标题： 支持向量回归的统计力学 显示英文标题

标题： Statistical Mechanics of Support Vector Regression

摘要： 深度学习和计算神经科学中的一个关键问题是将神经表征的几何特性与任务性能联系起来。 在这里，我们考虑连续解码任务中的这一问题，其中神经可变性可能影响任务精度。 使用统计力学的方法，我们研究了$\varepsilon$-不敏感支持向量回归（$\varepsilon$-SVR）的平均情况学习曲线，并讨论其作为线性解码能力度量的性能。 我们的分析揭示了在临界负载下的训练误差相变，捕捉了容忍参数$\varepsilon$与神经可变性之间的相互作用。 我们在泛化误差中发现了一个双下降现象，表明$\varepsilon$起到了正则化器的作用，同时抑制并移动了这些峰值。 理论预测通过玩具模型和深度神经网络得到了验证，将支持向量机的理论扩展到了具有固有神经可变性的连续任务。 显示英文摘要

摘要： A key problem in deep learning and computational neuroscience is relating the geometrical properties of neural representations to task performance. Here, we consider this problem for continuous decoding tasks where neural variability may affect task precision. Using methods from statistical mechanics, we study the average-case learning curves for $\varepsilon$-insensitive Support Vector Regression ($\varepsilon$-SVR) and discuss its capacity as a measure of linear decodability. Our analysis reveals a phase transition in training error at a critical load, capturing the interplay between the tolerance parameter $\varepsilon$ and neural variability. We uncover a double-descent phenomenon in the generalization error, showing that $\varepsilon$ acts as a regularizer, both suppressing and shifting these peaks. Theoretical predictions are validated both with toy models and deep neural networks, extending the theory of Support Vector Machines to continuous tasks with inherent neural variability.