Learning of matrix-valued data has recently surged in a range of scientific and business applications. Trace regression is a widely used method to model effects of matrix predictors and has shown great success in matrix learning. However, nearly all existing trace regression solutions rely on two assumptions: (i) a known functional form of the conditional mean, and (ii) a global low-rank structure in the entire range of the regression function, both of which may be violated in practice. In this article, we relax these assumptions by developing a general framework for nonparametric trace regression models via structured sign series representations of high dimensional functions. The new model embraces both linear and nonlinear trace effects, and enjoys rank invariance to order-preserving transformations of the response. In the context of matrix completion, our framework leads to a substantially richer model based on what we coin as the "sign rank" of a matrix. We show that the sign series can be statistically characterized by weighted classification tasks. Based on this connection, we propose a learning reduction approach to learn the regression model via a series of classifiers, and develop a parallelable computation algorithm to implement sign series aggregations. We establish the excess risk bounds, estimation error rates, and sample complexities. Our proposal provides a broad nonparametric paradigm to many important matrix learning problems, including matrix regression, matrix completion, multi-task learning, and compressed sensing. We demonstrate the advantages of our method through simulations and two applications, one on brain connectivity study and the other on high-rank image completion.
翻译:最近,在一系列科学和商务应用中,对基于矩阵的数据进行了大量研究。跟踪回归是用来模拟矩阵预测器效应的一种广泛使用的方法,在矩阵学习中也取得了巨大成功。然而,几乎所有现有的痕量回归解决方案都依赖于两个假设:(一) 有条件平均值的已知功能形式,和(二) 整个回归功能范围内的全球低级结构,两者在实践中都可能遭到违反。在本篇文章中,我们通过对高维功能进行结构化的标志系列展示,为非参数回归模型制定非参数回归模型的一般框架,放松这些假设。新模型包含线性和非线性跟踪效应,并享有对响应的顺序保存转换的不一致性。在矩阵完成过程中,我们的框架导致一个大大更丰富的模式,其依据是我们所发现的矩阵的“指示级别”,两者都可能在实践中遭到违反。我们通过加权分类任务,我们建议采用学习减少模式的方法,通过一系列叙级员学习回归模型学习回归模型,并开发可平行的计算算法,以实施标志性序列矩阵组合。在矩阵完成过程中,我们确立了超额风险的模型的模型,通过多项学习模型的模型,包括:我们的重要的升级模型的模型,我们学习方法,通过许多的模型的模型的模型的模型,通过一系列,通过一系列,通过一系列,通过一系列的模型进行,我们的许多学习方法,通过一系列的模型的跨级的模型,我们提供了多项学习的模型的模型的模型的模型的模型的模型的模型的模型的模型的模型的模型,通过学习。