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Bi-stochastic normalization provides an alternative normalization of graphLaplacians in graph-based data analysis and can be computed efficiently bySinkhorn-Knopp (SK) iterations. This paper proves the convergence ofbi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacianwith rates, when $n$ data points are i.i.d. sampled from a general$d$-dimensional manifold embedded in a possibly high-dimensional space. Undercertain joint limit of $n \to \infty$ and kernel bandwidth $\epsilon \to 0$,the point-wise convergence rate of the graph Laplacian operator (under 2-norm)is proved to be $ O( n^{-1/(d/2+3)})$ at finite large $n$ up to log factors,achieved at the scaling of $\epsilon \sim n^{-1/(d/2+3)} $. When the manifolddata are corrupted by outlier noise, we theoretically prove the graph Laplacianpoint-wise consistency which matches the rate for clean manifold data plus anadditional term proportional to the boundedness of the inner-products of thenoise vectors among themselves and with data vectors. Motivated by ouranalysis, which suggests that not exact bi-stochastic normalization but anapproximate one will achieve the same consistency rate, we propose anapproximate and constrained matrix scaling problem that can be solved by SKiterations with early termination. Numerical experiments support ourtheoretical results and show the robustness of bi-stochastically normalizedgraph Laplacian to high-dimensional outlier noise.

Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise