46 problems

Let ${\cal H}$ be a class of base classifiers and let ${\rm conv}({\cal H})$ denote its convex hull. Consider classifiers obtained in consecutive rounds of boosting, and let…

Combinatorial-dimension conjecture. For every uniformly bounded class $F$,

Countability necessity conjecture. Neither the countability of $I$ nor the countability of the languages can be dispensed with without additional hypotheses.

Let $F$ be the distribution function of the transformed observations, let $\hat F_n$ be the empirical distribution function based on the rank-based pseudo-observations, and define…

Let $\mathrm{\oldXi}$ be a spatio-temporal point process satisfying Assumption … is sharp, with errors of order … This conjecture concerns the effect of temporal dependence on empi…

Let $\mathcal{X} \subseteq \mathbb{R}^d$, let $\nu$ have a density, and let $P$ and $Q$ be measures on $\mathcal{X}$ with densities bounded away from zero, satisfying…

Let $P$ be the uniform distribution on the unit square $[0,1]^2$, and let $P_n$ be the empirical distribution based on $n$ observations from $P$. Write $\mathcal{T}_2(P_n,P)$ for t…

Norm-comparison conjecture. One has

Let $F$ be a class of functions, let $D$ denote the unit ball of $L_2(\mu)$, and let $r_{\mathbb{Q}}(\kappa)$ and $r_{\mathbb{M}}(\kappa)$ be the Gaussian fixed points defined by ……

Let … still holds under reasonable assumptions, since the leakage is limited. The paper notes that this does not follow from standard cross-fitting arguments for the modified proce…

Asymptotic equicontinuity conjecture. Similar arguments could also be used to verify $$ for the sequential process $\mathbb Z_n$.

Let $k\geq1$, and consider probability measures on $\{0,1\}^{\mathbb{N}}$ together with their coordinate moments of order $k$. Finite-moment conjecture. For every $k\geq1$, similar…

Let $\mu$ be a probability measure on $\{0,1\}^{\mathbb{N}}$, let $X\sim\mu$, and let $\Delta_n(\mu)$ be the expected uniform absolute deviation of the associated centered, normali…

Let $\mu$ be a probability measure on $\{0,1\}^{\mathbb{N}}$, let $X\sim\mu$, and let $\Delta_n(\mu)$ denote the expected supremum norm of the centered empirical mean. Pairwise-cor…

Let $b0,1d^d$ be the Boolean cube, let $bmu$ be a probability distribution on it with mean $p$, and let $X^{(1)},dots,X^{(n)}$ be an iid sample from $bmu$. Define the empirical mea…

Talagrand's small-cover conjecture. The result proved for $T\subset\{0,1\}^d$ should remain true for every $T\subset\mathbb{R}^d_+$.

Let $Z_1,Z_2,\ldots,Z_n$ be independent and identically distributed random variables, with each $Z_i$ representing a cluster of $m>1$ observations, and suppose that $c(i)=1$ for al…

The set $\mathcal{K}_{q,c}$ has Gaussian width $O\big(s\log\frac{n}{s}\big)$ but does not admit a low covering dimension. Conjecture. More advanced concentration inequalities or ge…

Let $\mathcal{U}$ be the class of sets … and let $\mu_m$ and $\mu$ denote the empirical and true distributions, respectively. Theorem 1 gives the uniform estimate … with high proba…

Log-factor removal conjecture. The factor $\log n$ multiplying $T(\mu)$ can be removed, yielding the improved asymptotic rate

Let $c$ be a cost function and let $\hatT_{c,n}$ and $T_c(\mu,\nu)$ denote the empirical and population optimal transport quantities considered in the paper. The preceding examples…

Canonical-gradient class conjecture. This inclusion should automatically follow from $\mathcal{Q}\subset\mathcal{F}_{1,M,L}$. The conjecture would likely be established using stand…

Giné–Véber conjecture. The random variable $\sqrt{n}f$ has sub-Gaussian tails with variance $1-n/N$.

Consider empirical process results whose analysis separates the size of the function class from the stochastic properties of the underlying process. Extension conjecture. The theor…

Let $s$ be the exponent appearing in the functional-dependence and entropy conditions for the empirical process theory developed in the paper. Truncation-exponent conjecture. The q…