{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:21:51Z","timestamp":1760059311465,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11671213"],"award-info":[{"award-number":["11671213"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We study the stability of Weak Rescaled Pure Greedy Algorithms for convex optimization, WRPGA(co), in general Banach spaces. We obtain the convergence rates of WRPGA(co) with noise and errors under a weaker assumption for the modulus of smoothness of the objective function. The results show that the rate is almost the same as that of WRPGA(co) without noise and errors, which is optimal and independent of the spatial dimension. This makes WRPGA(co) more practically applicable and scalable for high-dimensional data. Furthermore, we apply WRPGA(co) with errors to the problem of m-term approximation and derive the optimal convergence rate. This indicates the flexibility of WRPGA(co) and its wide utility across machine learning and signal processing. Our numerical experiments verify the stability of WRPGA(co). Thus, WRPGA(co) is a desirable choice for practical implementation.<\/jats:p>","DOI":"10.3390\/axioms14060446","type":"journal-article","created":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T06:11:08Z","timestamp":1749190268000},"page":"446","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stability of Weak Rescaled Pure Greedy Algorithms"],"prefix":"10.3390","volume":"14","author":[{"given":"Wan","family":"Li","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Man","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peixin","family":"Ye","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenhui","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/BF02124742","article-title":"Some remarks on greedy algorithms","volume":"5","author":"DeVore","year":"1996","journal-title":"Adv. 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