最优贝叶斯推理中的重叠矩阵集中
Overlap matrix concentration in optimal Bayesian inference
摘要：
我们考虑模型的贝叶斯推断的信号与向量分量的有限维。我们表明，在适当的扰动下，这些模型是副本对称的意义上，重叠矩阵集中。重叠矩阵是这些模型中的序参数，并且与误差度量（例如最小均方误差）直接相关。我们的证明是有效的最佳贝叶斯推理设置。这意味着它依赖于模型及其所有超参数已知的假设，以便可以准确地写出后验分布。我们的结果适用于高维推理和学习中的重要问题的例子是低秩张量因子分解，教师-学生场景中具有有限数量隐藏神经元的委员会机器神经网络，或多层版本的广义线性模型。
英文摘要：
We consider models of Bayesian inference of signals with vectorial components of finite dimensionality. We show that, under a proper perturbation, these models are replica symmetric in the sense that the overlap matrix concentrates. The overlap matrix is the order parameter in these models and is directly related to error metrics such as minimum mean-square errors. Our proof is valid in the optimal Bayesian inference setting. This means that it relies on the assumption that the model and all its hyper-parameters are known so that the posterior distribution can be written exactly. Examples of important problems in high-dimensional inference and learning to which our results apply are low-rank tensor factorization, the committee machine neural network with a finite number of hidden neurons in the teacher-student scenario, or multi-layer versions of the generalized linear model.
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