从非线性混合模型推导出子空间聚类的线性框架，揭示了隐藏维度与展开阶数的关系。
Nonlinear mixture model motivated subspace clustering
- 通过泰勒展开将非线性混合模型近似为线性子空间结构
- 发现展开阶数K与子空间维数d相等，且锚点数量为KC
- 为子空间维度估计提供理论依据，适合研究聚类算法原理者阅读
我们从盲源分离中使用的非线性混合模型（NMM）推导出子空间聚类（SC）所依赖的线性子空间并集（UoS）模型。将观测向量视为C个潜变量的未知多元非线性映射，在假设该映射可微至未知阶数K的前提下，用K阶泰勒展开逼近NMM，得到与线性UoS框架等价的模型。由此确立三重关系：(i) 平滑阶数K对应未知子空间维数d；(ii) KC等于锚点数量；(iii) 表示向量的稀疏度等于K（即d）。这些关系可用于估计子空间维数的上下界，并在六个基准数据集上使用五种主流SC算法验证。理论结果对自表示矩阵的后处理具有重要意义。
原文摘要 · Abstract (English)
We derive the linear union-of-subspaces (UoS) model for subspace clustering (SC) from the nonlinear mixture model (NMM) used in blind source separation (BSS) to represent a D-dimensional observation vector as an unknown multivariate nonlinear mapping of C latent variables. Assuming the mapping is differentiable up to an unknown order K, we approximate NMM by a K-th order Taylor expansion, yielding a model equivalent to the linear UoS framework underlying SC. This establishes that: (i) the smoothness order K corresponds to the unknown subspace dimension d; (ii) KC equals the number of anchors; and (iii) the sparsity of the representation vector equals K (i.e., d). These relationships enable estimation of bounds on subspace dimension, and that is validated on six benchmark datasets using five established SC algorithms. Established theoretical results are important for post-processing of self-representation matrices estimated by SC algorithms.
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