随机图上高斯过程学习曲线的非线性理论
Replica theory for learning curves for Gaussian processes on random graphs
摘要：
统计物理学方法可以用于从潜在的噪声数据中学习推理方法的性能，如通过定义为平均误差与训练示例数量的学习曲线所量化的。我们分析了一个具有挑战性的问题，在该地区的非参数推断，有效地无限数量的参数必须学习，特别是高斯过程回归。当输入是一个随机图上的顶点和输出噪声函数值，我们表明，副本技术可以用来获得准确的性能预测的限制大图。高斯过程先验的协方差由随机游动核定义，该核是连续空间上平方指数核的离散模拟。传统上，这个内核只在全局上进行归一化，因此先验方差在顶点之间可能不同;作为一个更有原则的替代方案，我们考虑局部归一化，其中先验方差是均匀的。
英文摘要：
Statistical physics approaches can be used to derive accurate predictions for the performance of inference methods learning from potentially noisy data, as quantified by the learning curve defined as the average error versus number of training examples. We analyse a challenging problem in the area of non-parametric inference where an effectively infinite number of parameters has to be learned, specifically Gaussian process regression. When the inputs are vertices on a random graph and the outputs noisy function values, we show that replica techniques can be used to obtain exact performance predictions in the limit of large graphs. The covariance of the Gaussian process prior is defined by a random walk kernel, the discrete analogue of squared exponential kernels on continuous spaces. Conventionally this kernel is normalised only globally, so that the prior variance can differ between vertices; as a more principled alternative we consider local normalisation, where the prior variance is uniform.
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