Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Grassmanian Interpolation of Low-Pass Graph Filters: Theory and Applications

Anton Savostianov, Michael T. Schaub, Benjamin Stamm

TL;DR

This work addresses the computational bottleneck of obtaining low-frequency subspaces for evolving or parametric graphs by proposing a Grassmannian, Riemannian-normal-coordinate interpolation scheme for eigenspaces and the corresponding low-pass filters. It derives an error bound showing tangent-space interpolation errors propagate linearly to the manifold via the exponential map, and provides a practical interpolation algorithm that updates spectra and realigns subspaces. The paper demonstrates two illustrative applications: (i) a similarity-corrected, time-varying graph family to capture evolving homophily and (ii) dot product graphs derived from a static graph to improve node classification, enabling efficient topology search through filter interpolation. These methods offer substantial computational savings and provide principled tools for spectral processing in parametric graph families with potential extensions to higher-order topologies and GNN architectures.

Abstract

Low-pass graph filters are fundamental for signal processing on graphs and other non-Euclidean domains. However, the computation of such filters for parametric graph families can be prohibitively expensive as computation of the corresponding low-frequency subspaces, requires the repeated solution of an eigenvalue problem. We suggest a novel algorithm of low-pass graph filter interpolation based on Riemannian interpolation in normal coordinates on the Grassmann manifold. We derive an error bound estimate for the subspace interpolation and suggest two possible applications for induced parametric graph families. First, we argue that the temporal evolution of the node features may be translated to the evolving graph topology via a similarity correction to adjust the homophily degree of the network. Second, we suggest a dot product graph family induced by a given static graph which allows to infer improved message passing scheme for node classification facilitated by the filter interpolation.

Grassmanian Interpolation of Low-Pass Graph Filters: Theory and Applications

TL;DR

This work addresses the computational bottleneck of obtaining low-frequency subspaces for evolving or parametric graphs by proposing a Grassmannian, Riemannian-normal-coordinate interpolation scheme for eigenspaces and the corresponding low-pass filters. It derives an error bound showing tangent-space interpolation errors propagate linearly to the manifold via the exponential map, and provides a practical interpolation algorithm that updates spectra and realigns subspaces. The paper demonstrates two illustrative applications: (i) a similarity-corrected, time-varying graph family to capture evolving homophily and (ii) dot product graphs derived from a static graph to improve node classification, enabling efficient topology search through filter interpolation. These methods offer substantial computational savings and provide principled tools for spectral processing in parametric graph families with potential extensions to higher-order topologies and GNN architectures.

Abstract

Low-pass graph filters are fundamental for signal processing on graphs and other non-Euclidean domains. However, the computation of such filters for parametric graph families can be prohibitively expensive as computation of the corresponding low-frequency subspaces, requires the repeated solution of an eigenvalue problem. We suggest a novel algorithm of low-pass graph filter interpolation based on Riemannian interpolation in normal coordinates on the Grassmann manifold. We derive an error bound estimate for the subspace interpolation and suggest two possible applications for induced parametric graph families. First, we argue that the temporal evolution of the node features may be translated to the evolving graph topology via a similarity correction to adjust the homophily degree of the network. Second, we suggest a dot product graph family induced by a given static graph which allows to infer improved message passing scheme for node classification facilitated by the filter interpolation.
Paper Structure (22 sections, 3 theorems, 42 equations, 4 figures, 2 algorithms)

This paper contains 22 sections, 3 theorems, 42 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction.
  2. Graph Filters
  3. Graphs and matrix representations.
  4. Filters.
  5. Eigenspace interpolation
  6. Parametric graph families and graph filters
  7. Grassmann manifold and interpolation
  8. Interpolation in normal coordinates
  9. Filter interpolation.
  10. Error Analysis
  11. Sensitivity of the Exponential Map
  12. Lagrange Interpolation Error
  13. Illustrative application I: Similarity correction and CSBM
  14. Homophily
  15. Contextual Stochastic Block Model
  16. ...and 7 more sections

Key Result

Lemma 3.1

Let $\Delta, \widetilde{\Delta} \in T_{ [V_\bullet] } \mathrm{Gr}(n,k)$ for a given base point $[V_\bullet]$; then where $C > 0$ is a fixed constant and $\sigma_k(\Delta)$ is the smallest positive singular value of $\Delta$.

Figures (4)

  • Figure 1: Riemannian interpolation in normal coordinates: scheme on $T_{D_\bullet} \mathrm{Gr}(n,k)$. Base point $V_\bullet$ and exact computations $\{ V_i\}_{i=1}^N$ are shown in magenta; the Lagrangian interpolant $\widetilde{\Delta} \in T_{D_\bullet} \mathrm{Gr}(n,k)$ of the corresponding Grassman logarithms $\Delta_i$ and the consequent exponential map $\widetilde{V}$ back to $\mathrm{Gr}(n,k)$ are shown in blue.
  • Figure 2: CSBM setup parameters: (a) adjacency matrix with colors corresponding to different cluster and inter-cluster edges; (b) evolution of the features means $\mu_1$ and $\mu_2$ with the cosine similarity of the corresponding vectors; (c) quantiles of the edge weights with similarity correction illustrating changing topology of $S(t)$; (d) distance between neighbouring low-frequency subspaces along the trajectory $\mathcal{V}(t)$, $k = 5$ supporting the fact that $\mathcal{V}(t)$ is slowly changing trajectory suitable for interpolation.
  • Figure 3: Interpolation results for CSBM: (a) distance between the exact computation of the extremal subspace $V(t)$ and the interpolant $\widetilde{V}(t)$ for base point $V_\bullet$ chosen on the trajectory $\mathcal{V}(t)$ and randomly (dips relate to Chebyshev nodes, $N = 10$); (b) decreasing subspace interpolation error, maximal and at different points times (error changes depending on the location of interpolation nodes), vs number of interpolation nodes $N$; (c) decreasing filter interpolation error, maximal and at different points times, vs number of interpolation nodes $N$; (d) gains in the computation time between the filter interpolation and the exact computation.
  • Figure 4: Improvement in the classification accuracy for KarateClub (left) and MNIST similarity graph (right). The low-pass filter retains approximately $k/n\approx 10\%$ of the spectrum; interpolation uses $N=10$ Chebyshev nodes.

Theorems & Definitions (3)

  • Lemma 3.1: Sensivity of the Exponential Map
  • Lemma 3.2: Lagrange Interpolation, szabados1990interpolation
  • Lemma 4.1: Expected temporal CSBM