提出一种通用方法，可在曲面上高效逼近马尔可夫高斯马特恩场。
Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds
- 基于离散外微积分，构造无需预设参数的收敛图模型近似。
- 精度矩阵为图拉普拉斯的谱函数，支持低秩压缩与测量降维。
- 适合处理曲面随机场建模，尤其适用于几何复杂的数据集。
马尔可夫型威特尔-马特恩场可通过有限元法对双参数族方程 $(κ^2 - Δ)^{α/2} u = \mathcal{W}$（$κ∈\mathbb{R}, α∈\mathbb{N}$）进行离散化，以稀疏精度矩阵的离散高斯马尔可夫随机场（GMRF）实现收敛逼近。借助离散外微积分（DEC）最新进展，本文提出一种新的、密切相关的收敛性GMRF逼近方法，适用于完整且无边界的黎曼流形，并以良好中心化的单纯复形进行离散。该方法（i）对参数 $α, κ$ 无关，可统一逼近整个$(α, κ)$族的精度与协方差矩阵，实现参数推断而非猜测；（ii）天然支持点态及分段光滑观测数据，两者近似效果均佳；（iii）计算不依赖插值器选择，更换收敛插值器无额外开销。此外，我们证明在充分连通且体积集中的一类离散网格上，精度矩阵为图拉普拉斯的谱函数。我们提供该马特恩GMRF族的低秩近似器，并指出一个应用场景：通过压缩感知减少建模所需的测量数量。
原文摘要 · Abstract (English)
Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}. \] of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $α, κ$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(α, κ)$-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.
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