18 problems

Let $f_n(x)$ be the density of $\sqrt{n}|S_{ij}|$, let $\bar{F}_n(x)=\int_x^{\sqrt{n}}f_n(x)$, and let $G_{n,p}(x)$ be the cumulative distribution function of the maximum of an ind…

Consider the Gaussian linear regression model with responses $\mathbf{y}\in\mathbb{R}^N$, design matrix $\mathbf{x}\in\mathbb{R}^{N\times p}$, coefficient vector…

Let $\ell_{\alpha}(u)=\alpha u$ for $u\geq 0$ and $(\alpha-1)u$ for $u<0$ be the pinball loss, let $\hat{\eta}_{n+1}$ be the dual quantile-regression score, and let…

Let $(X_i,Y_i)_{i=1}^{n+1}$ follow the data-generating process described in Section 2, let $\hat{\beta}$ be the LASSO estimator fit on all $n+1$ observations, and let…

Consider the logistic regression model with intercept … where $\theta_0\in\mathbb{R}$, the covariates are independent realizations…

Let $n,d$ be positive parameters, let $q\in[1,\infty]$, and consider testing … against … Assume $dq^3=\omega(n^3)$. The signed triangle test is the signed subgraph-count test based…

Let $n,d$ be positive parameters, let $q\in[1,\infty]$, and consider testing … against … Here the signed four-cycle test is the signed subgraph-count test based on the four-cycle…

Spectrum-aware debiasing conjecture. Under suitable conditions, there is a unique solution $\widehat{\mathsf{adj}}$ of this equation and

Let $\Upsilon_{\gamma,\beta}(y)$ be the shrinker defined in equation (Upsilon), and let $\mathsf{z}^{+}_{\gamma,\beta}$ and $\mathsf{z}^{-}_{\gamma,\beta}$ be the associated spectr…

Error-order conjecture. The expected coordinatewise error satisfies

Fix $k\in\mathbb{N}$ and $C>0$. Suppose that the data-generating process satisfies … where the innovations are independent and identically distributed as $\mathcal{N}(0,\Lambda)$,…

Let $(\bX,y) \in \mathbb{R}^{n \times d} \times \mathbb{R}^n$ be an isotropic Gaussian sample, and let SVP denote support vector proliferation, namely coincidence between the solut…

Let $t: \mathbb{N} \to \mathbb{N}$. Consider a “natural” high-dimensional testing problem specified by distributions $\mathbb{P}_N$ and $\mathbb{Q}_N$ on an observation space…

Fix $\pi\in\mathcal P_2(\mathbb R)$, $\delta\in(0,\infty)$, and $\sigma>0$. Consider sequences of dimensions, Gaussian design matrices, coefficient vectors, and noise vectors satis…

Phase-transition conjecture. If $A(\beta_i)\leq 0$, then

Let $\Gamma$ be the set of parameter pairs $(\alpha,\beta)$ for which the replica-symmetric formula vanishes, and let $\Gamma'$ be its interior. For the likelihood ratio…

Let $n,d\to\infty$ with $n/d\to c>0$ and $n_\ell/n\to p_\ell$ for $\ell\in\{1,\dots,L\}$. Let $A$ and $\beta_i$ be the quantities defined in the paper, and let $\hat{u}_i$ be the…

Let $LR_{0,p,T}$ be Johansen's likelihood-ratio statistic for testing no cointegrating relationships, with $p,T\to\infty$ and $p/T\to c<1/2$ in the simultaneous regime. Scaled LR l…