QTT分解及其在高维数字信号处理上的应用

QTT decomposition and its applications to high dimensional signal process

近几年张量列(TT)和量子化张量列(QTT)分解方法被证明是一种非常有效的特征降维工具,并已广泛应用于PDE、算法加速和信号处理等领域.给出了关于QTT分解的一些新结果.首先用分块张量的方法扩展了QTT的定义,使之适用于更加复杂的降维问题.同时指出新定义的QTT分解也是一种基于流形学习的降维工具.其次讨论了QTT与小波变换和卷积在结构上的联系与区别,并指出QTT也是一种特征提取工具.最后将QTT分解应用于三维数据(MRI图像)的去噪和边缘检测,取得了不错的效果.

In recent years, tensor train （TT） and quantized tensor train （QTT） decomposition have proved to be a very effective method in feature dimension reduction, and it has been widely used in the field of PDE, algorithm acceleration and signal processing, etc. We present some new results about the QTT decomposition. Firstly, we extend the definition of the QTT through block-tensor, which can be applied to more complex dimension reduction problems. At the same time, it is pointed out that the new definition of the QTT decomposition is a dimension reduction tool based on manifold learning. Secondly, we discuss the relationship and difference between QTT and wavelet transformation and convolution in struc- ture, and point out that QTT is also a feature extraction tool. Finally, we apply the QTT decomposition to the denoising and edge detection of MRI images, and achieve good results.