提出固定半径距离带评估降维保真度,更准确衡量全局与局部结构保持情况。
A Fixed-Radius Distance-Band Benchmark for Dimensionality-Reduction Fidelity

- 用固定半径距离带替代可变邻域,避免对近邻方法的偏倚。
- 发现高全局相关性下簇内尺度压缩达93倍,传统指标无法察觉。
- 适用于关注结构保真、异常检测和少数群体表征的研究者。
降维方法通常通过点在2维嵌入中保留其k近邻的程度来评估(如recall@k、信任度、连续性)。我们指出这类指标存在偏差:其基于每点可变半径与硬阈值,偏好近邻图方法(t-SNE、UMAP),而惩罚保留绝对距离的方法。本文改用固定半径距离带的谢尔德ρ(Shepard rho)评估保真度,即在累积距离带内计算高维与低维成对距离的斯皮尔曼相关性,分别报告近邻与全局结构,并确保每个点在相同绝对半径下被评估。在具有已知几何结构的合成数据集上(非均匀密度、密集聚类、闭环过渡、离群点、不平衡双群体数据,信噪比=1,维度D=768,样本数N=1000),我们评测了八种方法(PCA、Isomap、t-SNE、UMAP、PyMDE、PCC、DREAMS、闭源工具toorPIA),结果表明:(i) 高全局谢尔德ρ可与约93倍的簇内尺度压缩共存,此现象在基于秩的指标中不可见,但可通过基于值的过压缩度量发现;(ii) recall@k与固定半径带指标系统性分歧,方向符合预期偏倚;(iii) 限定成员的谢尔德ρ能解决单点及少数群体问题,而多对统计量无法处理——即使最近的局部+全局混合方法DREAMS也在此类问题上沉默失效。附加的外样本测试(addplot)评估未见异常是否落在正常区域外,以及其方向能否标识来源。所有指标均基于全部成对距离精确计算,不依赖方法内部机制,所有数值均可离线复现:仅闭源方法的输出坐标(非算法)提交至附录资源。
原文摘要 · Abstract (English)
Dimensionality-reduction (DR) methods are routinely judged by how well each point's k nearest neighbors survive the 2-D embedding (recall@k, trustworthiness, continuity). We argue this family is a biased measure of distance fidelity: its per-point variable radius and hard inclusion threshold favor neighbor-graph methods (t-SNE, UMAP) and penalize methods that preserve absolute distances. We instead score DR fidelity with a fixed-radius distance-band Shepard rho: the Spearman correlation between high-D and 2-D pairwise distances, restricted to cumulative distance bands so that near and global structure are reported separately, with every point judged on the same absolute radius. On synthetic datasets with known ground-truth geometry (non-uniform density, dense clusters, a closed-loop transition, off-subspace outliers, imbalanced two-population data) at realistic noise (SNR=1, D=768, N=1000), we benchmark eight methods -- PCA, Isomap, t-SNE, UMAP, PyMDE, PCC, DREAMS, and the closed-source toorPIA -- and show that (i) high global Shepard rho can coexist with a ~93x collapse of within-cluster scale, invisible to rank-based scores but obvious in a value-based over-compression metric; (ii) recall@k and the fixed-radius band disagree systematically, in the direction the bias predicts; (iii) a membership-restricted Shepard rho resolves single-point and minority-population questions that many-pair statistics cannot -- questions on which even DREAMS, a recent local-plus-global hybrid, fails silently. A supplementary out-of-sample (addplot) test asks whether a never-seen anomaly lands outside the normal region and whether its direction identifies its source. All metrics are computed exactly on all pairwise distances, independently of any method's internals, and every number is reproducible offline: the closed-source method's output coordinates (not its algorithm) are committed to the artifact.
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