Fast Randomized Singular Value Decomposition For Third-order Tensor

With the rapid development of science and technology,the size and type of data become larger and more complex.In order to store and process these large and complex data,there is a need to improve the form of data storage to process the data,such that it can be stored and analyzed data with less memory and less time.It is well known that gray scale photos can be stored in the form of matrices,so how about the storage of color pictures and the recognition of multiple gray scale photos ? Therefore,we introduce tensor under this background.This paper mainly introduces tensor's basic knowledge and studies the singular value decomposition and applications of the third-order tensor.we first introduce some basic knowledge of tensor and the tensor-related operations to have a basic understanding of tensor.Then the third-order tensor and some related theorems are introduced in detail.Based on the tensor product of the third-order tensor,tensor can be decomposed by Fourier transform,and the singular value decomposition algorithm of k-term truncated tensor is also introduced.So tensor's storage can be greatly reduced.Thirdly,we introduce the fast randomized singular value decomposition of a matrix,and put forward tensor's idea,as well as a concrete algorithm of fast randomized singular value decomposition based on this and tensor singular value decomposition.At the same time,the expected error and computational complexity of the algorithm are analyzed and given.Finally,experiments on four sets of data are carried out according to our algorithm and error analysis.This picture is a color photo selected from the network(size is 629 × 800 × 3),and then we do the k-term truncated tensor singular value decomposition and tensor fast random singular value decomposition for the photo under different k values.It is found that when k = 50,the image is basically the same as the original one,which reduces the size of the image to less than1/3 of the original amount of storage.Using the same image,we add random noises to it and decompose it with two algorithms.When we take k = 50,we can find that we have done a preliminary de-noising on the picture.In this paper,tensor's fast randomized singular value decomposition(SVD)is also used to carry out the experiment of face recognition.We select two data sets.One is Cropped Yale Face B data set,which includes 1140 pictures of different lighting conditions,each picture has 192 × 168 pixels and can be expressed as a 192 × 1140 × 168 tensor;The other is the ATT data set,which includes 400 images of 40 people with different expressions,each of which is a pixel of 112 × 92 and can be represented as a 112 × 400 × 92 tensor.We give the concrete process of using tensor singular value decomposition to deal with these tensors.By using different truncation terms and power iterations,we compare the results and find that the two algorithms have the same effect.The calculation time of our tensor's fast singular value decomposition(SVD)has been greatly improved.