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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 …

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 …

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 …

It turns out that Neyman-Pearson theory helps get a nontrivial inequality. Notations. For a p.s.d matrix $M$ of size $p$, consider the inner product on $\mathbb R^p$ defined by $\langle x,z \rangle_M …

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 …

For a subcollection $\mathcal S$ of $k$-element subsets of $[n]$, consider the random variable $Z_{\mathcal S} := \sup_{A \in \mathcal S}|X_A|$, where $X_A:=\sum_{i \in A}X_i$, and the $X_i$'s are iid …

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 …

Here is my solution without the reduction trick to $1$D gaussian. Let $U := X/\|X\|$. Since $U$ is uniformly distributed on the unit $n$-sphere, it follows that the random variable $U^Tz$ has the sam …

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 …

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 < …