{"id":{"repo_id":"duke","oai_identifier":"oai:dukespace.lib.duke.edu:10161/3863"},"canonical_url":"https://search.dev.ndltd.org/etd/duke/oai:dukespace.lib.duke.edu:10161/3863","repository":{"repo_id":"duke","name":"Duke University","base_url":"https://dukespace.lib.duke.edu/server/oai/request"},"display":{"title":"Estimating the Intrinsic Dimension of High-Dimensional Data Sets: A Multiscale, Geometric Approach","abstract":"<p>This work deals with the problem of estimating the intrinsic dimension of noisy, high-dimensional point clouds. A general class of sets which are locally well-approximated by <italic>k</italic> dimensional planes but which are embedded in a <italic>D</italic>>><italic>k</italic> dimensional Euclidean space are considered. Assuming one has samples from such a set, possibly corrupted by high-dimensional noise, if the data is linear the dimension can be recovered using PCA. However, when the data is non-linear, PCA fails, overestimating the intrinsic dimension. A multiscale version of PCA is thus introduced which is robust to small sample size, noise, and non-linearities in the data.</p>","abstract_html":"&lt;p&gt;This work deals with the problem of estimating the intrinsic dimension of noisy, high-dimensional point clouds. A general class of sets which are locally well-approximated by &lt;italic&gt;k&lt;/italic&gt; dimensional planes but which are embedded in a &lt;italic&gt;D&lt;/italic&gt;&gt;&gt;&lt;italic&gt;k&lt;/italic&gt; dimensional Euclidean space are considered. Assuming one has samples from such a set, possibly corrupted by high-dimensional noise, if the data is linear the dimension can be recovered using PCA. However, when the data is non-linear, PCA fails, overestimating the intrinsic dimension. A multiscale version of PCA is thus introduced which is robust to small sample size, noise, and non-linearities in the data.&lt;/p&gt;","abstract_has_math":false,"creators":["Little, Anna Victoria"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Maggioni, Mauro"],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011","date_published":"2011","updated_at":"2026-07-24T02:07:23Z","subjects":["Applied Mathematics","dimension estimation","geometric measure theory","multiscale analysis","point cloud data"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10161/3863","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Maggioni, Mauro"]},{"key":"dc:creator","label":"Author","values":["Little, Anna Victoria"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2011-05-20T19:35:35Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2011-11-15T05:30:16Z"]},{"key":"dc:date.issued","label":"Date","values":["2011"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Applied Mathematics","dimension estimation","geometric measure theory","multiscale analysis","point cloud data"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10161/3863"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["<p>This work deals with the problem of estimating the intrinsic dimension of noisy, high-dimensional point clouds. A general class of sets which are locally well-approximated by <italic>k</italic> dimensional planes but which are embedded in a <italic>D</italic>>><italic>k</italic> dimensional Euclidean space are considered. Assuming one has samples from such a set, possibly corrupted by high-dimensional noise, if the data is linear the dimension can be recovered using PCA. However, when the data is non-linear, PCA fails, overestimating the intrinsic dimension. A multiscale version of PCA is thus introduced which is robust to small sample size, noise, and non-linearities in the data.</p>"]},{"key":"dc:title","label":"Title","values":["Estimating the Intrinsic Dimension of High-Dimensional Data Sets: A Multiscale, Geometric Approach"]}]}],"canonical_facts":{"dc:contributor.advisor":["Maggioni, Mauro"],"dc:creator":["Little, Anna Victoria"],"dc:date.accessioned":["2011-05-20T19:35:35Z"],"dc:date.available":["2011-11-15T05:30:16Z"],"dc:date.issued":["2011"],"dc:description.abstract":["<p>This work deals with the problem of estimating the intrinsic dimension of noisy, high-dimensional point clouds. A general class of sets which are locally well-approximated by <italic>k</italic> dimensional planes but which are embedded in a <italic>D</italic>>><italic>k</italic> dimensional Euclidean space are considered. Assuming one has samples from such a set, possibly corrupted by high-dimensional noise, if the data is linear the dimension can be recovered using PCA. However, when the data is non-linear, PCA fails, overestimating the intrinsic dimension. A multiscale version of PCA is thus introduced which is robust to small sample size, noise, and non-linearities in the data.</p>"],"dc:identifier.uri":["https://hdl.handle.net/10161/3863"],"dc:subject":["Applied Mathematics","dimension estimation","geometric measure theory","multiscale analysis","point cloud data"],"dc:title":["Estimating the Intrinsic Dimension of High-Dimensional Data Sets: A Multiscale, Geometric Approach"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T02:07:23Z"}