Minimax Rates for Sparse Varying-Coefficient Models

Definitions and mathematical statement

The supplied problem asks: "Determine the minimax-optimal rates of convergence for estimation in sparse, high-dimensional varying-coefficient models."

A minimax formulation requires a statistical experiment $\{P_{\theta,n}:\theta\in\Theta_n\}$, where $n$ denotes sample size, a parameter class $\Theta_n$, an estimand $T_n(\theta)$, and a nonnegative loss $L_n$. Once these are specified, define the minimax risk by \[ R_n(\Theta_n,T_n,L_n) =\inf_{\widehat T_n}\sup_{\theta\in\Theta_n} \mathbb E_{\theta,n}\!\left[L_n\left(\widehat T_n,T_n(\theta)\right)\right], \] where the infimum is over all measurable estimators based on the observations and taking values in the specified estimation space. A minimax-optimal rate is a positive sequence $r_n$ for which there are constants $0<c\leq C<\infty$, independent of $n$ and of any dimension, sparsity, or regularity parameters over which uniformity is required, such that \[ c\,r_n\leq R_n(\Theta_n,T_n,L_n)\leq C\,r_n \] throughout the specified asymptotic regime, for all sufficiently large $n$. Any permitted dependence of the constants on fixed model parameters must also be specified.

Requested determination. Specify the intended sparse, high-dimensional varying-coefficient experiment and parameter class, and determine the corresponding minimax-optimal rate, including its dependence on sample size, dimension, sparsity, and coefficient regularity where applicable. The display above is a general framework for interpreting the request, not a statistical model supplied by the source record.

Unresolved qualifications. The record does not specify the observation equation, sampling and noise assumptions, covariate design, coefficient-function domain or boundary conditions, smoothness class, meaning of sparsity, estimation target, loss, or growth regime. Consequently it does not determine a unique minimax risk or rate. Its references to homogeneous cases and to a general exact minimax-rate question are also not accompanied by definitions of those cases or their hypotheses.

Short English statement

Determine the best possible estimation rates for sparse, high-dimensional varying-coefficient models. A precise rate requires an explicit statistical model, parameter class, estimation target, and loss; these are missing from the supplied record. Its supplied partially solved status and dated literature assessment require independent mathematical review.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002550$, source problem code catalogue entry, title "Minimax Rates for Sparse Varying-Coefficient Models." The record belongs to the Oberwolfach Reports Open Problems collection and lists proposed year 2014; no individual proposer is supplied.
• Supplied workshop provenance: Adaptive Statistical Inference (2014), Oberwolfach Reports, pp. 721--779, DOI 10.4171/owr/2014/13. https://doi.org/10.4171/owr/2014/13. The record describes these reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page or named contributor is given.
• Supplied literature reference: Klopp--Pensky (2015). https://doi.org/10.1214/15-AOS1309.
• Supplied literature reference: Klopp--Pensky preprint. https://arxiv.org/abs/1312.4087.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem partially solved and states that Klopp and Pensky established minimax lower bounds and adaptive estimators attaining them up to constants in homogeneous cases or a logarithmic factor otherwise, leaving the general exact minimax-rate question not fully settled. This is supplied dated curation derived from the OWR collection, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.