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Fast matrix computations for functional additive models

机译：功能加性模型的快速矩阵计算

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摘要

It is common in functional data analysis to look at a set of related functions: a set of learning curves, a set of brain signals, a set of spatial maps, etc. One way to express relatedness is through an additive model, whereby each individual function g_i (x) is assumed to be a variation around some shared mean f(x). Gaussian processes provide an elegant way of constructing such additive models, but suffer from computational difficulties arising from the matrix operations that need to be performed. Recently Heersink & Furrer have shown that functional additive model give rise to covari-ance matrices that have a specific form they called quasi-Kronecker (QK), whose inverses are relatively tractable. We show that under additional assumptions the two-level additive model leads to a class of matrices we call restricted quasi-Kronecker (rQK), which enjoy many interesting properties. In particular, we formulate matrix factorisations whose complexity scales only linearly in the number of functions in latent field, an enormous improvement over the cubic scaling of naieve approaches. We describe how to leverage the properties of rQK matrices for inference in Latent Gaussian Models.
机译：在功能数据分析中，通常要查看一组相关功能：一组学习曲线，一组脑信号，一组空间图等。表达相关性的一种方法是通过加性模型，其中每个人假设函数g_i（x）是围绕某个共享均值f（x）的变化。高斯过程提供了构建此类加法模型的理想方法，但由于需要执行的矩阵运算而遭受计算难题。最近，Heersink＆Furrer表明，功能加法模型会产生协方差矩阵，它们具有一种称为准克罗内克（QK）的特定形式，其逆相对易处理。我们表明，在附加假设下，两级加法模型导致一类称为受限准克罗内克（rQK）的矩阵，该矩阵具有许多有趣的属性。尤其是，我们制定了矩阵分解，其复杂度仅在潜场中的函数数量上呈线性比例缩放，这与单纯方法的三次比例缩放相比有了巨大的改进。我们描述了如何在潜在高斯模型中利用rQK矩阵的属性进行推理。

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