Generalized Spectral Coarsening

Every basis of the vector space $\mathcal{H}_{k}$ of a complex $K$ is spanned by a representative of equivalence classes of $k$-dimensional nontrivial loops, and is isomorphic to the kernel of the $k$-Hodge Laplacian (Eckmann, 1944): $\ker L_{k}\simeq\mathcal{H}_{k}(K)$. The eigenvectors of $L_{k}$ corresponding to zero eigenvalues (the harmonic part of the spectrum) are minimal representatives of their respective homology classes. An eigenvector corresponding to a non-zero eigenvalue of $L^{\text{rw}}_{k}$, must either be an eigenvector of $L_{k}^{up}$ or $L_{k}^{\text{down}}$ with the same eigenvalue. Furthermore, an element of the nullspace of $L_{k}^{\text{rw}}$ must be in the kernel of both of its components. Given an eigenpair $(\lambda_{i},v_{i})$ of $L_{k}^{\text{down}}$ then $(\lambda_{i},\vartheta_{k}v_{i})$ is an eigenpair of $L_{k-1}^{\text{up}}$ (Torres and Bianconi, 2020; Horak and Jost, 2013). The Hodge decomposition ties everything together, by expressing the space of $k$-simplices $K_{k}$ as a direct sum of gradients, curls, and harmonics:

We refer to several excellent introductions (Lim, 2020; Chen et al., 2021) to Hodge decompositions.

Due to the interplay of spectra of $L_{k}$ and $L_{k+1}$ discussed in Section 3.2, constraints on Laplacians for multiple $k$ can potentially introduce ‘spectral conflicts’. We avoid this by considering only the $L_{k}^{\text{up}}$ components of the chosen Laplacians for each $k$. For wide spectral bands and appropriate choice of $k$, the spectral region of interest will be a subspace of the spectrum of $L_{k}^{\text{up}}$ and thus preserved.

During contraction, let $z$ be the index of a contracted simplex in $K$, $t$ be the index of the target simplex in $\hat{K}$ and $|\phi|$ bet the cardinality of the candidate family being contracted. $P_{k}$ is then a simplicial map $K_{k}\rightarrow\hat{K}_{k}$ where every $k-$simplex of $K$ maps to a valid simplex of $\hat{K}$, or to zero. For consistency, it is required that $\hat{\vartheta}_{k}P_{k}=P_{k-1}\vartheta_{k}$. To satisfy desirable spectral approximation guarantees (Loukas, 2019) and for $P^{+}=P^{T}D^{-2}$ to hold, we set the elements of each contraction set as $P_{t,z}=\frac{1}{|\phi|}$, for all $z\in\phi$, where $\phi\mapsto t$. We resolve the ambiguity of the choice of target simplex by consistently selecting the target simplex based on its index. e.g. for edge contractions, both vertices are merged into one with the smaller index.

Our algorithm is agnostic of the choice of combinations of simplices $\Phi$ to be contracted. We tested with various candidate families: edges (pairs of vertices), faces (triplets of vertices) and more general vertex neighborhoods consisting of closed-stars. Larger contraction sets result in aggressive coarsening at each iteration which leads to larger spectral error. All results in this paper use only edge collapses. This also simplifies comparison to related work (which are restricted to edge collapses).

For $k>0$, the harmonic portion of the spectrum of $L_{k}$ is non-trivial and encodes information about non-trivial cycles called homology generators. A homology group $\mathcal{H}_{k}$ is a vector space of cycles that are not bounding higher dimensional simplices and therefore manifest as the null space of $L_{k}$. It is often desirable to preserve these eigenvectors, despite their coresponding zero eigenvalue, to maintain topological consistency. In practice, for all experiments we use modified eigenvalues $\tilde{S}=\mathbb{I}+S$ when dealing with spectral of higher-dimensional Laplacians.

Edge contractions may produce self-intersections, particularly if a fixed vertex-placement policy is followed, say, at the midpoint of an edge. While there are methods to avoid this effectively (Sacht et al., 2013), we adopted a simple vertex positioning scheme for tetrahedral meshes to prevent intersections.

