Geometry of the Fisher–Rao metric on the space of smooth densities on a compact manifold

Bruveris Martins; Michor Peter W.

摘要

Abstract > It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form <displayedItem xml:id="mana201600523-disp-0001" type="mathematics" numbered="no"> <mat:math display="block" altimg="urn:x-wiley:0025584X:media:mana201600523:mana201600523-math-0001" wiley:location="equation/mana201600523-math-0001.png"> <mat:mstyle scriptlevel="0" displaystyle="true"> <mat:mrow> <mat:mspace width="147.60004pt"/> <mat:msub> <mat:mi>G</mat:mi> <mat:mi>μ</mat:mi> </mat:msub> <mat:mrow> <mat:mo stretchy="false">(</mat:mo> <mat:mi>α</mat:mi> <mat:mo>,</mat:mo> <mat:mi>β</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> <mat:mo>=</mat:mo> <mat:msub> <mat:mi>C</mat:mi> <mat:mn>1</mat:mn> </mat:msub> <mat:mrow> <mat:mo stretchy="false">(</mat:mo> <mat:mi>μ</mat:mi> <mat:mrow> <mat:mo stretchy="false">(</mat:mo> <mat:mi>M</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> <mat:msub> <mat:mo>∫</mat:mo> <mat:mi>M</mat:mi> </mat:msub> <mat:mfrac> <mat:mi>α</mat:mi> <mat:mi>μ</mat:mi> </mat:mfrac> <mat:mfrac> <mat:mi>β</mat:mi> <mat:mi>μ</mat:mi> </mat:mfrac> <mat:mspace width="0.16em"/> <mat:mi>μ</mat:mi> <mat:mo>+</mat:mo> <mat:msub> <mat:mi>C</mat:mi> <mat:mn>2</mat:mn> </mat:msub> <mat:mrow> <mat:mo stretchy="false">(</mat:mo> <mat:mi>μ</mat:mi> <mat:mrow> <mat:mo stretchy="false">(</mat:mo> <mat:mi>M</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> <mat:msub> <mat:mo>∫</mat:mo> <mat:mi>M</mat:mi> </mat:msub> <mat:mi>α</mat:mi> <mat:mo>·</mat:mo> <mat:msub> <mat:mo>∫</mat:mo> <mat:mi>M</mat:mi> </mat:msub> <mat:mi>β</mat:mi> </mat:mrow> </mat:mstyle> </mat:math> </displayedItem> for some smooth functions <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201600523:mana201600523-math-0002" wiley:location="equation/mana201600523-math-0002.png"> <mat:mrow> <mat:msub> <mat:mi>C</mat:mi> <mat:mn>1</mat:mn> </mat:msub> <mat:mo>,</mat:mo> <mat:msub> <mat:mi>C</mat:mi> <mat:mn>2</mat:mn> </mat:msub> </mat:mrow> </mat:math> of the total volume <mat:math display="inline" altimg="urn:x-wiley:0025584 </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract Type =“Main”XML：Lang =“en”> <标题类型=“main”>抽象</ title> > 众所周知，在大于一个尺寸的封闭歧管上，每个平滑弱的黎曼度量在扩散族组的作用下不变的光滑正密度的空间，是形式 <lightsitem xml：id =“mana201600523-disp-0001”type =“数学”编号=“否” <mat：数学显示=“块”Altimg =“URN：X-Wiley：0025584X：MANA201600523：MANA201600523-MANA201600523-MATH-0001”WIley：Location =“等式/ MANA201600523-MATH-0001.PNG”> <mat：mstyle scriptlevel =“0”displaystyle =“true”> <MAT：MROW> <mat：mspace width =“147.60004pt”/> <mat：msub> <MAT：MI> G </ MAT：MI> <MAT：MI>μ</ MAT：MI> </ mat：msub> <MAT：MROW> <mat：mo stractry =“false”>（</ mat：mo> <mat：mi>α</ mat：mi> <mat：mo>，</ mat：mo> <MAT：MI>β</ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> <mat：mo> = </ mat：mo> <mat：msub> <MAT：MI> C </ MAT：MI> <MAT：MN> 1 </ MAT：MN> </ mat：msub> <MAT：MROW> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MI>μ</ MAT：MI> <MAT：MROW> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MI> M </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> <mat：msub> <mat：mo>∫</ mat：mo> <MAT：MI> M </ MAT：MI> </ mat：msub> <MAT：MFRAC> <mat：mi>α</ mat：mi> <MAT：MI>μ</ MAT：MI> </ mat：mfrac> <MAT：MFRAC> <MAT：MI>β</ MAT：MI> <MAT：MI>μ</ MAT：MI> </ mat：mfrac> <mat：mspace width =“0.16em”/> <MAT：MI>μ</ MAT：MI> <mat：mo> + </ mat：mo> <mat：msub> <MAT：MI> C </ MAT：MI> <MAT：MN> 2 </ MAT：MN> </ mat：msub> <MAT：MROW> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MI>μ</ MAT：MI> <MAT：MROW> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MI> M </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> <mat：msub> <mat：mo>∫</ mat：mo> <MAT：MI> M </ MAT：MI> </ mat：msub> <mat：mi>α</ mat：mi> <MAT：MO>·</ MAT：MO> <mat：msub> <mat：mo>∫</ mat：mo> <MAT：MI> M </ MAT：MI> </ mat：msub> <MAT：MI>β</ MAT：MI> </ mat：MROW> </ mat：mstyle> </ mat：math> </ lightsitem> 对于一些顺利的功能 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584X：MANA201600523：MANA201600523-MANA201600523-MATH-0002”WIley：Location =“等式/ MANA201600523-MATH-0002.PNG”> <MAT：MROW> <mat：msub> <MAT：MI> C </ MAT：MI> <MAT：MN> 1 </ MAT：MN> </ mat：msub> <mat：mo>，</ mat：mo> <mat：msub> <MAT：MI> C </ MAT：MI> <MAT：MN> 2 </ MAT：MN> </ mat：msub> </ mat：MROW> </ mat：math> 总量 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584 </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" >著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-24615/'>《Mathematische Nachrichten》</a> <b style="margin: 0 2px;">|</b><span>2019年第3期</span><b style="margin: 0 2px;">|</b><span>共13页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Bruveris Martins&option=202" target="_blank" rel="nofollow">Bruveris Martins;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Michor Peter W.&option=202" target="_blank" rel="nofollow">Michor Peter W.;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Department of MathematicsBrunel University LondonUxbridge UB8 3PH United Kingdom;</p> <p>Fakult?t für MathematikUniversit?t WienOskar‐Morgenstern‐Platz 1 A‐1090 Wien Austria;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li > <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span><a href="https://www.zhangqiaokeyan.com/clc/156.html" title="数学">数学;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Fisher–Rao metric&option=203" rel="nofollow">Fisher–Rao metric;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=information geometry&option=203" rel="nofollow">information geometry;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=space of densities&option=203" rel="nofollow">space of densities;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=surfaces of revolution&option=203" rel="nofollow">surfaces of revolution;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Primary: 58B20&option=203" rel="nofollow">Primary: 58B20;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=58D15&option=203" rel="nofollow">58D15;</a> </p> <div class="translation"> 机译：Fisher-Rao公制;信息几何;密度的空间;革命的表面;初级：58b20;58d15; </div> </li> </ul> </div> </div> <div class="literature cardcommon" id="literaturereference" style="display:none"> <div class="similarity "> <h3 class="all_title" id="enpatent111">引文网络</h3> <div class="referencetab clearfix"> <ul id="referencedaohang"> <li dataid="referenceul">参考文献</li> <li dataid="citationul">引证文献</li> <li dataid="commonreferenceul">共引文献</li> <li dataid="commoncitationul">同被引文献</li> <li dataid="tworeferenceul">二级参考文献</li> <li dataid="twocitationul">二级引证文献</li> </ul> </div> <div class="reference_details" id="referenceList"> <ul id="referenceul"></ul> <ul id="citationul"></ul> <ul id="commonreferenceul"></ul> <ul id="commoncitationul"></ul> <ul id="tworeferenceul"></ul> <ul id="twocitationul"></ul> </div> </div> </div> <div class="literature cardcommon"> <div class="similarity "> <h3 class="all_title" id="enpatent66">相似文献</h3> <div class="similaritytab clearfix"> <ul> <li class="active" >外文文献</li> <li >中文文献</li> <li >专利</li> </ul> </div> <div class="similarity_details"> <ul > <li> <div> <b>1. </b><a class="enjiyixqcontent" href="/journal-foreign-detail/0704023595759.html">Geometry of the Fisher–Rao metric on the space of smooth densities on a compact manifold</a> <b>[J]</b> . <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Bruveris 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