通过距离和角度的分量校准进行可解释的本征维数估计
Interpretable intrinsic dimension estimation through componentwise calibration of distance and angle
- National Sun Yat-sen University（国立中山大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种可解释的本征维数估计方法，通过分量校准距离和角度统计量，改进DANCo并识别估计偏差来源，在多种基准上验证了有效性。
AI中文摘要：
DANCo（基于角度和范数集中度的维数估计）联合校准最近邻距离和角度统计量，在干净的本征维数（ID）基准上持续达到最先进的精度。然而，实际数据会引入邻域相对噪声和样本幅度异质性，这些因素可能扭曲这些几何信号。我们将DANCo重新表述为分量形式，保留独立的距离和角度差异曲线，以便识别和解释估计的来源。对于距离分量，我们推导了广义比率ID估计器（Gride）的通用阶比率的闭式Kullback-Leibler散度；当两个角度参数都匹配时（Full），在24个流形上，当噪声等于典型邻域间距的40%时，Gride的平均百分比误差从27.7%降至17.6%。对于角度分量，两种采样机制促使在保持浓度匹配的同时对齐平均方向（Profiled）。在生成维度为70、嵌入维度为100的高斯尺度混合上，剖面化将最小邻域距离（MiND）估计从22.8提高到66.7，而移除已知幅度后，MiND-Full恢复到71.9；因此，对照实验将Full的不足归因于幅度异质性。在CIFAR-10和ImageNet上，降低幅度的归一化使角度位置向参考值移动，并缩小了Full-Profiled差距，这是受控混合的观测对应物。在四个预训练的卷积神经网络中，Gride-Profiled、两最近邻估计器（TWO-NN）和最大似然估计器（MLE）表现出相似的上升-下降曲线，而Full-Profiled差异识别出对角度校准最敏感的层。
英文摘要：
DANCo (Dimensionality from Angle and Norm Concentration) jointly calibrates nearest-neighbor distance and angular statistics and consistently reaches state-of-the-art accuracy on clean intrinsic-dimension (ID) benchmarks. Practical data, however, introduce neighborhood-relative noise and sample-amplitude heterogeneity that can distort these geometric signals. We reformulate DANCo componentwise, retaining separate distance and angular discrepancy curves so that the source of an estimate can be identified and interpreted. For the distance component, we derive a closed-form Kullback-Leibler divergence for the generic-order ratios of the generalized ratios ID estimator (Gride); when both angular parameters are matched (Full), Gride reduces mean percentage error from $27.7\%$ to $17.6\%$ at noise equal to $40\%$ of typical neighbor spacing on 24 manifolds. For the angular component, two sampling regimes motivate aligning mean direction while retaining concentration matching (Profiled). On a Gaussian scale mixture with generating dimension 70 embedded in 100 dimensions, profiling raises the Minimum Neighbor Distance (MiND) estimate from $22.8$ to $66.7$, while removing the known amplitudes restores MiND-Full to $71.9$; the control thus attributes the Full shortfall to amplitude heterogeneity. On CIFAR-10 and ImageNet, amplitude-reducing normalizations move angular location toward the references and narrow the Full-Profiled gap, an observational counterpart to the controlled mixture. Across four pretrained convolutional neural networks, Gride-Profiled, the two-nearest-neighbor estimator (TWO-NN), and the maximum-likelihood estimator (MLE) exhibit similar rise-and-fall profiles, while Full-Profiled differences identify the layers most sensitive to angular calibration.