Our scheme operates in four steps: (1) identify the faces in the link of the edge being collapsed; (2) construct the Delaunay tetrahedralization of the vertices of these faces; (3) choose barycenters of the new tetrahedra that are on the same side of the faces in 1 as the corresponding vertex of the edge being collapsed; (4) the output vertex is the centroid of all vertices that pass the test in 3. If no vertices pass the test in 3 then we do not perform the collapse.

To enable comparisons of $L_{k},\;k=1$ for triangle meshes we extend previous work (Lescoat et al., 2020) as a baseline. Their method minimizes spectral error $E=\|PM^{-1}LF-\tilde{M}^{-1}\tilde{L}PF\|_{\tilde{M}}^{2},$ where $P$ is a coarsening projection matrix, $M$ is a mass matrix, $L$ is a differential operator, and $F$ is the spectrum of interest as a matrix of eigenvectors. A tilde above the respective notations denotes their coarsened versions. From their output coarsening matrix $P_{0}$ we additionally infer coarsening matrices $P_{1}$ and $P_{2}$ which operate on the space of edges and faces respectively. However, they do not consider higher dimensional mappings in their error metric. Since they only consider one Laplacian $L_{0}^{\mathrm{cot}}$, we include this as one of the targets in all our comparisons, unless stated otherwise. We visualize spectral preservation via functional maps (Liu et al., 2019; Lescoat et al., 2020; Chen et al., 2020) $C=U_{c}^{T}PU$ which is a diagonal matrix when the input and output spectra match perfectly. Unfortunately, it is not straightforward to compare with prior non-spectal tetrahedral coarsening methods, or to extend previous spectral coarsening methods to operate on tetrahedral meshes (Section 2).

We visualize coarsened triangular (Figure 8) and tetrahedral (Figure 11) meshes along with a few eigenvectors (as heat maps on vertices). The results are reassuring that for the special case of coarsening triangle meshes our algorithm produces similar spectral results to previous work. Using $L_{1}$ subtly improves the preservation of structures such as the jaw-line, the mouth and the nose of the Suzanne model (Figure 8). The boundary loops around the eyes (eigenvectors of the null space of $L_{1}$) aim to preserve their original size compared to when only $L_{0}$ is used. The bottom row of Figure 8 contains quantitative comparisons of our eigenvalues against the baseline and reference. It appears that the approximation is good for $L_{0}$, but curiously spectral divergence is observed for our method and the baseline in $L_{1}$ .

This is also observed on a tetrahedral mesh (Fertility model) as shown in the bottom row of Figure 9. However, the profile of eigenvalues for $L_{1}$ and the eigenvalues of $L_{2}$ are approximated well. We also show functional maps to support these observations. The Laplacians considered were $L_{0}^{\mathrm{cot}}$ and $L_{1}$ for the triangle mesh (2D), and additionally $L_{2}$ for the tetrahedral mesh. We report errors in Table 4 using the following error metrics: a local volume-preserving measure $\|C\|_{\Pi^{\perp}}=\|C^{T}C-\mathbb{I}\|_{F}^{2}$, an isometry measure $\|C\|_{L_{\text{c}}}=\|S_{c}C-CS\|_{F}^{2}/\|C\|_{F}^{2}$, a conformality measure $\|C\|_{\text{conf}}=\|C^{T}SC-Sc\|_{F}^{2}$, a subspace approximation measure $\|C\|_{\text{sub}}=|\|S^{1/2}U^{T}\Pi US^{-1/2}\|_{2}-1|$, subspace alignment $\|C\|_{\theta}=\|\sin\Theta(U,P^{+}U_{c})\|_{F}^{2}$ and relative eigenvalue error $\|S\|_{\lambda}=\|(S-Sc)/S\|_{2}$.

The above results indicate comparable performance against the baseline for for triangle meshes and the added capability of handling higher dimensional Laplacians for tetrahedral (3D) meshes. Additional examples are included in the supplementary material. We illustrate the evolution of spectral errors for the Tree root model over three levels of coarsening (Figure 3).

