学习径向基函数Schönberg测度的平均域论
A Mean-Field Theory for Learning the Schönberg Measure of Radial Basis Functions
摘要：
我们开发并分析了投影粒子Langevin最优化方法，从训练样本中学习径向基函数的Schönberg整表示中的分布。更具体地说，我们的特征分布鲁棒最优化方法相对于Wasserstein距离，以优化Schönberg整表示中的分布。为了提供理论性能保证，我们分析了平均场制度的投影粒子在线（随机）最优化方法的缩放限制。特别是，我们证明，在标度限制，经验措施的朗之万粒子收敛到法律的反映伊藤扩散漂移过程。此外，漂移也是基本过程的规律的函数。利用半鞅的Itô引理和Wiener过程的Grisanov测度变换，我们得到了一个具有Robin边界条件的Mckean-Vlasov型偏微分方程，它描述了平均场区域中投射Langevin粒子的经验测度的演化.此外，我们建立了弱意义下的偏微分方程定常解的存在唯一性。我们应用我们的学习方法来训练径向核的内核局部敏感哈希（LSH）函数，其中的训练数据集是通过一个小的数据库子集上的$k$ -平均聚类方法生成的。随后，我们将我们的内核LSH与经过训练的内核一起应用于MNIST数据集上的图像检索任务，并证明了我们的内核学习方法的有效性。我们还应用我们的内核学习方法结合内核支持向量机（SVM）的分类基准数据集。
英文摘要：
We develop and analyze a projected particle Langevin optimization method to learn the distribution in the Schönberg integral representation of the radial basis functions from training samples. More specifically, we characterize a distributionally robust optimization method with respect to the Wasserstein distance to optimize the distribution in the Schönberg integral representation. To provide theoretical performance guarantees, we analyze the scaling limits of a projected particle online (stochastic) optimization method in the mean-field regime. In particular, we prove that in the scaling limits, the empirical measure of the Langevin particles converges to the law of a reflected Itô diffusion-drift process. Moreover, the drift is also a function of the law of the underlying process. Using Itô lemma for semi-martingales and Grisanov's change of measure for the Wiener processes, we then derive a Mckean-Vlasov type partial differential equation (PDE) with Robin boundary conditions that describes the evolution of the empirical measure of the projected Langevin particles in the mean-field regime. In addition, we establish the existence and uniqueness of the steady-state solutions of the derived PDE in the weak sense. We apply our learning approach to train radial kernels in the kernel locally sensitive hash (LSH) functions, where the training data-set is generated via a $k$-mean clustering method on a small subset of data-base. We subsequently apply our kernel LSH with a trained kernel for image retrieval task on MNIST data-set, and demonstrate the efficacy of our kernel learning approach. We also apply our kernel learning approach in conjunction with the kernel support vector machines (SVMs) for classification of benchmark data-sets.
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