Random projections reduce the dimension of a set of vectors while preserving structural information, such as distances between vectors in the set. This paper proposes a novel use of row-product random matrices in random projection, where we call it Tensor Random Projection (TRP). It requires substantially less memory than existing dimension reduction maps. The TRP map is formed as the Khatri-Rao product of several smaller random projections, and is compatible with any base random projection including sparse maps, which enable dimension reduction with very low query cost and no floating point operations. We also develop a reduced variance extension. We provide a theoretical analysis of the bias and variance of the TRP, and a non-asymptotic error analysis for a TRP composed of two smaller maps. Experiments on both synthetic and MNIST data show that our method performs as well as conventional methods with substantially less storage.
翻译:随机预测会减少一组矢量的尺寸,同时保存结构信息,例如集中矢量之间的距离。本文提议在随机预测中新使用行产品随机矩阵,我们称之为“Tensor随机投影(TRP) ” 。它要求的内存大大少于现有尺寸减少地图。 TRP地图是几个较小随机预测的Khatri-Rao产物,它与任何基本随机预测相容,包括稀少的地图,它能够以非常低的查询成本和没有浮动点操作来降低尺寸。我们还开发了一个缩小差异的扩展。我们对TRP的偏差和差异进行理论分析,并对由两个较小的地图组成的TRP进行非抽误分析。关于合成和MNIST数据的实验显示,我们的方法以及储存量要少得多的传统方法运行。