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Let $d$ and $N$ be two large comparable integers, for example assume $$ N,d \to \infty, \quad d/N \to \gamma \in (0,\infty). $$ Let $w_1,\dotsc,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \ …

Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a rando …

Let $g:\mathbb R \to \mathbb R $ be a continuous function which is "sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and "sufficiently integrable" (e.g integrable w.r.t $N(0,1)$). Let $d'$ an …

Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\S …

Define the piecewise-linear function $\psi(t):=\max(t,0)$ for all $t \in \mathbb R$. Let $d,n,k \to \infty$ at the same rate (i.e $n \asymp k \asymp d$). Let $y_1,\ldots,y_n \in \{-1,1\}$ uniformly …

Let $p=(p(1),\ldots,p(N))$ be a discrete distribution on $[N]:=\{1,2,\ldots,N\}$ with full support (i.e all the $p(i)$'s are strictly positive and sum to $1$). Let $i_1,i_2,\ldots,i_T$ be an iid sampl …

Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R …

Let $n,d,s$ be positive integers with $s \le d$, and let $B_0(d,s)$ be the set of all (real) $d$-dimensional vectors with at most $s$ nonzero components. Given an $n \times d$ matrix $X$ with rows $x_ …

Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R …

Let $P$ be a probability distribution on $\mathbb R^d \times \mathbb R$, and let $(x_1,y_1), \ldots, (x_n,y_n)$ be an iid sample of size $n$ from $P$. Fix $\epsilon,t\gt 0$. For any unit-vector $w \in …

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$, $A \subseteq \mathbb R^n$ be Borel and $c \in \mathbb R^n$. Define the translate $A_c := c + A := \{c+a \mid a \in A\} = \{x \in \math …

Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, an …

Let $N$ be a large positive integer and let $[N] := \{1,2,\ldots,N\}$. For any $k$, let $K_{N,k}$ denote the collection of $k$-element subsets of $[N]$. Let $x=(x_1,\ldots,x_N)$ be a uniformly random …

Let $k$ and $N_1$ be positive integers and set $N=kN_1$. Partition $[N] := \{1,2,\ldots,N\}$ $k$ disjoint from $G_1,\ldots,G_k$ of each of size $N_1$, and let $\mathcal T(k,N_1)$ be a transversal of t …

Let $X=(x_1,\ldots,x_p)$ be an $p \times n$ random matrix with iid entries from $\{\pm 1\}$, distributed so that $\mathbb P(x_{ij} = 1) \equiv 1/2$, where $x_i=(x_{i1},\ldots,x_{in})$. Let $y$ be a ra …