Delaunay-weighted two-sample test uses geometric direction information to detect principal-direction covariance differences in high-dimensional manifold data

Introduces the Delaunay weight, a new geometric proximity measure built on the Delaunay triangulation over a manifold that uses both geodesic distance and relative direction, whereas existing kernel, local regression, and k-NN approaches use only local distances. Prior similarity measures rely only on pairwise distances; this weight depends on the logarithmic map vectors log_{z*_i}(z_j) rather than merely their norms, so equal distances with different directions yield different weights. Definition 2 gives the formal manifold definition and shows conditions (c†) and (c‡) coincide when M=R^d; the R^2 example in Figures 2 and 3 shows the three nearest neighbors of z2 all lie to the upper right, while the Delaunay simplex formed by z5, z7, z8 covers all directions.

Constructs the test statistic T_DW as the average of all within-group Delaunay weights in the pooled sample and provides a permutation procedure. It follows the within-group proximity averaging idea of the k-NN test of Schilling (1986) but replaces the proximity measure with the Delaunay weight, thereby incorporating manifold structure and direction information into the statistic. Equation (4) defines T_DW; the permutation step sets the p-value to {number of B permuted statistics not smaller than the observed value + 1}/(B+1), and the text notes that under the null this p-value is uniform on {1/(B+1),…,1}, converging to Unif[0,1] as B→∞.

Establishes asymptotic normality under the null and consistency under the alternative. Because the discrete nature of the Delaunay triangulation makes the unconditional variance hard to express explicitly, the authors first derive the expectation and variance conditional on Z and then prove asymptotic normality of the standardized statistic. Theorem 1 gives explicit expressions for E_{H0}(T_DW|Z) and Var_{H0}(T_DW|Z); Theorem 2 gives asymptotic normality; Theorem 3 proves that for any significance level α∈(0,1) and F≠G the permutation test rejects the null with asymptotic probability one, following the idea of Henze and Penrose (1999).

Develops the stereographic projected DELAUNAYSPARSE algorithm to approximate the Delaunay weight matrix when the manifold is unknown, and verifies robustness to the manifold-learning steps. The original DELAUNAYSPARSE algorithm handles only points inside the convex hull; the authors add an inverse stereographic projection onto a sphere so that points outside the convex hull also get a Delaunay simplex. The algorithm has complexity O(n^{2+1/d}d^2); numerical experiments show the weight matrix differs only by floating-point error once η≥15, so η=15 is recommended; in the sensitivity analysis the 'known manifold', 'fixed d̂=10=d', and 'estimated d̂' implementations give similar results, with power lost only when the estimated intrinsic dimension falls below the truth.

The test targets high-dimensional data satisfying the manifold hypothesis: both distributions are supported on a d-dimensional geodesically convex Riemannian manifold M embedded in R^D with d much smaller than D, and the sample size must exceed the estimated intrinsic dimension. When the manifold is unknown, the method obtains a low-dimensional Euclidean representation via Isomap-style manifold learning, Dijkstra geodesic distance estimation, and classical MDS, then approximates the weight matrix with the stereographic projected DELAUNAYSPARSE algorithm, so it applies directly to bioinformatics, image analysis, and natural language processing data believed to have low-dimensional structure. In the real data analysis the authors use mouse-level permutation to control type I error from within-mouse cell correlation, indicating the method extends to settings with grouped correlation structure. The authors also note the test could be combined with the kernel or e-distance test through a Cauchy combination test for more robust performance under unknown alternative types, and that the Delaunay weight matrix could serve statistical inferences beyond hypothesis testing.

Under the location alternative the test has lower power than the kernel and e-distance tests, and the authors themselves note their method is not the most powerful under location differences; so in practice, when the alternative type is unknown, a single test's performance depends on whether the difference comes from location or direction. In the real data the intrinsic dimension is estimated as 3, far below the 68 proteins, but this estimate relies on Algorithm S1, and the authors note that within-mouse cell correlation may bias the low-dimensional representation and the Delaunay weights, causing power loss, although mouse-level permutation controls type I error. The theoretical results rest on regularity conditions, including a regularization condition on the Delaunay triangulation (satisfied, the authors say, if no hub exists), and whether these hold in a given real dataset still needs case-by-case judgment. The future directions the authors raise—noise-contaminated data, unions of manifolds of different dimensions, and combinations with other tests—are not yet validated in this paper.