We demonstrate the versatility of our method by coarsening an input complex with two different spectral bands to be preserved: $\beta_{1}+1$ lowest, and 10 largest eigenpairs of $L_{1}$. Here, $\beta_{1}$ is the betti number (rank of the homology group $\mathcal{H}_{1}$). Figure 5 presents these results along with the functional maps $C$ as explained above. As expected (by design), Gudhi exactly preserves the homology rank $\beta_{1}$, with no regard for other features. On the contrary, our method can be tuned to a spectral band of choice. Either the nullspace-encoded harmonic information (first row of functional maps) or the high frequency band(second row of maps).

We constructed a diverse dataset of 100 simplicial complexes (Keros et al., 2022) with non-trivial topology by randomly sampling 400 points repeatedly on multi-holed tori, and subsequently constructing alpha complexes at various thresholds. It contains from zero up to 66 homology cycles. On this dataset, we computed mean spectral error metrics (Table 1), alongside a homology preservation error $E_{\beta}=|\beta_{1}^{\mathrm{fine}}-\beta_{1}^{\mathrm{coarse}}|$. Intuitively, this is the error in the number of 1-cycles destroyed by coarsening. The number of simplices reduced by Gudhi was consistent across both experiments. We ran two variants of our method which preserved the first $30$ eigenpairs of $L_{1}$, reducing the input complexes by a factor of 0.8, and the first $\beta_{1}+1$ eigenpairs of $L_{1}$, matching the target number of vertices to the result of Gudhi. The results are summarized in Table 1 with standard deviations in parentheses. Gudhi, as designed, preserves $\beta_{1}$ exactly but exhibits spectral leak elsewhere. With random contractions, the leak is amortized across spectral bands but it destroys about 1 cycle on average. Our method is controllable, highlighting that it can be particularly effective if the spectral constraints are known for a particular application.

A straightforward application of our method is to suppress frequencies associated with noise. We filtered a noisy Bunny model (43K vertices) using a very narrow low-pass filter: only the first three eigenvectors of $L_{0}$ and $L_{1}$. Figure 6 visualizes our result along with the baseline filters unevenly, possibly due to noisy geometric information in its cotan weighting.

We solve the Poisson equation $-\Delta u=1$ with Dirichlet boundary condition $u=0$ on the popular Plate-hole model, using piece-wise linear (triangular) elements. In addition to solving it on the discretized mesh, we test robustness by applying planar pertubations to the vertices with increasing levels of Gaussian noise. For each setting, we coarsen the noisy mesh with $\rho=0.75$ (8000 to 2000 vertices) using $L_{0}^{\text{cot}}$ and $L^{1}$, run a standard FEM solver on the coarse mesh and lift the solution to the input mesh. Figure 10 shows a plot of error vs noise (top left), the reference solution (top right) and error maps for different noise settings (columns) and methods (rows). Also shown are previous work (Lescoat et al., 2020), with (optpos) and without (noopt) vertex position optimization, and a quadric-based method (Garland and Heckbert, 1997). The coarse mesh is overlaid on the error maps and can be viewed by magnifiying the figure. Our method exhibits robustness against noise and improved approximation performance against element reduction.

We evaluate the fidelity (Figure 12) of spectral distance measures computed on the coarse mesh and “lifted” to the fine mesh, between vertices $w,v$ of a triangle surface mesh (Suzanne) and a volume tetrahedral mesh (Fertility), using the same parameters as Figures 8 and 11. Figure 1 illustrates the spectral distance approximation process on the Engine tetrahedral model, coarsened with $\rho=-.8$ (from 46220 to 10000 vertices) while preserving the first 50 eigenvectors of $L_{0}^{\text{cot}}$ and the first 25 eigenvectors of $L_{1}$ and $L_{2}$. The model has 20 holes that are retained in the coarse mesh. The metrics used and their parameters are tabulated below.

Spectral distance approximation on Suzanne outperforms the baseline by orders on magnitude in most cases, with error remaining low even for the tetrahedral Fertility mesh. The qualitative comparison indicates agreement between our “lifted” version and the reference, despite some localized distortions.