16 problems

Replica prediction for the single-index global maximum. Under this double limit,

Let $d$ be the ambient dimension, let $k$ be the sparsity level satisfying $d^{\iota}<k<o(d)$ for some small $\iota>0$, and let $s^\star=1$. Consider the sparse warm-up algorithm w…

Consider the single-index model with fixed latent dimension $k=1$, sample-to-dimension ratio $\frac{n}{d}\to\delta$, and an even link function $h$. Let weak recovery mean estimatin…

Let $\mathcal{L}_n(\boldsymbol{\theta})$ be the loss function for the single-index model, let $\delta$ denote the aspect ratio, and let $\mathcal{T}_s(\cdot)$ be the preprocessing…

Let $d$ be the ambient dimension, let $k^\star$ denote the information exponent, and consider minibatch stochastic gradient descent for the empirical loss. Minibatch-SGD conjecture…

Let $d$ be the ambient dimension, let $k^\star$ denote the information exponent, and let … be the empirical loss. Abbe–Sandon's empirical-loss conjecture. Gradient descent on $L_n$…

Let $d$ be the ambient dimension, let $k^\star$ denote the information exponent of the learning problem, and let $L_n(w):= \frac{1}{n}\sum_{i=1}^n L(w;x_i;y_i)$ be the empirical lo…

Let $a_* = \bm{\hat\beta}^T\bm\Sigma\bm w$ be the random correlation parameter, and let $\bar a$ be its deterministic limit in probability obtained by solving the nonlinear system…

Let $F(x)=f(\langle v,x\rangle)$ be a single-index regression function on $\mathbb{R}^d$, where $v\in\mathbb{R}^d$ is an unknown unit index vector and $f:\mathbb{R}\to\mathbb{R}$ i…

Equivalence conjecture. Both estimators have the same asymptotic properties.

Let $X$ be a covariate vector whose components may be continuous or discrete, and consider the bundled least-squares estimator in the monotone single index model. Mixed-design conv…

Let the sliced stability condition be the regularity assumption on the inverse regression curve described above, and let the conditions proposed by Hsing et al. be imposed, with th…

The partially linear single-index longitudinal data model is estimated using the SGEE-based procedure and compared with the PULS estimators in terms of asymptotic efficiency. Effic…

Minimax-rate conjecture. The presented lower bound is correct, and consequently the upper-bound result should be improved. The critical case $(2\beta+1)p=r$ is the only case in whi…

In a single-index regression, the linearity condition is the condition under which the conditional mean of the covariates given the index is linear in the index, and an adaptive es…

Single-index distribution conjecture. The results of the paper should continue to hold under this assumption, without specifying the conditional distribution beyond its dependence…