<?xml version="1.0" encoding="utf-8"?>
<TEI xmlns="http://www.tei-c.org/ns/1.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:hal="http://hal.archives-ouvertes.fr/" xmlns:gml="http://www.opengis.net/gml/3.3/" xmlns:gmlce="http://www.opengis.net/gml/3.3/ce" version="1.1" xsi:schemaLocation="http://www.tei-c.org/ns/1.0 http://api.archives-ouvertes.fr/documents/aofr-sword.xsd">
  <teiHeader>
    <fileDesc>
      <titleStmt>
        <title>HAL TEI export of hal-00981634v3</title>
      </titleStmt>
      <publicationStmt>
        <distributor>CCSD</distributor>
        <availability status="restricted">
          <licence target="https://creativecommons.org/publicdomain/zero/1.0/">CC0 1.0 - Universal</licence>
        </availability>
        <date when="2026-05-19T17:34:53+02:00"/>
      </publicationStmt>
      <sourceDesc>
        <p part="N">HAL API Platform</p>
      </sourceDesc>
    </fileDesc>
  </teiHeader>
  <text>
    <body>
      <listBibl>
        <biblFull>
          <titleStmt>
            <title xml:lang="en">The Degrees of Freedom of Partly Smooth Regularizers</title>
            <author role="aut">
              <persName>
                <forename type="first">Samuel</forename>
                <surname>Vaiter</surname>
              </persName>
              <email type="md5">5ac6602d81d93dccbee3ff5e790f0e84</email>
              <email type="domain">math.cnrs.fr</email>
              <idno type="idhal" notation="string">samuel-vaiter</idno>
              <idno type="idhal" notation="numeric">1995</idno>
              <idno type="halauthorid" notation="string">38917-1995</idno>
              <idno type="IDREF">https://www.idref.fr/182993116</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-4077-708X</idno>
              <idno type="VIAF">https://viaf.org/viaf/313572753</idno>
              <idno type="GOOGLE SCHOLAR">https://scholar.google.fr/citations?user=HkXkm7IAAAAJ</idno>
              <affiliation ref="#struct-60"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Charles-Alban</forename>
                <surname>Deledalle</surname>
              </persName>
              <email type="md5">dda4865822409dd4312053db40da9d8d</email>
              <email type="domain">math.u-bordeaux1.fr</email>
              <idno type="idhal" notation="numeric">935353</idno>
              <idno type="halauthorid" notation="string">509128-935353</idno>
              <affiliation ref="#struct-27730"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Jalal M.</forename>
                <surname>Fadili</surname>
              </persName>
              <email type="md5">0b809ac6a6c687916595dac45770836b</email>
              <email type="domain">ensicaen.fr</email>
              <idno type="idhal" notation="string">jalal-fadili</idno>
              <idno type="idhal" notation="numeric">15510</idno>
              <idno type="halauthorid" notation="string">20367-15510</idno>
              <idno type="IDREF">https://www.idref.fr/118253999</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-8165-7578</idno>
              <affiliation ref="#struct-406734"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Gabriel</forename>
                <surname>Peyré</surname>
              </persName>
              <email type="md5">8f7426d5d9183245fe9b02d1e972a597</email>
              <email type="domain">ens.fr</email>
              <idno type="idhal" notation="string">gpeyre</idno>
              <idno type="idhal" notation="numeric">1211</idno>
              <idno type="halauthorid" notation="string">31235-1211</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-4477-0387</idno>
              <idno type="IDREF">https://www.idref.fr/077078128</idno>
              <idno type="VIAF">https://viaf.org/viaf/39616547</idno>
              <idno type="ISNI">http://isni.org/isni/000000011625330X</idno>
              <affiliation ref="#struct-60"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Charles</forename>
                <forename type="middle">H</forename>
                <surname>Dossal</surname>
              </persName>
              <email type="md5">a58547c5dd19438aa6ada606842728e0</email>
              <email type="domain">math.u-bordeaux1.fr</email>
              <idno type="idhal" notation="numeric">1136514</idno>
              <idno type="halauthorid" notation="string">1859633-1136514</idno>
              <orgName ref="#struct-116255"/>
              <affiliation ref="#struct-27730"/>
            </author>
            <editor role="depositor">
              <persName>
                <forename>Samuel</forename>
