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Isotropic Case The case $\Sigma=I_d$ is treated. For $z \sim N(0,I_d)$, let $D$ be the distribution of $z$ conditioned on $|z^\top u| \le \theta$. Then, the rows of $Y$ are iid from $D$. It is clear t …

Disclaimer. This is just a partial solution to the auxiliary problem (estimating $N_T$). It appears $N_T$ is related to The Coupon Collector's Problem with unequal probabilities https://en.wikipedia. …

Disclaimer. It turns out that as pointed out by user @Jason Gaitonde, the idea I presented at the end of my question eventually solves my problem with the right choise of $N_1$, namely $N_1 = C \log …

Claim. If $X$ has density, then $L(S_p^n,t) \longrightarrow 0$ in the limit $t \to 0^+$. Indeed, if $X$ has density, then so does $F(X)$, for any continuous function $F:\mathbb R^n \to \mathbb R^m$. …

For any $c \ge 0$, define $\theta(s,c)$ by \begin{eqnarray} \theta(X,s,c) := \inf_{\delta \in \mathrm{CRE}(s,c)}\dfrac{\|X\delta\|_2}{\sqrt{n}\|\delta\|_2}, \end{eqnarray} where $\mathrm{CRE}(s,c) := …

Here we bound the entire spectrum of $\Sigma'$, from below and above. This post is inspired by a comment of user @BrendanMcKay. Claim. $\lambda_\max(\Sigma') = O(m/n)$ and $\lambda_\min(\Sigma') = \O …

The following VC dimension bound was established in this answer to VC dimension of a certain derived class of binary functions, $$ \operatorname{VCdim}(H) \le 2\cdot \operatorname{VCdim}(\operatorname …

Consider the "loss function" $\ell_t:\mathbb R^2 \to \{0,1\}$ defined by $\phi_t(y,y') := 1_{yy' \le t}$, and let consider the function class on $X \times \{\pm 1\}$ given by $$ S_t(F):= \ell_t \circ …

This post solves the problem (hopefully) in case where $A$ is a closed convex set with "sufficiently smooth" boundary. Preliminaries Let $S_{n-1}$ be the unit-sphere in $\mathbb R^n$ and consider the …

I provide a complete solution for the case where $A$ is the intersection of $N = \mathcal O(\mathrm{poly}(n))$ half-spaces $H_i := \{x \in \mathbb R^n \mid x^\top w_i \le b_i\}$, where each $w_i$ is a …

It turns our the problem has a simple answer, once the easy case has been solved (see the OP). Indeed, we write $\overline{f} = f + \zeta_0(f)$, so that $\zeta_0(f) = 0$. Now, one has $$ T = \overlin …

Claim (Nonasymptotic result under smoothenss condition). Suppose $g$ is $\mathcal C^5$ at $0$ and that $d'$ and $d$ are sufficiently large with $c_1 \le n'/d \le c_2$ for some absolute constants $0 < …

Solution with added restriction that $h$ is Lipschitz continuous Below, I do some computations which seem to suggest the result is true under some additional smoothness constraints on $h$. I'm not 10 …

It turns out that the optimal value of the problem can be computed arbitrarily well using basic probability arguments. Of course, this post doesn't answer my question, since the only motivation of the …

It turns out that one can get a stronger result than demanded in the question: compute $\Delta(a,b)$ for any $a,b \in S_{d-1}$, perpendicular or not. Indeed, Claim. If $f(\rho)=a_0 + a_1 \rho + a_2 \ …