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Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning

Duc Thien Nguyen, Konstantinos Slavakis, Dimitris Pados

TL;DR

The paper addresses imputation of time-varying edge flows on graphs with missing observations by extending the Multilinear Kernel Regression with Imputation via Manifold learning (MultiL-KRIM) to the simplicial-complex setting. It embeds edge-flows in a reproducing kernel Hilbert space and enforces graph topology through Hodge Laplacians, while employing a low-rank, multilinear factorization X ≈ $\mathbf{U}_1 \mathbf{U}_2 \mathbf{K} \mathbf{V}_1 \mathbf{V}_2$ to achieve scalable, nonlinear regression without training data. A parallel successive-convex-approximation (SCA) algorithm solves the resulting non-convex objective, with complexity per iteration expressed in terms of $N_1$, $N_l$, $T$, and low-rank dimensions. Empirical results on the Cherry Hills water network and Sioux Falls transportation network show that the proposed method yields consistently lower MAE than FlowSSL, S-VAR, and MMF, while using substantially fewer parameters. This work integrates topology, manifold geometry, and kernel methods to robustly recover edge flows in real networks.

Abstract

This paper extends the recently developed framework of multilinear kernel regression and imputation via manifold learning (MultiL-KRIM) to impute time-varying edge flows in a graph. MultiL-KRIM uses simplicial-complex arguments and Hodge Laplacians to incorporate the graph topology, and exploits manifold-learning arguments to identify latent geometries within features which are modeled as a point-cloud around a smooth manifold embedded in a reproducing kernel Hilbert space (RKHS). Following the concept of tangent spaces to smooth manifolds, linear approximating patches are used to add a collaborative-filtering flavor to the point-cloud approximations. Together with matrix factorizations, MultiL-KRIM effects dimensionality reduction, and enables efficient computations, without any training data or additional information. Numerical tests on real-network time-varying edge flows demonstrate noticeable improvements of MultiL-KRIM over several state-of-the-art schemes.

Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning

TL;DR

The paper addresses imputation of time-varying edge flows on graphs with missing observations by extending the Multilinear Kernel Regression with Imputation via Manifold learning (MultiL-KRIM) to the simplicial-complex setting. It embeds edge-flows in a reproducing kernel Hilbert space and enforces graph topology through Hodge Laplacians, while employing a low-rank, multilinear factorization X ≈ to achieve scalable, nonlinear regression without training data. A parallel successive-convex-approximation (SCA) algorithm solves the resulting non-convex objective, with complexity per iteration expressed in terms of , , , and low-rank dimensions. Empirical results on the Cherry Hills water network and Sioux Falls transportation network show that the proposed method yields consistently lower MAE than FlowSSL, S-VAR, and MMF, while using substantially fewer parameters. This work integrates topology, manifold geometry, and kernel methods to robustly recover edge flows in real networks.

Abstract

This paper extends the recently developed framework of multilinear kernel regression and imputation via manifold learning (MultiL-KRIM) to impute time-varying edge flows in a graph. MultiL-KRIM uses simplicial-complex arguments and Hodge Laplacians to incorporate the graph topology, and exploits manifold-learning arguments to identify latent geometries within features which are modeled as a point-cloud around a smooth manifold embedded in a reproducing kernel Hilbert space (RKHS). Following the concept of tangent spaces to smooth manifolds, linear approximating patches are used to add a collaborative-filtering flavor to the point-cloud approximations. Together with matrix factorizations, MultiL-KRIM effects dimensionality reduction, and enables efficient computations, without any training data or additional information. Numerical tests on real-network time-varying edge flows demonstrate noticeable improvements of MultiL-KRIM over several state-of-the-art schemes.
Paper Structure (10 sections, 8 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 8 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed method
  4. A short recap of MultiL-KRIM
  5. Extended MultiL-KRIM for edge-flow imputation
  6. Computational complexity
  7. Numerical Tests
  8. Cherry Hills water network
  9. Sioux Falls transportation network
  10. Conclusions

Figures (3)

  • Figure 1: A "collaborative-filtering" approach: Points $\Set{ \varphi (\mathbfit{l}_{k_j}) }_{j=1}^3$, which lie into or close to the unknown-to-the-user manifold $\mathscr{M} \subset \mathscr{H}$, collaborate affinely to approximate $\varphi (\check{\bm{\mu}}_{t})$. All affine combinations of $\Set{ \varphi (\mathbfit{l}_{k_j})}_{j=1}^3$ define the approximating "linear patch" (gray-colored plane), which mimics the concept of a tangent space to $\mathscr{M}$RobbinSalamon:22.
  • Figure 2: Mean MAE value curves in log scale (the lower the better) w.r.t. the groud truth signals vs. sampling ratios. The shaded areas indicate a range of one standard deviation above and below the mean.
  • Figure 3: Visualization of coefficient matrices $\hat{\mathbf{U}}_1^{(*)}, \hat{\mathbf{U}}_2^{(*)}, \hat{\mathbf{V}}_1^{(*)}$ and $\hat{\mathbf{V}}_2^{(*)}$, output by \ref{['alg:multil.general']} for Cherry Hills water network data, at sampling ratio $s = 0.5$. Heat maps show the entry-wise absolute values of the matrices, rescaled to be in $[0, 1]$ range.