                <surname>Vaiter</surname>
              </persName>
              <email type="md5">05a4cb85e7a6e3d423b983e881fd9108</email>
              <email type="domain">cnrs.fr</email>
            </editor>
            <funder>ERC Sigma-Vision</funder>
          </titleStmt>
          <editionStmt>
            <edition n="v1">
              <date type="whenSubmitted">2014-04-22 15:37:09</date>
            </edition>
            <edition n="v2">
              <date type="whenSubmitted">2014-07-23 11:49:09</date>
            </edition>
            <edition n="v3" type="current">
              <date type="whenSubmitted">2015-10-27 13:44:28</date>
              <date type="whenWritten">2015-10-23</date>
              <date type="whenModified">2025-06-13 11:20:02</date>
              <date type="whenReleased">2015-10-27 15:26:37</date>
              <date type="whenProduced">2015-10-23</date>
              <date type="whenEndEmbargoed">2015-10-27</date>
              <ref type="file" target="https://hal.science/hal-00981634v3/document">
                <date notBefore="2015-10-27"/>
              </ref>
              <ref type="file" subtype="author" n="1" target="https://hal.science/hal-00981634v3/file/dof-partly-smooth.pdf" id="file-1221079-1299555">
                <date notBefore="2015-10-27"/>
              </ref>
              <ref type="externalLink" target="http://arxiv.org/pdf/1404.5557"/>
            </edition>
            <edition n="v4">
              <date type="whenSubmitted">2016-02-10 12:50:30</date>
            </edition>
            <respStmt>
              <resp>contributor</resp>
              <name key="165194">
                <persName>
                  <forename>Samuel</forename>
                  <surname>Vaiter</surname>
                </persName>
                <email type="md5">05a4cb85e7a6e3d423b983e881fd9108</email>
                <email type="domain">cnrs.fr</email>
              </name>
            </respStmt>
          </editionStmt>
          <publicationStmt>
            <distributor>CCSD</distributor>
            <idno type="halId">hal-00981634</idno>
            <idno type="halUri">https://hal.science/hal-00981634</idno>
            <idno type="halBibtex">vaiter:hal-00981634</idno>
            <idno type="halRefHtml">2015</idno>
            <idno type="halRef">2015</idno>
            <availability status="restricted">
              <licence target="https://about.hal.science/hal-authorisation-v1/">HAL Authorization<ref corresp="#file-1221079-1299555"/></licence>
            </availability>
          </publicationStmt>
          <seriesStmt/>
          <notesStmt/>
          <sourceDesc>
            <biblStruct>
              <analytic>
                <title xml:lang="en">The Degrees of Freedom of Partly Smooth Regularizers</title>
                <author role="aut">
                  <persName>
                    <forename type="first">Samuel</forename>
                    <surname>Vaiter</surname>
                  </persName>
                  <email type="md5">5ac6602d81d93dccbee3ff5e790f0e84</email>
                  <email type="domain">math.cnrs.fr</email>
                  <idno type="idhal" notation="string">samuel-vaiter</idno>
                  <idno type="idhal" notation="numeric">1995</idno>
                  <idno type="halauthorid" notation="string">38917-1995</idno>
                  <idno type="IDREF">https://www.idref.fr/182993116</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-4077-708X</idno>
                  <idno type="VIAF">https://viaf.org/viaf/313572753</idno>
                  <idno type="GOOGLE SCHOLAR">https://scholar.google.fr/citations?user=HkXkm7IAAAAJ</idno>
                  <affiliation ref="#struct-60"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Charles-Alban</forename>
                    <surname>Deledalle</surname>
                  </persName>
                  <email type="md5">dda4865822409dd4312053db40da9d8d</email>
                  <email type="domain">math.u-bordeaux1.fr</email>
                  <idno type="idhal" notation="numeric">935353</idno>
                  <idno type="halauthorid" notation="string">509128-935353</idno>
                  <affiliation ref="#struct-27730"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Jalal M.</forename>
                    <surname>Fadili</surname>
                  </persName>
                  <email type="md5">0b809ac6a6c687916595dac45770836b</email>
                  <email type="domain">ensicaen.fr</email>
                  <idno type="idhal" notation="string">jalal-fadili</idno>
