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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 …

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x …

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 …

Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a positi …

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 $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid r …

Let $X=(X_1,\ldots,X_d)$ be a standard normal random vector in $\mathbb R^d$, let $m:\mathbb R^d \to \mathbb R$ be a function, and let $E=E_m$ denote the expectation operator conditioned on $m(X) > 0$ …

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 $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that $$ d/n_k \to \phi_k \in (0,\infty). $$ Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent rando …

Let $p \ge 2$ be a positive integer, and let $Q \in \mathcal P(\mathbb Z_p)$ be a probability distribution on $\mathbb Z_p$. Question. What are necessary and sufficient conditions on $Q$ to ensure th …