用连续拉普拉斯算子近似离散拉普拉斯算子：求证与误差估计

Great question—this is a super common gap in applied manifold learning papers, but there’s a rigorous mathematical foundation behind that spectral convergence claim. Let’s break down the key references and proof frameworks you’re seeking:

• Foundational Spectral Convergence Proofs
  The core link between discrete graph Laplacians and continuous manifold Laplacians (including convergence as graphs grow denser) was formalized in Belkin and Niyogi’s 2003 work. They proved that for a smooth manifold sampled uniformly with sufficiently dense points, the normalized graph Laplacian converges spectrally to the manifold’s Laplace-Beltrami operator. Their paper includes explicit error bounds for eigenvalues and eigenfunctions, showing convergence rates tied to sampling density and graph connectivity.

• Arbitrary-Order Expansion Justification
  For the claim about using arbitrary-order expansions of the continuous Laplacian to approximate the discrete version, look to work on spectral graph theory and meshless PDE approximations:

  • Chung’s book Spectral Graph Theory covers generalizations of Laplacian approximations, including higher-order expansions that bridge discrete and continuous settings.
  • Coifman and Lafon’s diffusion maps work extends this by framing the graph Laplacian as a discrete diffusion operator. They show that as the graph becomes a better manifold approximation, the discrete operator can be expanded in terms of the continuous Laplacian, with error estimates that decay as point distances shrink.

• Tight Error Estimates
  If you need precise error bounds for the approximation, check out Singer and co-authors’ work on manifold sampling and graph Laplacian convergence. They proved that under mild sampling conditions, the discrete graph Laplacian converges to the continuous one in operator norm, with explicit rates for how quickly eigenvalues and eigenfunctions align. This directly validates the use of arbitrary-order expansions as the graph scales.

Most papers skip citing these sources because they’ve become standard in the field, but the rigor is absolutely there. Start with Belkin & Niyogi’s paper if you want to dig into the core proofs—it’s the bedrock for nearly all subsequent work on this topic.

内容的提问来源于stack exchange，提问作者Michael Jarret