                  <idno type="idhal" notation="numeric">15510</idno>
                  <idno type="halauthorid" notation="string">20367-15510</idno>
                  <idno type="IDREF">https://www.idref.fr/118253999</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-8165-7578</idno>
                  <affiliation ref="#struct-406734"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Gabriel</forename>
                    <surname>Peyré</surname>
                  </persName>
                  <email type="md5">8f7426d5d9183245fe9b02d1e972a597</email>
                  <email type="domain">ens.fr</email>
                  <idno type="idhal" notation="string">gpeyre</idno>
                  <idno type="idhal" notation="numeric">1211</idno>
                  <idno type="halauthorid" notation="string">31235-1211</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-4477-0387</idno>
                  <idno type="IDREF">https://www.idref.fr/077078128</idno>
                  <idno type="VIAF">https://viaf.org/viaf/39616547</idno>
                  <idno type="ISNI">http://isni.org/isni/000000011625330X</idno>
                  <affiliation ref="#struct-60"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Charles</forename>
                    <forename type="middle">H</forename>
                    <surname>Dossal</surname>
                  </persName>
                  <email type="md5">a58547c5dd19438aa6ada606842728e0</email>
                  <email type="domain">math.u-bordeaux1.fr</email>
                  <idno type="idhal" notation="numeric">1136514</idno>
                  <idno type="halauthorid" notation="string">1859633-1136514</idno>
                  <orgName ref="#struct-116255"/>
                  <affiliation ref="#struct-27730"/>
                </author>
              </analytic>
              <monogr>
                <imprint/>
              </monogr>
              <idno type="arxiv">1404.5557</idno>
            </biblStruct>
          </sourceDesc>
          <profileDesc>
            <langUsage>
              <language ident="en">English</language>
            </langUsage>
            <textClass>
              <keywords scheme="author">
                <term xml:lang="en">Degrees of freedom</term>
                <term xml:lang="en">Sparsity</term>
                <term xml:lang="en">Model selection</term>
                <term xml:lang="en">o-minimal structures</term>
                <term xml:lang="en">Semi-algebraic sets</term>
                <term xml:lang="en">Group Lasso</term>
                <term xml:lang="en">Total variation</term>
                <term xml:lang="en">Partial smoothness</term>
              </keywords>
              <classCode scheme="halDomain" n="math.math-st">Mathematics [math]/Statistics [math.ST]</classCode>
              <classCode scheme="halDomain" n="math.math-it">Mathematics [math]/Information Theory [math.IT]</classCode>
              <classCode scheme="halDomain" n="info.info-ts">Computer Science [cs]/Signal and Image Processing</classCode>
              <classCode scheme="halDomain" n="spi.signal">Engineering Sciences [physics]/Signal and Image processing</classCode>
              <classCode scheme="halDomain" n="stat.th">Statistics [stat]/Statistics Theory [stat.TH]</classCode>
              <classCode scheme="halDomain" n="info.info-it">Computer Science [cs]/Information Theory [cs.IT]</classCode>
              <classCode scheme="halTypology" n="UNDEFINED">Preprints, Working Papers, ...</classCode>
              <classCode scheme="halOldTypology" n="UNDEFINED">Preprints, Working Papers, ...</classCode>
              <classCode scheme="halTreeTypology" n="UNDEFINED">Preprints, Working Papers, ...</classCode>
            </textClass>
            <abstract xml:lang="en">
              <p>In this paper, we are concerned with regularized regression problems where the prior regularizer is a proper lower semicontinuous and convex function which is also partly smooth relative to a Riemannian submanifold. This encompasses as special cases several known penalties such as the Lasso ($\ell^1$-norm), the group Lasso ($\ell^1-\ell^2$-norm), the $\ell^\infty$-norm, and the nuclear norm. This also includes so-called analysis-type priors, i.e. composition of the previously mentioned penalties with linear operators, typical examples being the total variation or fused Lasso penalties.We study the sensitivity of any regularized minimizer to perturbations of the observations and provide its precise local parameterization.Our main sensitivity analysis