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6 exercise(s) shown, 0 hidden
BBy Bot
Jun 01'24

Replicate the results of Figure and Table by running the corresponding experiments.

BBy Bot
Jun 01'24

Let $X$, $Y\sim\mathcal{N}(0,1,\mathbb{R}^d)$. Show the following.

• $\forall\:d\geqslant1\colon\E(\|X-Y\|-\sqrt{2d})\leqslant1/\sqrt{2d}$.
• $\forall\:d\geqslant1\colon\V(\|X-Y\|)\leqslant 3$.

Hint: Check firstly $\V((X_i-Y_i)^2)=3$ by establishing that $X_i-Y_i\sim\mathcal{N}(0,2,\mathbb{R})$ and by using a suitable formula for computing the fourth moment. Conclude then that $\V(\|X-Y\|^2)\leqslant3d$. Adapt finally the arguments we gave above for $\E(\|X\|-\sqrt{d})$ and $\V(\|X\|)$.

BBy Bot
Jun 01'24

Replicate the results of Table. Make additionally a plot of the distribution of mutual distances (this should give a picture similar to Figure).

BBy Bot
Jun 01'24

Replicate the results of Table. Let your code also compute the averages and variances of scalar products of the normalized vectors, i.e., $\langle{}x/\|x\|,y/\|y\|\rangle{}$.

BBy Bot
Jun 01'24

Compute in a simulation norm, distance and scalar product of points that are drawn from the hypercube $H_d$ (coordinate-wise) uniformly, i.e., $x=(x_1,\dots,x_d)$ is drawn such that $x_i\sim\mathcal{U}([-1,1])$ for $i=1,\dots,d$. Make plots and tables similar to Figure and [[#TAB-1 |Table\,--\,]]. Compare the experimental data with our theoretical results above.

BBy Bot
Jun 01'24

If you enjoy horror movies, then watch 2002's Hypercube.