论文标题
关于连续最大熵分布的可计算性：谎言组的隔离轨道
On the Computability of Continuous Maximum Entropy Distributions: Adjoint Orbits of Lie Groups
论文作者
论文摘要
Given a point $A$ in the convex hull of a given adjoint orbit $\mathcal{O}(F)$ of a compact Lie group $G$, we give a polynomial time algorithm to compute the probability density supported on $\mathcal{O}(F)$ whose expectation is $A$ and that minimizes the Kullback-Leibler divergence to the $ g $ -INVARIANT MEATURIAINT MATER在$ \ MATHCAL {O}（f）$上。这大大扩展了作者的最新工作（Stoc 2020），后者为等级$ k $ -projections的歧视提供了这样的结果，这是统一组$ \ mathrm {u}（n）$的特定伴随轨道。我们的结果依赖于先前工作中提出的基于椭圆形方法的框架。但是，要将其应用于紧凑型谎言组的一般环境，我们需要谎言理论的工具。例如，伴随表示的属性用于查找包含$ \ Mathcal {o}（f）$的凸壳的最小仿射空间的定义平等性，并在最佳双重解决方案上建立界限。同样，Harish-Chandra积分公式用于获得双重目标函数的评估Oracle。尽管Harish-Chandra积分公式使我们能够在谎言组的伴随轨道上编写某些积分作为少数决定因素的总和，但仅针对lie代数$ \ Mathfrak $ \ Mathfrak {g} $g。$G。$G。$G。$的元素的cartan subalgebra的元素定义。此外，伴随轨道的凸壳是一种轨道，本文中研究的轨道是光谱的。因此，我们的主要结果可以看作是将最大熵框架扩展到一类Spectrahedra。
Given a point $A$ in the convex hull of a given adjoint orbit $\mathcal{O}(F)$ of a compact Lie group $G$, we give a polynomial time algorithm to compute the probability density supported on $\mathcal{O}(F)$ whose expectation is $A$ and that minimizes the Kullback-Leibler divergence to the $G$-invariant measure on $\mathcal{O}(F)$. This significantly extends the recent work of the authors (STOC 2020) who presented such a result for the manifold of rank $k$-projections which is a specific adjoint orbit of the unitary group $\mathrm{U}(n)$. Our result relies on the ellipsoid method-based framework proposed in prior work; however, to apply it to the general setting of compact Lie groups, we need tools from Lie theory. For instance, properties of the adjoint representation are used to find the defining equalities of the minimal affine space containing the convex hull of $\mathcal{O}(F)$, and to establish a bound on the optimal dual solution. Also, the Harish-Chandra integral formula is used to obtain an evaluation oracle for the dual objective function. While the Harish-Chandra integral formula allows us to write certain integrals over the adjoint orbit of a Lie group as a sum of a small number of determinants, it is only defined for elements of a chosen Cartan subalgebra of the Lie algebra $\mathfrak{g}$ of $G.$ We show how it can be applied to our setting with the help of Kostant's convexity theorem. Further, the convex hull of an adjoint orbit is a type of orbitope, and the orbitopes studied in this paper are known to be spectrahedral. Thus our main result can be viewed as extending the maximum entropy framework to a class of spectrahedra.