result shows that the predictor moves locally stably along the same active submanifold as the observations undergo small perturbations. This local stability is a consequence of the smoothness of the regularizer when restricted to the active submanifold, which in turn plays a pivotal role to get a closed form expression for the variations of the predictor w.r.t. observations. We also show that, for a variety of regularizers, including polyhedral ones or the group Lasso and its analysis counterpart, this divergence formula holds Lebesgue almost everywhere.When the perturbation is random (with an appropriate continuous distribution), this allows us to derive an unbiased estimator of the degrees of freedom and of the risk of the estimator prediction.Our results hold true without requiring the design matrix to be full column rank.They generalize those already known in the literature such as the Lasso problem, the general Lasso problem (analysis $\ell^1$-penalty), or the group Lasso where existing results for the latter assume that the design is full column rank.</p>
            </abstract>
          </profileDesc>
        </biblFull>
      </listBibl>
    </body>
    <back>
      <listOrg type="structures">
        <org type="laboratory" xml:id="struct-60" status="VALID">
          <idno type="IdRef">088064239</idno>
          <idno type="ISNI">0000 0004 0645 4046</idno>
          <idno type="RNSR">199712592E</idno>
          <idno type="ROR">https://ror.org/03s0gj002</idno>
          <idno type="Wikidata">Q2913547</idno>
          <orgName>CEntre de REcherches en MAthématiques de la DEcision</orgName>
          <orgName type="acronym">CEREMADE</orgName>
          <date type="start">1971-01-01</date>
          <desc>
            <address>
              <addrLine>Place du Maréchal de Lattre de Tassigny 75775 - Paris Cedex 16</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.ceremade.dauphine.fr/index.html</ref>
          </desc>
          <listRelation>
            <relation active="#struct-300302" type="direct"/>
            <relation active="#struct-564132" type="indirect"/>
            <relation name="UMR7534 / URA749" active="#struct-441569" type="direct"/>
          </listRelation>
        </org>
        <org type="laboratory" xml:id="struct-27730" status="VALID">
          <idno type="IdRef">155051342</idno>
          <idno type="ISNI">0000000123024783</idno>
          <idno type="RNSR">200711916B</idno>
          <idno type="ROR">https://ror.org/05m3r1b84</idno>
          <orgName>Institut de Mathématiques de Bordeaux</orgName>
          <orgName type="acronym">IMB</orgName>
          <date type="start">2003-01-01</date>
          <desc>
            <address>
              <addrLine>351 cours de la Libération 33405 TALENCE CEDEX</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.math.u-bordeaux.fr/IMB/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-259761" type="direct"/>
            <relation active="#struct-409745" type="direct"/>
            <relation name="UMR5251" active="#struct-441569" type="direct"/>
          </listRelation>
        </org>
        <org type="researchteam" xml:id="struct-406734" status="VALID">
          <orgName>Equipe Image - Laboratoire GREYC - UMR6072</orgName>
          <desc>
            <address>
              <country key="FR"/>
            </address>
          </desc>
          <listRelation>
            <relation active="#struct-150" type="direct"/>
            <relation active="#struct-7127" type="indirect"/>
            <relation active="#struct-455934" type="indirect"/>
            <relation active="#struct-246955" type="indirect"/>
            <relation name="UMR6072" active="#struct-441569" type="indirect"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-300302" status="VALID">
          <idno type="IdRef">027787109</idno>
          <idno type="ISNI">0000000120977052</idno>
          <idno type="ROR">https://ror.org/052bz7812</idno>
          <idno type="Wikidata">Q1546437</idno>
          <orgName>Université Paris Dauphine-PSL</orgName>
          <desc>
            <address>
              <addrLine>Place du Maréchal de Lattre de Tassigny75775 PARIS Cedex 16</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://dauphine.psl.eu/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-564132" type="direct"/>
          </listRelation>
        </org>
        <org type="regroupinstitution" xml:id="struct-564132" status="VALID">
          <idno type="IdRef">241597595</idno>
          <idno type="ISNI">0000 0004 1784 3645</idno>
          <idno type="ROR">https://ror.org/013cjyk83</idno>
