249 problems

The procedure uses randomly weighted sums of centered estimated vectors $\widehat{\mathbf U}_{j,t}$ to approximate the distribution of the leading term in the debiasing decompositi…

Consider the high-dimensional global testing procedures developed in the paper, including tests based on studentized $L_q$-norm statistics and their monotonically indexed U-statist…

Tight-bound conjecture. The tight upper bound for this quantity is actually $p^5/n^4$.

Bridgeless limit-interchange conjecture. In general, the same result remains true when the order of the limits is swapped, at least for bridgeless estimators for every $q\in[1,2]$…

Data-independent-prior extension conjecture. With additional effort, the Laplace approximation and the model-selection consistency results for excluding supersets and avoiding fals…

Let $\mathbf{a}(\bm{\theta})$ denote the asymptotic bias function, and let Theorem … remains true for a larger class of asymptotic bias functions. The conjecture concerns extending…

Restricted eigenvalue conjecture. For Poisson regression models, condition (III) holds with high probability under very mild conditions.

Sharper-threshold conjecture. More accurate bounds for $\lambda_1$ and $\lambda_2$ can be obtained, for example, using techniques from the cited literature.

The conjecture that Property C holds for spherically symmetric distributions. Property C, a concentration condition on the extreme singular values of transformed design matrices, h…

The rodeo estimator is studied in a triangular-array setting: the regression function $m$, design density $f$, dimension $d$, and number $r$ of relevant variables may depend on the…

Peak-location conjecture. Under —and, by the stated reduction, under —the location of the peak is exactly the same as in the isotropic case, namely when

Let $b$ be the parameter in the high-dimensional geometric matching model with dimension $d=\omega(\log n)$ and noise variance $\sigma^2=d/(b\log n)$, and let the maximum likelihoo…

Laplace detection-threshold conjecture. The parameter ranges considered in the experiments lie in the regime $n\gtrsim\sqrt{p}$, where the term $\kappa\propto p^{0.25}n^{-0.5}$ dom…

Multivariate constant-delay conjecture. This strong-signal regime corresponds to Regime 4 in the univariate setting, where the detection delay is also of constant order up to logar…

Let $T^{(D)}_{n}(\alpha,\hat j_n)$ and $Q^{(D)}_{n}(\alpha)$ denote the $L_\infty$-type and $L_2$-type sample-splitting self-normalized (SS-SN) statistics, respectively. Suppose th…

Low-degree conjecture. For many average-case high-dimensional testing problems, low-degree polynomials of degree

Let $d$ denote the ambient dimension, and consider the semi-parametric nonlinear regression setting in which the covariate-target relation is a linear dimensionality reduction foll…

The one-pass setting requires controlling the smoothness of the empirical risk at each iteration, which can impose substantially stronger dimension-dependent conditions than in bat…

Conjectured improvement. The condition could be improved to

Generalization conjecture. It may be possible to derive a generalization of the proposition to an arbitrary regularization schedule in which the term

While the analysis concerns analytically tractable overparameterized linear regression, let signal-noise dynamics denote the interaction between denoising and signal forgetting dur…

Let $p$ and $n$ denote the dimension and sample size, respectively, and let the limit referred to as the normal limit hold for the debiased estimator after correcting the likelihoo…

Consider the gradient flow dynamics in the absence of regularization, under Assumption 2, and let $\alpha_\mathrm{inter}(\kappa,\kappa^*,\Delta)$ be the interpolation threshold. Un…

Let $\alpha_\mathrm{inter}(\kappa,\kappa^*,\Delta)$ denote the interpolation threshold characterized in Proposition 1. When $\kappa\geq 1$, this threshold is the smallest value of…

Dimension-free robustness conjecture. The dimension-dependent factor in the lower bound of Theorem 2.6 is unnecessary; equivalently, the corresponding asymptotic lower-bound expres…