{"id":166,"date":"2008-03-31T08:26:38","date_gmt":"2008-03-31T08:26:38","guid":{"rendered":"http:\/\/ahay.org\/blog\/?p=166"},"modified":"2015-08-04T23:51:42","modified_gmt":"2015-08-04T23:51:42","slug":"new-insights-into-one-norm-solvers-from-the-pareto-curve","status":"publish","type":"post","link":"https:\/\/ahay.org\/blog\/2008\/03\/31\/new-insights-into-one-norm-solvers-from-the-pareto-curve\/","title":{"rendered":"New insights into one-norm solvers from the Pareto curve"},"content":{"rendered":"<p>A new paper has been added to the <a href=\"\/wiki\/Reproducible_Documents\">collection of reproducible papers<\/a>:<\/p>\n<ul>\n<li><a href=\"\/RSF\/book\/slim\/geo2008NewInsightsPareto\/paper_html\/\">New insights into one-norm solvers from the Pareto curve<\/a><\/li>\n<\/ul>\n<p><strong>Abstract:<\/strong><br \/>\nGeophysical inverse problems typically involve a trade off between data misfit and some prior. Pareto curves trace the optimal trade off between these two competing aims. These curves are commonly used in problems with two-norm priors where they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one norm, Pareto curves remain relatively unexplored. We show how these curves lead to new insights into one-norm regularization. First, we confirm the theoretical properties of smoothness and convexity of these curves from a stylized and a geophysical example. Second, we exploit these crucial properties to approximate the Pareto curve for a large-scale problem. Third, we show how Pareto curves provide an objective criterion to gauge how different one-norm solvers advance towards the solution.<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"\/RSF\/book\/slim\/geo2008NewInsightsPareto\/pareto\/Fig\/plot.png\" width=\"414\" height=\"272\" alt=\"\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A new paper has been added to the collection of reproducible papers: New insights into one-norm solvers from the Pareto curve Abstract: Geophysical inverse problems typically involve a trade off between data misfit and some prior. Pareto curves trace the optimal trade off between these two competing aims. These curves are commonly used in problems [&hellip;]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_import_markdown_pro_load_document_selector":0,"_import_markdown_pro_submit_text_textarea":"","activitypub_content_warning":"","activitypub_content_visibility":"","activitypub_max_image_attachments":4,"activitypub_interaction_policy_quote":"anyone","activitypub_status":"","footnotes":""},"categories":[5],"tags":[],"class_list":["post-166","post","type-post","status-publish","format-standard","hentry","category-documentation"],"_links":{"self":[{"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/posts\/166","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/comments?post=166"}],"version-history":[{"count":1,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/posts\/166\/revisions"}],"predecessor-version":[{"id":749,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/posts\/166\/revisions\/749"}],"wp:attachment":[{"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/media?parent=166"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/categories?post=166"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/tags?post=166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}