{"id":72,"date":"2016-06-01T23:07:19","date_gmt":"2016-06-01T23:07:19","guid":{"rendered":"http:\/\/www.keithdillon.com\/?p=72"},"modified":"2023-05-15T03:54:21","modified_gmt":"2023-05-15T03:54:21","slug":"imposing-uniqueness-to-achieve-sparsity","status":"publish","type":"post","link":"https:\/\/www.keithdillon.com\/index.php\/2016\/06\/01\/imposing-uniqueness-to-achieve-sparsity\/","title":{"rendered":"Imposing uniqueness to achieve sparsity"},"content":{"rendered":"<p id=\"sp0065\">In this paper we take a novel approach to the regularization of underdetermined linear systems. Typically, a prior distribution is imposed on the unknown to hopefully force a sparse solution, which often relies on uniqueness of the regularized solution (something which is typically beyond our control) to work as desired. But here we take a direct approach, by imposing the requirement that the system takes on a unique solution. Then we seek a minimal residual for which this uniqueness requirement holds.<\/p>\n<p><!--more--><\/p>\n<p>When applied to systems with non-negativity constraints or forms of regularization for which sufficient sparsity is a requirement for uniqueness, this approach necessarily gives a sparse result. The approach is based on defining a metric of distance to uniqueness for the system, and optimizing an adjustment that drives this distance to zero. We demonstrate the performance of the approach with numerical experiments.<\/p>\n<p>K. Dillon and Y.-P. Wang, \u201c<a href=\"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0165168415004363\">Imposing uniqueness to achieve sparsity<\/a>,\u201d <em>Signal Processing<\/em>, vol. 123, pp. 1\u20138, Jun. 2016. <a href=\"https:\/\/www.keithdillon.com\/papers_preprints\/Imposing Uniqueness to Achieve Sparsity.pdf\">(pdf)<\/a><\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-100\" src=\"https:\/\/www.keithdillon.com\/wp-content\/uploads\/2016\/06\/imposing_uniqueness_paper_fig2.png\" alt=\"\" width=\"770\" height=\"702\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this paper we take a novel approach to the regularization of underdetermined linear systems. Typically, a prior distribution is imposed on the unknown to hopefully force a sparse solution, which often relies on uniqueness of the regularized solution (something<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/posts\/72"}],"collection":[{"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/comments?post=72"}],"version-history":[{"count":3,"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/posts\/72\/revisions"}],"predecessor-version":[{"id":101,"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/posts\/72\/revisions\/101"}],"wp:attachment":[{"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/media?parent=72"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/categories?post=72"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.keithdillon.com\/index.php\/wp-json\/wp\/v2\/tags?post=72"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}