查看“︁辛克宏定理”︁的源代码

'''辛克宏定理'''（{{lang-en|Sinkhorn's theorem}}）是[[线性代数]]中的一个定理，指所有元素为正的[[方块矩阵]]都可以写成一个正[[对角矩阵]]、一个{{le|双随机矩阵|Doubly stochastic matrix}}与另一个正对角矩阵之积的形式。

== 定理 ==
如果<math>A</math>是所有元素都严格为正的<math>n\times n</math>[[矩阵]]，则存在元素严格为正的对角矩阵<math>D_1</math>和<math>D_2</math>，使得<math>D_1AD_2</math>是双随机矩阵，即每一行或列之和均为1。矩阵<math>D_1</math>和<math>D_2</math>在前者乘以一个正数、后者除以相同正数的情况下是唯一的。<ref name="Sinkhorn">Sinkhorn, Richard. (1964). "A relationship between arbitrary positive matrices and doubly stochastic matrices." ''Ann. Math. Statist.'' '''35''', 876–879. {{Doi|10.1214/aoms/1177703591}}</ref><ref name="Marshall">Marshall, A.W., & Olkin, I. (1967). "Scaling of matrices to achieve specified row and column sums." ''Numerische Mathematik''. '''12(1)''', 83–90. {{Doi|10.1007/BF02170999}}</ref>

== 辛克宏-诺普算法 ==
一个逼近双随机矩阵的简单迭代方法是交替地缩放<math>A</math>中所有行与列的比例，使其元素之和为1。辛克宏和诺普（Knopp）提出了该算法并分析了其收敛性。这一算法与调查统计学中的{{le|迭代比例拟合|Iterative proportional fitting}}本质上相同。

== 扩展 ==
在[[酉矩阵]]中存在类似的结论：对于任意酉矩阵<math>U</math>，都存在两个对角酉矩阵<math>L</math>和<math>R</math>，使得<math>LUR</math>的每一列和每一行的元素之和均为1。<ref name="IdelWolf">{{Cite journal |last=Idel |first=Martin |last2=Wolf |first2=Michael M. |title=Sinkhorn normal form for unitary matrices |journal=Linear Algebra and Its Applications |date=2015 |volume=471 |page=76–84 |arxiv=1408.5728 |doi=10.1016/j.laa.2014.12.031 |s2cid=119175915}}</ref>

对矩阵间映射也有类似的扩展结论<ref name="GeoPav">{{Cite journal |last=Georgiou |first=Tryphon |last2=Pavon |first2=Michele |title=Positive contraction mappings for classical and quantum Schrödinger systems |journal=Journal of Mathematical Physics |date=2015 |volume=56 |issue=3 |page=033301–1–24 |arxiv=1405.6650 |bibcode=2015JMP....56c3301G |doi=10.1063/1.4915289 |s2cid=119707158}}</ref><ref name="Gur">{{Cite journal |last=Gurvits |first=Leonid |title=Classical complexity and quantum entanglement |journal=Journal of Computational Science |date=2004 |volume=69 |issue=3 |page=448–484 |doi=10.1016/j.jcss.2004.06.003 |doi-access=free}}</ref>：给定一个克劳斯（Kraus）算子<math>\Phi</math>表示将一个[[密度矩阵]]映射到另一个密度矩阵的[[量子操作]]，即

: <math> S \mapsto \Phi(S) = \sum_i B_i S B_i^*, </math>

其满足迹不变

: <math> \sum_i B_i^* B_i = I, </math>

并且其范围在正定锥内部（严格正定）。在此条件下，存在正定的缩放因子<math>x_j</math>（<math>j\in\{0,1\}</math>），使得缩放后的克劳斯算子

: <math> S \mapsto x_1\Phi(x_0^{-1}Sx_0^{-1})x_1 = \sum_i (x_1B_ix_0^{-1}) S (x_1B_ix_0^{-1})^* </math>

是双随机的，即

: <math> x_1\Phi(x_0^{-1}I x_0^{-1})x_1 = I, </math>

: <math> x_0^{-1}\Phi^*(x_1I x_1)x_0^{-1} = I, </math>

其中<math>I</math>表示恒等算子。

== 应用 ==
2010年代，辛克宏定理开始被用于求解熵正则化的[[最优传输]]问题。<ref>{{Cite conference |last=Cuturi |first=Marco |date=2013 |title=Sinkhorn distances: Lightspeed computation of optimal transport |url= |pages=2292–2300 |booktitle=Advances in neural information processing systems}}</ref>这一方法在[[机器学习]]领域引起了关注，因为由此得到的辛克宏距离（Sinkhorn distance）可用于评估数据分布之间的差异。<ref>{{Cite conference |last=Mensch, Arthur |last2=Blondel, Mathieu |last3=Peyre, Gabriel |date=2019 |title=Geometric losses for distributional learning |arxiv=1905.06005 |booktitle=Proc ICML 2019}}</ref><ref>{{Cite conference |last=Mena, Gonzalo |last2=Belanger, David |last3=Munoz, Gonzalo |last4=Snoek, Jasper |date=2017 |title=Sinkhorn networks: Using optimal transport techniques to learn permutations |booktitle=NIPS Workshop in Optimal Transport and Machine Learning}}</ref><ref>{{Cite conference |last=Kogkalidis, Konstantinos |last2=Moortgat, Michael |last3=Moot, Richard |date=2020 |title=Neural Proof Nets |url=https://aclanthology.org/2020.conll-1.3 |booktitle=Proceedings of the 24th Conference on Computational Natural Language Learning}}</ref>在一些[[最大似然估计|最大似然]]训练效果不佳的情况下，这种方法可以改进机器学习算法的训练效果。

== 参考文献 ==
{{Reflist}}

[[Category:线性代数定理]]
[[Category:矩陣論]]