• 在每个维度上的期望边长为$E_p(r)=r^{1/p}$


• $E_{10}(0.01)=0.63,\ E_{10}(0.1)=0.8$


• 如果取1%或10%的数据做一个“本地”平均的化，需要覆盖63%或80%的数据范围


• 大幅度降低$r$也不能很好的缓解这个问题

• 1D case: sample density was 10/5=2 samples/interval
• 2D case: sample density was 10/25 = 0.4 samples/interval
• 3D case: sample density was 10/125 = 0.08 samples/interval

Get sparser and sparser

Simple classifier generalizes much better to unseen data because it did not learn specific exceptions that were only in our training data by coincidence

By using less features, the curse of dimensionality was avoided such that the classifier did not overfit the training data

Training samples that fall outside the unit circle are in the corners of the feature space and are more difficult to classify than samples near the center of the feature space.

$$V(d)=\frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}0.5^d$$

With $radius=0.5$

The volume of the hypersphere tends to zero as the dimensionality tends to infinity