          <orgName>Université Paris Sciences et Lettres</orgName>
          <orgName type="acronym">PSL</orgName>
          <desc>
            <address>
              <addrLine>60 rue Mazarine 75006 Paris</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.psl.eu/</ref>
          </desc>
        </org>
        <org type="regroupinstitution" xml:id="struct-441569" status="VALID">
          <idno type="IdRef">02636817X</idno>
          <idno type="ISNI">0000000122597504</idno>
          <idno type="ROR">https://ror.org/02feahw73</idno>
          <orgName>Centre National de la Recherche Scientifique</orgName>
          <orgName type="acronym">CNRS</orgName>
          <date type="start">1939-10-19</date>
          <desc>
            <address>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.cnrs.fr/</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-259761" status="VALID">
          <idno type="IdRef">175206562</idno>
          <idno type="ROR">https://ror.org/057qpr032</idno>
          <orgName>Université de Bordeaux</orgName>
          <orgName type="acronym">UB</orgName>
          <desc>
            <address>
              <addrLine>35, place Pey Berland - 33076 Bordeaux</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.u-bordeaux.fr/</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-409745" status="VALID">
          <orgName>Institut Polytechnique de Bordeaux</orgName>
          <orgName type="acronym">Bordeaux INP</orgName>
          <desc>
            <address>
              <addrLine>Avenue des facultés33405 Talence</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.bordeaux-inp.fr/fr</ref>
          </desc>
        </org>
        <org type="laboratory" xml:id="struct-150" status="VALID">
          <idno type="IdRef">035827335</idno>
          <idno type="ISNI">0000 0000 9466 2590</idno>
          <idno type="RNSR">201722568L</idno>
          <idno type="ROR">https://ror.org/043749971</idno>
          <idno type="Wikidata">Q3117752</idno>
          <orgName>Groupe de Recherche en Informatique, Image et Instrumentation de Caen</orgName>
          <orgName type="acronym">GREYC</orgName>
          <date type="start">1995-01-01</date>
          <desc>
            <address>
              <addrLine>Boulevard du Maréchal Juin - 14050 CAEN Cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.greyc.fr</ref>
          </desc>
          <listRelation>
            <relation active="#struct-7127" type="direct"/>
            <relation active="#struct-455934" type="indirect"/>
            <relation active="#struct-246955" type="direct"/>
            <relation name="UMR6072" active="#struct-441569" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-7127" status="VALID">
          <idno type="IdRef">026403064</idno>
          <idno type="ISNI">0000000121864076</idno>
          <idno type="ROR">https://ror.org/051kpcy16</idno>
          <orgName>Université de Caen Normandie</orgName>
          <orgName type="acronym">UNICAEN</orgName>
          <date type="start">1432-01-01</date>
          <desc>
            <address>
              <addrLine>Esplanade de la Paix - CS 14032 - 14032 CAEN Cedex 5</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.unicaen.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-455934" type="direct"/>
          </listRelation>
        </org>
        <org type="regroupinstitution" xml:id="struct-455934" status="VALID">
          <idno type="IdRef">190906332</idno>
          <idno type="ISNI">0000000417859671 </idno>
          <idno type="ROR">https://ror.org/01k40cz91</idno>
          <orgName>Normandie Université</orgName>
          <orgName type="acronym">NU</orgName>
          <date type="start">2015-01-01</date>
          <desc>
            <address>
              <addrLine>Esplanade de la Paix - CS 14032 - 14032 Caen Cedex 5</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.normandie-univ.fr/</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-246955" status="VALID">
          <idno type="IdRef">190907991</idno>
          <idno type="ROR">https://ror.org/01fpqqe90</idno>
          <orgName>École Nationale Supérieure d'Ingénieurs de Caen</orgName>
          <orgName type="acronym">ENSICAEN</orgName>
          <desc>
            <address>
              <addrLine>6, Boulevard du Maréchal Juin CS 45053 14050 CAEN cedex 04</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.ensicaen.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-455934" type="direct"/>
          </listRelation>
        </org>
      </listOrg>
    </back>
  </text>
</TEI>