Infinite-dimensional Log-Determinant divergences between positive definite Hilbert–Schmidt operators
Infinite-dimensional Log-Determinant divergences between positive definite Hilbert–Schmidt operators
复制标题
正定希尔伯特-施密特算子之间的无限维对数行列式散度
DOI：
10.1007/s11117-019-00701-4
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发表时间：
2017
期刊：
影响因子：
1
通讯作者：
H. Q. Minh
中科院分区：
文献类型：
--
作者：
H. Q. Minh
本文将作者以前关于定义在Hilbert空间上的正定单位化迹类算子集上的无穷维Alpha对数-行列式(Log-Det)发散和Alpha-Beta对数-Det发散的工作推广到了正定单位化Hilbert-Schmidt算子的整个Hilbert流形上。这一推广是通过引入针对单元化Hilbert-Schmidt算子的扩展Hilbert-Carleman行列式来实现的，此外还引入了针对单元化迹类算子的扩展Fredholm型行列式。所得到的Alpha-Beta Log-Det散度的参数化族是一般的，并且包含了正定单位化Hilbert-Schmidt算子之间的许多特殊情况的散度，包括无限维仿射不变黎曼距离和对称Stein散度的无限维推广。
The current work generalizes the author’s previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, defined on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive definite unitized Hilbert–Schmidt operators. This generalization is carried out via the introduction of the extended Hilbert–Carleman determinant for unitized Hilbert–Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive definite unitized Hilbert–Schmidt operators as special cases, including the infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional generalization of the symmetric Stein divergence.