{"657047":{"#nid":"657047","#data":{"type":"event","title":"Ph.D. Dissertation Defense - Andrew McRae","body":[{"value":"\u003Cp\u003E\u003Cstrong\u003ETitle\u003C\/strong\u003E\u003Cem\u003E:\u0026nbsp; \u003C\/em\u003E\u003Cem\u003EStructured Statistical Estimation via Optimization\u003C\/em\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Cstrong\u003ECommittee:\u003C\/strong\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Mark Davenport, ECE, Chair, Advisor\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Justin Romberg, ECE\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Vladimir Koltchinskii, Math\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Vidya Muthukumar, ECE\u003C\/p\u003E\r\n\r\n\u003Cp\u003EDr. Arkadi Nemirovski, ISyE\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Cstrong\u003EAbstract:\u0026nbsp;\u003C\/strong\u003EIn this thesis, I show how we can exploit low-dimensional structure in high-dimensional statistics and machine learning problems via optimization. I show several settings where, with an appropriate choice of optimization algorithm, we can perform useful estimation with a complexity that scales not with the original problem dimension but with a much smaller intrinsic dimension. In the low-rank matrix completion and denoising problems, we can exploit low-rank structure to recover a large matrix from noisy observations of some or all of its entries. I prove state-of-the-art results for this problem in the case of Poisson noise and show that these results are minimax-optimal. Next, I study the problem of recovering a sparse vector from nonlinear measurements. I present a lifted matrix framework for the sparse phase retrieval and sparse PCA problems that includes a novel atomic norm regularizer. I prove that solving certain convex optimization problems in this framework yields estimators with near-optimal performance. Although we do not know how to compute these estimators efficiently and exactly, we derive a principled heuristic algorithm for sparse phase retrieval that matches existing state-of-the-art algorithms. Third, I show how we can exploit low-dimensional manifold structure in supervised learning. In a reproducing kernel Hilbert space framework, I show that smooth functions on a manifold can be estimated with a complexity scaling with the manifold dimension rather than a larger embedding space dimension. Finally, I study the interaction between high ambient dimension and a lower intrinsic dimension in the harmless interpolation phenomenon (where learned functions generalize well despite interpolating noisy data). I present a general framework for this phenomenon in linear and reproducing kernel Hilbert space settings, proving that it occurs in many situations that previous work has not covered.\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Structured Statistical Estimation via Optimization "}],"uid":"28475","created_gmt":"2022-04-06 20:47:22","changed_gmt":"2022-04-06 20:47:22","author":"Daniela Staiculescu","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2022-04-15T13:00:00-04:00","event_time_end":"2022-04-15T15:00:00-04:00","event_time_end_last":"2022-04-15T15:00:00-04:00","gmt_time_start":"2022-04-15 17:00:00","gmt_time_end":"2022-04-15 19:00:00","gmt_time_end_last":"2022-04-15 19:00:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"434381","name":"ECE Ph.D. Dissertation Defenses"}],"categories":[],"keywords":[{"id":"100811","name":"Phd Defense"},{"id":"1808","name":"graduate students"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78771","name":"Public"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}