Dimension Changes the Scale-Signal Channel in Nearest-Neighbor Two-Sample Testing
Abstract
Nearest-neighbor two-sample tests are often analyzed with dimension fixed. We study the power of the generalized edge-count test on a directed 1-nearest-neighbor graph. This statistic combines a weighted sum and a difference of the two within-sample edge counts. For balanced isotropic Gaussian samples with d/log N→γ>0, we derive the connection probabilities under local location and scale alternatives. Under location alternatives, the two counts have equal mean shifts, which cancel in their difference. Under scale alternatives, the two counts instead have mean shifts of opposite signs. We also derive the growth rate of the null expectation 𝔼0C3, where C3 counts pairs of directed edges sharing exactly one vertex. On an explicit range of γ, we establish local-power limits strictly above the test level for scale alternatives in the difference component at the scale |εN|asymp (𝔼0C3)1/2/(Nd). Here 𝔼0C3 gives the variance order of the difference count under these local alternatives. At this scale, the standardized drift of the weighted sum vanishes. When d=1, the weighted component instead detects local Gaussian scale alternatives while the difference component has no limiting drift. Thus dimension changes both the magnitude of a detectable scale difference and the component of the test that detects it.
Related
- Vectorized Generalized Nearest Neighbor Decoding for In-block Memory Channel
- One-Layer Transformer Provably Learns Multiclass One-Nearest Neighbor in Context
- Stochastic dominance of first return times for nearest-neighbor random walks on ℤd
- The nearest neighbor interchange proximity measure for the tree-basedness of rooted binary phylogenetic networks
- Orthogonal Inference for Conditional Z-Estimation without Cross-Fitting: A Distributional Nearest-Neighbor Approach
BibTeX
@misc{chen2026dimension,
title = {{Dimension Changes the Scale-Signal Channel in Nearest-Neighbor Two-Sample Testing}},
author = {Hao Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41271},
eprint = {2610.41271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41271},
}
RIS
TY - UNPB
TI - Dimension Changes the Scale-Signal Channel in Nearest-Neighbor Two-Sample Testing
AU - Hao Chen
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41271
UR - https://arxiv.org/abs/2610.41271
AB - Nearest-neighbor two-sample tests are often analyzed with dimension fixed. We study the power of the generalized edge-count test on a directed 1-nearest-neighbor graph. This statistic combines a weighted sum and a difference of the two within-sample edge counts. For balanced isotropic Gaussian samples with d/log N→γ>0, we derive the connection probabilities under local location and scale alternatives. Under location alternatives, the two counts have equal mean shifts, which cancel in their difference. Under scale alternatives, the two counts instead have mean shifts of opposite signs. We also derive the growth rate of the null expectation 𝔼_0C_3, where C_3 counts pairs of directed edges sharing exactly one vertex. On an explicit range of γ, we establish local-power limits strictly above the test level for scale alternatives in the difference component at the scale |ε_N|asymp (𝔼_0C_3)^1/2/(Nd). Here 𝔼_0C_3 gives the variance order of the difference count under these local alternatives. At this scale, the standardized drift of the weighted sum vanishes. When d=1, the weighted component instead detects local Gaussian scale alternatives while the difference component has no limiting drift. Thus dimension changes both the magnitude of a detectable scale difference and the component of the test that detects it.
ER -