Complex Interpolation of Matrices with an application to Multi-Manifold Learning

Adi Arbel; Stefan Steinerberger; Ronen Talmon

cs.LGApr 15, 2026

Complex Interpolation of Matrices with an application to Multi-Manifold Learning

Authors: Adi Arbel, Stefan Steinerberger, Ronen Talmon

Organizations: Viterbi Faculty of Electrical and Computer Engineering, Technion – Israel Institute of Technology, Haifa, Israel · Department of Mathematics and Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA

Abstract

Given two symmetric positive-definite matrices $A, B \in \mathbb{R}^{n \times n}$ , we study the spectral properties of the interpolation $A^{1-x} B^x$ for $0 \leq x \leq 1$ . The presence of `common structures' in $A$ and $B$ , eigenvectors pointing in a similar direction, can be investigated using this interpolation perspective. Generically, exact log-linearity of the operator norm $\|A^{1-x} B^x\|$ is equivalent to the existence of a shared eigenvector in the original matrices; stability bounds show that approximate log-linearity forces principal singular vectors to align with leading eigenvectors of both matrices. These results give rise to and provide theoretical justification for a multi-manifold learning framework that identifies common and distinct latent structures in multiview data.

Figures & tables

Figure 1: Two 2D projections of a 3D point cloud of a cow on two different view angles: given only the two two-dimensional point sets (discretized via a kernel into two symmetric positive-definite matrices $A,B\in\mathbb{R}^{n\times n}$ ), can one automatically discover that the eigenstructure captures the same underlying ground truth? (Details in section 2 and section 4 ).

Panel of Figure 2 — Figure 2: Two aligned point clouds sampled from two 2D cylinders embedded in $\mathbb{R}^{3}$ . The height (red) axis is shared among the point clouds, whereas the azimuthal angle is distinct.

Figure 3: The SVFD of the point clouds depicted in fig. 2 . The empirical singular values of the interpolated kernels connecting the second common eigenvalues (shown left and right) are highlighted in yellow.

Figure 4: The same as fig. 3 , but highlighting the empirical singular values originating at $x=0$ from the fourth largest eigenvalue of $\mathcal{C}_{1}$ . The point clouds on the left and right are colored by the corresponding eigenvector. The curve is far from a line, the eigenvectors have little in common.

Panel of Figure 5 — Figure 5: A sketch of the domain $\mathcal{D}$ .

Panel of Figure 6 — Figure 6: A sketch of the multimanifold learning setup: three hidden manifolds and two observables, where at each observation only one manifold ( $\mathcal{M}_{3}$ ) is common.

Multi-View Learning

Multi-view learning combines complementary representations of the same objects to improve prediction, clustering, and robust data integration.

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Department of Statistics University of Wisconsin Madison Madison, WI 53706, USA

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We study eigenpair estimation for the tensorized, density-weighted Laplace--Beltrami operator from

n

independent data points sampled from a multi-manifold mixture distribution where manifolds may intersect. Let

d

denote the largest intrinsic dimension of the manifold components and

s\geq2

the smoothness of the sampling densities. For simple, nontrivial eigenpairs, we establish the minimax rate

n^{-s/(d+2s)}

for approximating the eigenvalue and eigenfunction of the tensorized Laplace--Beltrami operator, and show that a continuum plug-in estimator based on kernel density estimation attains the same rate when the supporting manifolds are known. We further analyze a manifold-agnostic graph Laplacian estimator and prove that its eigenpairs converge to those of the tensorized Laplace--Beltrami operator at rate

n^{-1/(2.5d-1)}

, up to logarithmic factors, improving upon previously known spectral convergence rates for multi-manifold learning. Numerical experiments on synthetic multi-manifold data corroborate the rates guaranteed by our theorems. These results characterize the information-theoretic difficulty of multi-manifold spectral estimation and quantify the additional cost of learning without oracle component information.

Abstract

Multi-View Learning

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