数据场论：用于信号检测的泛函重整化群理论与应用
Data Field Theory: Theory and Applications of the Functional Renormalization Group for Signal Detection
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中文总结 AI 辅助
该研究综述了适用于高维数据信号检测的泛函重整化群理论，其可处理信号不与噪声体分离的场景，通过检验有效场论高斯不动点稳定性识别信号存在，涉及维度相变相关内容。
中文摘要 AI 辅助
本文综述了适用于高维数据信号检测的重整化群框架，该框架针对信号可能具有广延秩、且不作为孤立尖峰与噪声体分离的场景。该框架在随机矩阵普适类附近的准连续谱区域内，提供了一种概念上简单的信号与噪声区分准则，而这一场景超出了Baik-Ben Arous-Péché阈值等标准方法的适用范围，后者要求本征值与噪声体清晰分离。相比之下，重整化群方法直接追踪谱形变，且无需依赖尖峰分离即可一致地给出检测下限。我们综述了通过检验信号所在谱尾自由度集体行为的有效场论高斯不动点的稳定性来识别信号存在性的结果，还讨论了由信号诱导的正则标度的标度依赖性如何表现为维度相变。
英文摘要
We review the renormalization group framework for signal detection in high-dimensional data, tailored to the regime where the signal may be of extensive rank and does not separate from the noise bulk as isolated spikes. The framework provides a conceptually simple criterion for distinguishing signal from noise within a quasi-continuous spectral region near a random-matrix universality class. This scenario lies beyond the reach of standard methods such as the Baik-Ben Arous-Péché threshold, which requires eigenvalues to be cleanly separated from the bulk. The renormalization group approach, by contrast, directly tracks spectral deformations and consistently yields a lower limit of detection without relying on spike separation. We review results that identify the presence of a signal by testing the stability of the Gaussian fixed point of an effective field theory for the collective behaviour of the degrees of freedom in the spectral tail, where the signal resides. We also discuss how the scale dependence of the canonical dimension, induced by the signal, manifests as a dimensional phase transition.
