Finite sample rates of convergence for the Bigraphical and Tensor graphical Lasso estimators
Bigraphical 与张量图 Lasso 估计量的有限样本收敛速率AI 解读 ↓
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Bigraphical 与张量图 Lasso 估计量的有限样本收敛速率
- 研究问题
- 摘要称，本文关注矩阵正态与张量值数据下精度矩阵估计的收敛速率问题，特别是双向 Kronecker 和模型此前缺乏有限样本理论保证的缺口。
- 方法与思路
- 作者提出对 Bigraphical Lasso 与 TeraLasso 估计量的相对误差界进行改进，在算子范数与 Frobenius 范数下将界收紧约 log p 倍，并据此推导双向 Kronecker 和模型的有限样本估计误差。
- 材料陈述的贡献
- 摘要称，改进后的界使双向 Kronecker 和模型的有限样本误差得以建立，弥补了 GZH(2019) 中该情形仅有单样本误差、缺乏理论保证的空白；作者还阐述了 Kronecker 和模型并给出全部主要定理的完整证明。
- 阅读关注点
- 阅读全文时应核对：所依赖的识别与正则化假设、改进界相对 GZH(2019) 的具体条件差异、log p 因子收紧的推导步骤，以及双向模型有限样本界的确切形式与适用范围。
- 阅读边界
- 摘要未说明具体假设、常数、证明细节与模拟或实证结果；本文为 arXiv 预印本，不能据此声称已经同行评审或期刊接收。
原文摘要
Many modern datasets exhibit dependencies among observations as well as variables. A decade ago, Kalaitzis et. al. (2013) proposed the Bigraphical Lasso, an estimator for precision matrices of matrix-normals based on the Cartesian product of graphs; they observed that the associativity of the Kronecker sum yields an approach to the modeling of datasets organized into 3 or higher-order tensors. Subsequently, Greenewald, Zhou and Hero (2019) explored this possibility to a great extent, by introducing the tensor graphical Lasso (TeraLasso) for estimating sparse $L$-way decomposable inverse covariance matrices for all $L \ge 2$, and showing the rates of convergence in the Frobenius and operator norms for estimating this class of inverse covariance matrices for sub-gaussian tensor-valued data. In this paper, we provide sharper rates of convergence for both Bigraphical and TeraLasso estimators for inverse covariance matrices. This improves upon the rates presented in GZH 2019. In particular, (a) we strengthen the bounds for the relative errors in the operator and Frobenius norm by a factor of approximately $\log p$; (b) Crucially, this improvement allows for finite sample estimation errors in both norms to be derived for the two-way Kronecker sum model. This closes the gap between the low single-sample error for the two-way model as observed in GZH 2019 and the lack of theoretical guarantee for this particular case. The two-way regime is important because it is the setting that is the most theoretically challenging, and simultaneously the most common in applications. In the current paper, we elaborate on the Kronecker Sum model, highlight the proof strategy and provide full proofs of all main theorems.