AI 中文总结
该研究针对高斯混合数据训练的有限温度连续自旋感知机，提出变分方法推导极限淬火压力的上下极小极大变分界，可统一计算基态能量、训练损失与泛化误差。
AI 中文摘要
我们针对在高斯混合数据上训练的有限温度连续自旋感知机，提出一种变分方法。该模型允许自旋具有广泛的凹效用函数及对数凹可分离先验测度。通过结合插值方法与对数凹性及集中度估计，我们推导了极限淬火压力的上下极小极大变分界。值得注意的是，两个界仅在两个变分参数的优化顺序上存在差异，其余极值均由变分势的凹-凸结构控制。当两次优化可交换时，两个界匹配并确定模型的解。该势同时给出作为平稳性条件的不动点方程，为基态能量、训练损失和泛化误差的计算提供统一途径。
英文摘要
We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave--convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.
Comments51 pages, 10 figures
