18 problems

Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be sets of integers with the same set of pairwise differences $\mathcal{D}$, and suppose that every pairwise difference in $\mathcal{D}$ is…

Let model selection mean adaptive selection of nonzero means, and let the underlying estimand be $bc$, belonging to one of the parameter spaces and considered under one of the loss…

Consider Sparse Polyak, the adaptive step-size rule studied in the paper, whose definition includes a multiplicative factor $1/5$. Sparse Polyak factor conjecture. The factor $1/5$…

ACLIME analogue conjecture. A similar result to Corollary should hold for ACLIME as well.

Let $(\lambda_n,\rho_n)$ be in the algorithmic AMP phase-transition regime, with $\lambda_n=w\rho_n^{-2}$ for a fixed constant $w\in\mathbb{R}_{\ge 0}$, and let AMP be run for…

Given samples from an equal mixture of $\mathcal N(\mu_1,I_d)$ and $\mathcal N(\mu_2,I_d)$, suppose that $\mu_1$ and $\mu_2$ have bounded $\ell_2$ norms and that $\mu_1-\mu_2$ is…

Let the risk function and empirical risk arise from a sparse, possibly non-convex M-estimation problem, with sparsity constraint represented by an $_ell_0$ restriction. Uniform con…

In the spiked transport model, the two probability distributions differ only on a low-dimensional subspace called the spike. A sparse spike is a spike whose representation has only…

Consider the noisy phase retrieval model … where $\bbeta\in\mathbb{R}^d$ is $s$-sparse, $\epsilon$ is noise, and $n$ is the sample size. The information-theoretic lower bound is at…

Adaptation conjecture. Adaptation to the unknown noise variance $\varepsilon^2$ can be achieved without loss of the estimation rate in the sparse zone $s\le \sqrt{d}$. In the prese…

Incoherence-relaxation conjecture. The incoherence condition could be relaxed for the trimmed $\ell_1$ problem, similarly to the case of non-convex $\mu$-amenable regularizers such…

In the sparse normal means model, consider heavy-tailed prior distributions such as the horseshoe, the normal-exponential-gamma, and the generalized double Pareto priors. Bhattacha…

Let $\ell_0[p_n]\subset\mathbb{R}^{N_n}$ be the class of nearly black vectors with at most $p_n$ nonzero coordinates, and suppose that the indirect problem has ordered noise levels…

Consider estimation of a sparse parameter vector using the horseshoe+ estimator, with risk measured in the $\ell_2$ sense. Asymptotic minimaxity conjecture. The asymptotic minimaxi…

Sparse estimation tradeoff conjecture. If some protocol estimates the mean for any distribution $P\in\mathcal{P}_s$ with mean-squared loss $R$ and communication cost $C$, then

Consider the fixed-effects panel model … where $c_i^*$ is the time-homogeneous unobserved ability of worker $i$, and let $c^*=(c_1^*,\ldots,c_N^*)$ denote the vector of individual…

Exponential-tail extension conjecture. The conclusions of Theorem $$ should remain valid if $g$ is assumed only to have exponential tails, together with some mild conditions on its…

Sparse two-point prior lower-bound conjecture. For $\pi=\pi[\eta,\lambda_f]\in\mathfrak{m}(\eta)$,