发表机构
University of Basel; IBM Research（巴塞尔大学; IBM研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过解析刻画经典线性回归的双下降曲线，并将其推广至量子极端学习机，揭示编码强度、输入重上传和测量次数三个设计参数可正则化输出协方差并稳定插值阈值，为QELMs提供超越插值的泛化设计原则。
AI 中文摘要
增加模型容量可能导致机器学习模型的泛化性能先变差，直到其完美插值训练数据，而在超越插值点之后性能又变好。这种反转现象被称为“双下降”，对于独立特征已得到充分理解。对于相关特征，其形状由特征协方差的谱决定，但现有分析通常仅给出隐式方程。我们首先针对经典线性回归弥合了这一差距，通过两个量（特征总体协方差的条件数和相关强度的谱测度）解析地表征了整个风险曲线。然后，我们将该理论应用于量子模型，重点关注量子极端学习机（QELMs）。我们展示了三个可调的设计旋钮，即编码强度、输入重上传和测量次数，各自对QELM输出协方差进行正则化，并推导了插值阈值保持稳定的条件。有限的测量散粒噪声可以被利用来保证这种稳定性。数值实验证实了每项预测，为QELMs提供了超越插值良好泛化的具体设计原则。
英文摘要
Increasing capacity can induce a machine learning model to generalize worse until it perfectly interpolates the training data, and better once it moves beyond interpolation. This inversion, known as \emph{double descent}, is well understood for independent features. For correlated features, its shape is governed by the spectrum of the feature covariance, but existing analyses typically yield only implicit equations. We first close this gap for classical linear regression, analytically characterizing the entire risk curve through two quantities, namely the condition number of the feature population covariance and a spectral measure of correlation strength. Then, we apply this theory on quantum models, focusing on quantum extreme learning machines (QELMs). We show that three tunable design knobs, \textit{i.e.,} encoding strength, input reuploads, and measurement shots, each regularize the QELM output covariance, and derive conditions under which the interpolation threshold remains stable. Finite measurement shot noise can be leveraged to guarantee this stability. Numerical experiments confirm each prediction, yielding concrete design principles for QELMs that generalize well beyond interpolation.
CommentsMain: 10 pages, 2 figures; Supplementary: 89 pages, 5 figures
