Convergence of Graph Laplacian with kNN Self-tuned Kernels

Kernelized Gram matrix $W$ constructed from data points $\{x_i\}_{i=1}^N$ as $W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {σ^2} )$ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $σ$, and a common practice called self-tuned kernel adaptively sets a $σ_i$ at each point $x_i$ by the $k$-nearest neighbor (kNN) distance. When $x_i$'s are sampled from a $d$-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $L_N$ to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels $W^{(α)}_{ij} = k_0( \frac{ \| x_i - x_j \|^2}{ ε\hatρ(x_i) \hatρ(x_j)})/\hatρ(x_i)^α\hatρ(x_j)^α$, where $\hatρ$ is the estimated bandwidth function {by kNN}, and the limiting operator is also parametrized by $α$. When $α= 1$, the limiting operator is the weighted manifold Laplacian $Δ_p$. Specifically, we prove the point-wise convergence of $L_N f$ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$ consistency for $\hatρ$ which bounds the relative estimation error $|\hatρ - \barρ|/\barρ$ uniformly with high probability, where $\barρ = p^{-1/d}$, and $p$ is the data density function. Our theoretical results reveal the advantage of self-tuned kernel over fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of $d$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data.

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