解析DPP与k-DPP的可识别性差异，揭示其参数不变性机制。
From DPPs to $k$-DPPs: identifiability analysis via spectral decomposition
- 通过谱分解分析DPP和k-DPP的参数结构
- k-DPP参数仅在尺度、符号相似性和特征空间旋转下可识别
- 当组合数小于矩阵维数时存在连续不可识别性
我们通过谱分解 $L=UΛU^{ op}$ 研究确定性点过程（DPP）的几何结构。谱 $Λ$ 通过初等对称多项式决定基数分布，而特征空间方向 $U$ 决定固定基数层内的条件分布。条件化基数 $k$ 得到 $k$-DPP，其可识别性结构发生根本变化：谱参数仅在共同尺度下可识别，特征向量矩阵的平方子式决定旋转参数。我们精确刻画了可识别性差距，通过三个显式不变性（尺度、符号相似性、特征空间旋转）及维度计数定理，证明当 $inom{N}{k}<N(N+1)/2$ 时必然存在额外连续不可识别性。相比之下，完整DPP的不可识别性仅来自离散符号相似性。
原文摘要 · Abstract (English)
We study the geometry of determinantal point processes (DPPs) through the spectral decomposition $L=UΛU^{\top}$. The spectrum $Λ$ governs the cardinality distribution via elementary symmetric polynomials, while the eigenspace orientation $U$ governs the conditional law within each fixed-cardinality stratum. Conditioning on cardinality $k$ yields the $k$-DPP, for which the identifiability structure changes fundamentally: the spectral parameter becomes identifiable only up to a common scale, and the eigenspace rotation parameter is identifiable only through squared minors of the eigenvector matrix. We characterize the identifiability gap precisely, via three explicit invariances (scale, sign similarity, and eigenspace rotation) and a dimension-counting theorem showing the existence of additional continuous non-identifiability whenever $\binom{N}{k}<N(N+1)/2$. In contrast, for the full DPP the non-identifiability comes only from the discrete sign similarity.
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