English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we focus on the high-dimensional double sparse structure,where the parameter of interest simultaneously encourages group-wise sparsityand element-wise sparsity in each group. By combining the Gilbert-Varshamovbound and its variants, we develop a novel lower bound technique for the metricentropy of the parameter space, specifically tailored for the double sparsestructure over $\ell_u(\ell_q)$-balls with $u,q \in [0,1]$. We prove lowerbounds on the estimation error using an information-theoretic approach,leveraging our proposed lower bound technique and Fano's inequality. Tocomplement the lower bounds, we establish matching upper bounds through adirect analysis of constrained least-squares estimators and utilize resultsfrom empirical processes. A significant finding of our study is the discoveryof a phase transition phenomenon in the minimax rates for $u,q \in (0, 1]$.Furthermore, we extend the theoretical results to the double sparse regressionmodel and determine its minimax rate for estimation error. To tackle doublesparse linear regression, we develop the DSIHT (Double Sparse Iterative HardThresholding) algorithm, demonstrating its optimality in the minimax sense.Finally, we demonstrate the superiority of our method through numericalexperiments.

Minimax Rates for High-dimensional Double Sparse Structure over $\ell_u(\ell_q)$-balls