Statistical-physics calculations in machine learning and theoretical neuroscience often involve lengthy derivations that obscure physical interpretation. We present concise, non-replica derivations of key results and highlight their underlying similarities. Using a cavity approach, we analyze high-dimensional learning problems: perceptron classification of points and manifolds, and kernel ridge regression. These problems share a common structure--a bipartite system of interacting feature and datum variables--enabling a unified analysis. For perceptron-capacity problems, we identify a symmetry that allows derivation of correct capacities through a na\"ive method.

翻译：机器学习和理论神经科学中的统计物理计算通常涉及冗长的推导过程，这些过程掩盖了物理解释的本质。我们提出了关键结果的简洁非复本推导，并强调了它们之间的内在相似性。通过采用空腔方法，我们分析了高维学习问题：点与流形的感知机分类，以及核岭回归。这些问题共享一个共同的结构——由相互作用的特征变量与数据变量组成的二部系统——这使得统一分析成为可能。对于感知机容量问题，我们发现了一种对称性，使得通过一种朴素方法即可推导出正确的容量。