, and DPFM~\cite{attaiki2021dpfm} extends the idea to partial-to-full shape matching. 

Nevertheless, these unsupervised methods still rely on off-the-shelf refinement methods~\cite{melzi_zoomout_2019,ovsjanikov_functional_2012} to obtain better matching results. AttentiveFMaps~\cite{li_learning_2022} proposes an attention mechanism to aggregate predicted functional maps with different resolutions and significantly improves the matching performance for non-isometric shapes. ULRSSM~\cite{cao_unsupervised_2023} utilises the concept of proper functional map space~\cite{ren2021discrete,attaiki_understanding_2023,sun_spatially_2023,cao_revisiting_2024} to establish a relationship between the soft correspondences based on feature similarity and the functional maps from~\cref{eq:fmap} and leads to state-of-the-art performance at the time of publishing for near-isometric, non-isometric as well as topologically noisy shapes. 
HybridFMaps~\cite{bastian2024hybrid} extends ULRSSM by incorporating the elastic basis~\cite{hartwig_elastic_2023} to further improve the matching performance, especially for non-isometric deformations.
DiffZoomOut~\cite{magnet_memory-scalable_2024} proposes a differentiable version of ZoomOut~\cite{melzi_zoomout_2019}, leading to more efficient inference, especially for high-resolution shapes. 
Another line of work combines functional maps with spatial alignment, leading to spatially smooth matching \cite {eisenberger2020deep,attaiki_ncp_2022,cao_spectral_2024}. 

Recent unsupervised methods~\cite{cao_unsupervised_2023,cao2024synchronous} have shown significant progress in shape matching for near-isometric, non-isometric shapes, even outperforming some supervised approaches~\cite{donati2020deep}. However, for partial shape matching (particularly the partial-to-partial setting), unsupervised methods struggle to achieve satisfactory results due to the absence of adequate regularization~\cite{rodola2017partial,ehm2025beyond}.
Additionally, if significant topological noise arises from self-intersections or poor reconstructions, unsupervised methods are likely to fail. 
These limitations severely hinder their applicability in real-world scenarios. 
Future work in this area should focus on improving matching performance for partial and topologically noisy shapes. 

\paragraph{Pointwise map recovery.}
An essential step in any spectral pipeline is the recovery of a pointwise map from the functional map. 
The most straightforward way to achieve this is via nearest-neighbour search in the spectral embedding space~\cite{ovsjanikov_functional_2012}.
However, this treats each vertex independently, and the resulting maps thus lack guarantees on smoothness or neighbourhood preservation~\cite{magnet2022smooth,vigano_adjoint_2023,ren2021discrete}, which limits their direct applicability in downstream tasks such as deformation transfer~\cite{sumner2004deformation,vigano2025nam} or statistical shape modelling~\cite{greenspan_s3m_2023}.
While the refinement methods discussed above mitigate these artefacts~\cite{melzi_zoomout_2019,magnet2022smooth}, they operate as post-processing and cannot fully recover information lost due to the low-rank approximation.
In \cref{sec:combinatorial}, we discuss combinatorial methods that partially address this limitation by encoding neighbourhood constraints directly into the matching formulation.

\subsection{Functional map estimation using image diffusion}
\label{subsec:fm_diffusion}

Diffusion models have emerged as the dominant paradigm in image synthesis, enabling high-quality generation through iterative denoising \cite{sohl2015deep,ho2020denoising,karras2022elucidating,po2024state}.
They operate by gradually corrupting data with Gaussian noise in a forward process and then learning to reverse this process step by step.
A popular approach is Denoising Diffusion Probabilistic Models~\cite{ho2020denoising}, where a generative model is trained by recovering content from a noise distribution, potentially conditioned on additional data.

Several works have observed that functional maps \cite{ovsjanikov_functional_2012} provide a suitable domain for image diffusion.  
Given the truncated Laplacian eigenbases $\phi_\mathcal{M}, \phi_\mathcal{N} \in \mathbb{R}^{n \times k}$ of shapes $\mathcal{M}$ and $\mathcal{N}$, the resulting functional map $C_{\mathcal{M} \mathcal{N}} \in \mathbb{R}^{k \times k}$ is a compact matrix, typically with $k \leqslant 200$.  
This compactness makes it feasible to repurpose architectures originally designed for image generation to predict functional maps.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.95\linewidth]{figures/06/6_1}
    \caption{Functional maps provide a suitable domain for \textbf{2D diffusion models} due to their compact matrix representation. Such spectral diffusion models, trained on large collections of registered shapes, introduce a learned prior for functional map solvers. Adapted from \cite{pierson2025diffumatch}.}
    \label{fig:6_2_classification}
\end{figure}

DenoisFM~\cite{zhuravlev_denoising_2025} trains a diffusion model to directly generate a functional map between a pair of shapes, addressing sign ambiguity of Laplacian eigenfunctions using an unsupervised sign-selection strategy based on learned surface features \cite{aubry2011wave}.  
FRIDU~\cite{rimon_fridu_2025} presents an approach for functional map refinement, where a diffusion model learns to generate refined maps conditioned on noisy initial maps, guiding learning objectives that encourage plausible output correspondences.
DiffuMatch~\cite{pierson2025diffumatch} is trained on the absolute values of functional maps $|C_{\mathcal{M} \mathcal{N}}|$ to overcome the sign ambiguity of Laplacian eigenbasis, replacing existing axiomatic constraints such as orthogonality \cref{eq:l_orth} and Laplacian commutativity \cref{eq:laplace} with the learned regularizer.
These early works demonstrate that diffusion models can serve as a substitute for an explicit functional map solver.

