Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we show that the behavior of the constructed function can be easily controlled by assumptions on the original convex function. We provide sparsity guarantees for the general family of functions via the proximity operator. Results related to the Fenchel Conjugate and Łojasiewicz exponent of these functions are also provided. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and linear combinations of the two. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases. We explore the effect of these functions on the penalized least squares problem and discuss several algorithms for solving this problem which rely on the particular structure of our functions. We then apply these methods to the total variation denoising problem from signal processing.
","abstract_html":"<p>Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we show that the behavior of the constructed function can be easily controlled by assumptions on the original convex function. We provide sparsity guarantees for the general family of functions via the proximity operator. Results related to the Fenchel Conjugate and Łojasiewicz exponent of these functions are also provided. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and linear combinations of the two. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases. We explore the effect of these functions on the penalized least squares problem and discuss several algorithms for solving this problem which rely on the particular structure of our functions. We then apply these methods to the total variation denoising problem from signal processing.</p>","abstract_has_math":false,"creators":["Tripp, Erin"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Lixin Shen"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-06-28T07:00:00Z","date_published":"2019-06-28T07:00:00Z","updated_at":"2026-07-24T04:55:50Z","subjects":["Convex Analysis","Moreau Envelope","Optimization","Sparse Optimization","Thresholding Operator","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/1094","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Lixin Shen"]},{"key":"dc:creator","label":"Author","values":["Tripp, Erin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Convex Analysis","Moreau Envelope","Optimization","Sparse Optimization","Thresholding Operator","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://surface.syr.edu/etd/1094"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we show that the behavior of the constructed function can be easily controlled by assumptions on the original convex function. We provide sparsity guarantees for the general family of functions via the proximity operator. Results related to the Fenchel Conjugate and Łojasiewicz exponent of these functions are also provided. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and linear combinations of the two. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases. We explore the effect of these functions on the penalized least squares problem and discuss several algorithms for solving this problem which rely on the particular structure of our functions. We then apply these methods to the total variation denoising problem from signal processing.
"]},{"key":"dc:title","label":"Title","values":["Structured Sparsity Promoting Functions: Theory and Applications"]}]}],"canonical_facts":{"dc:contributor":["Lixin Shen"],"dc:creator":["Tripp, Erin"],"dc:description.abstract":["Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we show that the behavior of the constructed function can be easily controlled by assumptions on the original convex function. We provide sparsity guarantees for the general family of functions via the proximity operator. Results related to the Fenchel Conjugate and Łojasiewicz exponent of these functions are also provided. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and linear combinations of the two. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases. We explore the effect of these functions on the penalized least squares problem and discuss several algorithms for solving this problem which rely on the particular structure of our functions. We then apply these methods to the total variation denoising problem from signal processing.
"],"dc:identifier":["https://surface.syr.edu/etd/1094"],"dc:subject":["Convex Analysis","Moreau Envelope","Optimization","Sparse Optimization","Thresholding Operator","Physical Sciences and Mathematics"],"dc:title":["Structured Sparsity Promoting Functions: Theory and Applications"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:55:50Z"}