AI 中文总结
该研究构建了不变学习的朗道理论框架，推导了有效自由能，经实验验证可预测正则化表型及模型学习内容，还推广至耦合集体模态并给出谱相边界准则。
AI 中文摘要
不变学习旨在寻找在不同环境下仍具有预测性的表示，但其目标在正则化路径上的行为通常尚不明确。我们通过将表示学习视为多模态磁化，从具体的不变学习目标推导出一种朗道型有效自由能，其低阶系数构成目标特征并诱导出不同的正则化表型。有效二次修正会移动相边界并实现有限强度的模态消除；四次修正调节 onset 后的振幅，通常在有限强度下留下残差加载；高阶结构控制非单调尾部、不稳定性及大正则化下的崩溃。在一个典型的双线性模型中，该理论给出了闭式相边界和稳态加载，以及定义选择性保留窗口的捷径模态和稳定模态的不同临界强度。受控实验证实了预测的相边界、加载和正则化表型。在单隐层和双隐层 ReLU 网络中，尽管尺度存在依赖深度的偏移，相同特征仍可定性预测正则化路径行为。矩阵扩展将该框架推广到耦合集体模态，并给出谱相边界准则。综上，该框架将低阶目标特征转化为正则化表型的预测，最终转化为正则化变化时模型所学习内容的预测。
英文摘要
Invariant-learning objectives pursue similar goals yet produce qualitatively different regularization paths, leaving unclear when shortcuts can be suppressed without damaging stable modes. Our starting point is simple: in the desired shortcut-suppressed regime, shortcut loading is small, so the objective's low-order expansion governs local stability and residual amplitude. This brings the problem into the domain of Landau phase-transition theory. We establish a mathematical isomorphism between the near-critical normal form of predictive-mode learning and Landau theory, identifying the learning objective as an effective free energy and critical-mode amplitude as an order parameter. The resulting low-order objective signatures predict distinct regularization phenotypes: quadratic $R_2$ terms shift phase boundaries and enable finite-strength elimination, whereas quartic $R_4$ terms continuously attenuate acquired modes while leaving nonzero residual loading at finite strengths. Higher-order terms may further shape nonlinear tails. In a canonical bilinear model, we derive exact phase boundaries and equilibrium loadings, identifying conditions for a selective-retention window in which shortcut suppression preserves stable structure. Controlled bilinear and ReLU experiments, including an MNIST construction, support the predicted signature-phenotype relation, while coupled-feature experiments validate its extension to collective modes. The framework connects the mathematical structure of invariant-learning objectives to what models learn as regularization varies.
