Regularized Laplacian Estimation and Fast Eigenvector Approximation

Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistical interpretation of their approximation procedure. Our interpretation will be analogous to the manner in which $\ell_2$-regularized or $\ell_1$-regularized $\ell_2$-regression (often called Ridge regression and Lasso regression, respectively) can be interpreted in terms of a Gaussian prior or a Laplace prior, respectively, on the coefficient vector of the regression problem. Our framework will imply that the solutions to the Mahoney-Orecchia regularized SDP can be interpreted as regularized estimates of the pseudoinverse of the graph Laplacian. Conversely, it will imply that the solution to this regularized estimation problem can be computed very quickly by running, e.g., the fast diffusion-based PageRank procedure for computing an approximation to the first nontrivial eigenvector of the graph Laplacian. Empirical results are also provided to illustrate the manner in which approximate eigenvector computation \emph{implicitly} performs statistical regularization, relative to running the corresponding exact algorithm.

💡 Research Summary

The paper builds on the recent work of Mahoney and Orecchia, who showed that three popular diffusion‑based procedures—Heat Kernel, Lazy Random Walk, and PageRank—do not merely approximate the first non‑trivial eigenvector of a graph Laplacian, but actually solve a regularized semi‑definite program (SDP) exactly. The authors extend this insight by providing a statistical, Bayesian interpretation of those diffusion methods, analogous to the way ridge regression corresponds to a Gaussian prior and the Lasso to a Laplace prior in linear regression.

First, the authors formalize a Bayesian model for graph Laplacian estimation. They treat the observed normalized Laplacian ( \hat L ) as a random “sample” matrix drawn from a scaled Wishart distribution with expectation equal to the true (population) Laplacian ( L ). The Wishart model captures the positive‑semidefinite nature of Laplacians and introduces a scale parameter ( m ) that plays the role of an inverse sample size. The conditional density of the sample Laplacian is \