这是一份southampton南安普顿大学 COMP6235作业代写的成功案
问题 1.
If $\mathbf{x}$ and $\mathbf{y}$ are two independent samples from the same spherical Gaussian with standard deviation ${ }^{1} \sigma$, then
$$
|\mathbf{x}-\mathbf{y}|^{2} \approx 2(\sqrt{d} \pm c)^{2} \sigma^{2}
$$
证明 .
- If $\mathbf{x}$ and $\mathbf{y}$ are samples from different spherical Gaussians each of standard deviation $\sigma$ and means separated by distance $\delta$, then
$$
|\mathbf{x}-\mathbf{y}|^{2} \approx 2(\sqrt{d} \pm c)^{2} \sigma^{2}+\delta^{2} .
$$
PHAS0038 COURSE NOTES :
\begin{aligned}
B=A A^{T} &=\left(\sum_{i} \sigma_{i} \mathbf{u}{\mathbf{i}} \mathbf{v}{\mathbf{i}}^{T}\right)\left(\sum_{j} \sigma_{j} \mathbf{u}{\mathbf{j}} v{j}^{T}\right)^{T} \
&=\sum_{i} \sum_{j} \sigma_{i} \sigma_{j} \mathbf{u}{\mathbf{i}} \mathbf{v}{\mathbf{i}}^{T} \mathbf{v}{\mathbf{j}} \mathbf{u}{\mathbf{j}}^{T} \
&=\sum_{i} \sigma_{i}^{2} \mathbf{u}{\mathbf{i}} \mathbf{u}{\mathbf{i}}^{T}
\end{aligned}