Quantic Tensor Train Decomposition And Its Application On Feature Dimension Reduction

With the advent of the era of big data, the tensor decomposition methods have attracted the attention of many scholars, and have gradually become a hot topic in the field of scientific computing, and have developed rapidly in recent decade. Tensor train (TT) and quantic tensor train (QTT) decomposition method have become very effective tools for dimension reduction of high dimensional data, and have been widely used in numerical solution of PDE, algorithm acceleration and signal processing etc.. In this dissertation, we mainly study the related theory and applications of the QTT decomposition.Firstly, we define the layer tensor, which is useful in the recommendation sys-tem, then extend the definition of QTT decomposition, deduction the equivalence relation of QTT and TT decomposition, and give some common tensor operation rules based on it. We have proved that these tensor operations (such as addition, tensor multiplication, Hadamard product. Kronecker product and matrix trans-pose) can be transferred to the corresponding operations of QTT cores. So that it can be realized in parallel. The concept of layer tensor makes the proof and presentations about tensor decomposition more intuitively and concisely. So it is more beneficial to deal with complex problems.Secondly, we discuss the construction methods of some common tensors on the explicit QTT representation, and an conquer algorithm for constructing the QTT-approximation with appointed accuracy is given, which is suitable for parallel computing and also improves the stability of the decomposition process.Finally, we discuss the relationship and differences between QTT and wavelet transformation and the convolution in structure, then apply it to the denoising and edge detection of MRI images. Numerical experiments have shown that QTT decomposition is a good feature extraction tool.