用于宇宙学推断的协方差矩阵去噪的多特征基方法
A multi-eigenbasis approach to covariance matrix denoising for cosmological inference
AI总结：
针对3D Lyα森林分析中数据驱动协方差矩阵不可逆的问题，提出一种多特征基去噪方法，通过投影到模拟参考特征基和残差分类特征基进行两步去噪，显著优于当前平滑方法。
摘要：
准确的协方差矩阵估计对于宇宙学分析至关重要，能够实现具有良好校准不确定性的无偏参数推断。获得可靠估计通常需要比数据向量维度多得多的独立样本，这并非总是可行。这一挑战对于3D Lyα森林分析尤为突出，该分析测量Lyα自相关及其与类星体的互相关，以共动分离的区间来联合约束宇宙学参数。其结果是数据向量非常大，从子样本测量的数据驱动协方差矩阵不可逆。当前方法对相关矩阵的非对角线元素进行平滑处理以建立可逆性，但这并未完全捕捉真实的关联结构。在这项工作中，我提出了一种新颖的多特征基去噪方法，用于数据驱动协方差矩阵，该方法是在3D Lyα森林分析的背景下开发的，并以DESI DR1模拟模拟为条件。首先将测量的含噪协方差投影到基于模拟的参考特征基上，得到初始去噪估计。然后通过将含噪残差投影到第二个由模拟训练的分类器导出的特征基上，构建加权残差校正，以捕捉初始重建中未恢复的关联结构。我在从分类器训练中保留的模拟协方差上验证了该方法，发现在矩阵级重建指标以及从许多模拟实现中测量的真实协方差获得的宇宙学参数后验恢复方面，该方法显著优于当前基于平滑的方法。
英文摘要：
Accurate covariance matrix estimation is crucial to cosmological analyses, enabling unbiased parameter inference with well-calibrated uncertainties. Obtaining a reliable estimate generally requires far more independent samples than the dimension of the data vector, which is not always feasible. This challenge is especially relevant for the 3D Ly$α$ forest analysis, which measures the Ly$α$ auto-correlation and its cross-correlation with quasars in bins of comoving separation to jointly constrain cosmological parameters. The consequence is a very large data vector, and the data-driven covariance measured from sub-samples is non-invertible. The current approach applies a smoothing procedure to the off-diagonals of the correlation matrix to establish invertibility, but this does not fully capture the true correlation structure. In this work, I present a novel multi-eigenbasis denoising method for the data-driven covariance matrix, developed in the context of the 3D Ly$α$ forest analysis and conditioned on DESI DR1 mock simulations. The measured noisy covariance is first projected onto the eigenbasis of a mock-based reference, yielding an initial denoised estimate. A weighted residual correction is then constructed by projecting the noisy residual onto a second eigenbasis derived from a mock-trained classifier, capturing correlation structure not recovered in the initial reconstruction. I validate the method on mock covariances withheld from classifier training and find significant improvements over the current smoothing-based approach in matrix-level reconstruction metrics and in the recovery of cosmological parameter posteriors when compared to those obtained from the true covariance measured from many mock realizations.
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