31 problems

Adaptive-rate conjecture. A result similar to the established bound

Let $f_0$ be an $b1$-Hölder continuous density, with $b1\in(0,1]$, and consider the Bayes histogram estimator and its convergence rate from Theorem. Sharpness conjecture. The rate…

Let $X_1,\ldots,X_n$ be i.i.d. random variables with unknown density $s$ with respect to a measure $\nu$ on $(E,\mathcal{E})$. Let $h$ denote the Hellinger distance between densiti…

Sugiyama et al.'s tuning conjecture. Optimal estimation of the density difference may be difficult when the kernel order $\gamma$ is unknown.

Ye et al.'s rate conjecture. A better rate than $\mathcal{O}(h+1/\sqrt{t}+1/ht)$ should not be attainable even when the mixing distribution has higher-order smoothness.

Universal testing-region conjecture. For any set, the testing region is always of the form

Let $m$ be a density model indexed by $\theta\in\Theta$, with projections $Q_\theta:m\to m_\theta$ satisfying the regularity assumptions described above. Let $k$ denote the number…

The estimated density flow map $t\mapsto \widehat R_t$ is compared with the ground-truth map $t\mapsto R_t^\ast$ through the integrated squared Hellinger distance … The preceding r…

Let $M$ be a Riemannian manifold under Assumptions M, let $X$ be a random point on $M$ with density function $f$ satisfying Assumptions L, and let $r_0(\lambda)$ be the population…

Let $M$ be a Riemannian manifold under Assumptions M, and let $f:M\rightarrow\mathbb{R}$ be a density function satisfying the conditions of Proposition: for some…

Let the data domain be a finite metric space, and consider private density estimation under Wasserstein distance with instance-optimal guarantees. Logarithmic-factor necessity conj…

Combined margin-density conjecture. Under this condition, the combined margin and density condition holds.

Sharp upper-bound conjecture. The lower bound may be loose, while the upper bound may be sharp. The bounds coincide for $\ep\leq n^{-(s+2)/(2s+1)}$ and otherwise leave a gap whose…

Smoother-representation conjecture. In most cases, there exists a different compositional representation of $f$ in which the functions $\psi_I$ are $\beta$-smooth.

Low-density Jacobian conjecture. Separatrices in the pushforward density follow from the intrinsic difficulty of capturing the correct slope, or Jacobian determinant, of $S$ in reg…

The paper considers posterior contraction rates in statistical models whose parameter spaces may be equipped with Wasserstein structure, and distinguishes parameter spaces with a l…

Density forecasts are probability density functions used to predict future random variables, and pdf fusion combines density forecasts from different sources. The emergence conject…

Compact-support conjecture. The condition that the density function of interest is supported on a compact set is critical for achieving optimality under both contamination and loca…

Natural density proxy conjecture. The natural density version of $\widehat{f}_{\mu}(\cdot)$ should serve as a good proxy for $\widetilde{f}_{\mu}^{M'}(\cdot)$ and lead to rates of…

Consider general cross-validation methods beyond the particular incomplete $V$-fold procedure studied in the paper. Their local behavior may be described in terms of the rescaled e…

Let $\hat{k}_T$ be the model selected by simple validation, let $k_*$ denote the oracle model index, and let $\Delta$ be the scale factor from Definition 3.defodgs. Local order con…

Optimal linear aggregation rates conjecture. The proposed approach can attain the optimal $K/n$ rates for linear density aggregation in the observational case.

Let $\widehat{\beta}_\psi$ denote the empirical mean of a wavelet basis function, and let $\beta_\psi^p$ denote its wavelet mean under the density $p$. Consider density estimators…

Consider the distribution learned by the paper's iterative boosting procedure for density estimation, whose full characterization is derived in the paper. Distribution-characteriza…

Let $c_t$ be a neural classifier used in an iterative boosting procedure for density estimation, and consider the negative log-likelihood as the procedure progresses. Early-stoppin…