Data dimensionality Reduction--low rank recovery

Data dimensionality Reduction--low rank recovery

In the actual signal or image acquisition and processing, the higher the dimension of the data, the greater the limit to the data collection and processing. For example, it is often difficult to acquire signals in three-dimensional or four-dimension (three spatial dimensions plus one spectral dimension or one time dimension). However, with the increase of data dimensionality, there are often more correlations and redundancy between these high-dimensional data. The higher the dimension of the signal, the more redundant the data becomes because the increase in the amount of information in the data itself is much slower than the data dimension. One obvious example is that the video signal is much larger than the compressible space of a single image.

For example, for an image, the correlation between pixels is sparse distributed in the image in a transform domain, and how to reasonably and fully utilize the sparsity and redundancy of high dimensional data is very important for collecting, representing and reconstructing these data efficiently.

The more challenging problem is that these large-scale data often contain vacancy elements, large errors, and damage, which further poses difficulties in analysing and processing these large-scale data. This phenomenon is very common in many practical applications. For example, in face recognition, the face images that are trained or to be recognized often contain shadows, highlights, occlusion, deformation, etc. in the motion recovery structure (Structure frommotion, SFM), there are often large matching errors in feature extraction and feature matching.

Sparsity is the point amount or most elements of the matrix is 0, the low rank of the matrix refers to the matrix's rank relative to the number of rows or columns of the matrix is very small. If the matrix is decomposed by singular value and all its singular values are arranged as a vector, then the sparsity of the vector corresponds to the low rank of the matrix.

The low rank can be regarded as the extension of sparsity on the matrix, and the matrix rank minimization mainly refers to the reconstruction of the matrix by using the low rank of the original data matrix, which involves minimizing the rank function of the matrix. The restoration of low rank matrix means that the data matrix is recovered by using the low rank of the original data matrix and the sparsity of the error matrix.

A typical application of matrix rank minimization is the low-rank matrix fill (Low-rankmatrix completion)

problem : The original data matrix is assumed to be low-rank, but the matrix contains many unknown elements. Recovering a complete low-rank matrix from an incomplete matrix is a low-rank matrix filling problem.

For example , the famous Netflix problem is a typical low-rank matrix fill problem. Netflix is a film leasing company in the United States. The recommended system (recommendation systems) is to recommend videos to users from only a handful of movie scores. If the recommendation is more in line with the user's preferences, the more it can improve the company's business of renting movies. To this end, the company has set up millions of dollars to reward the best way to improve the quality of the company's recommendation system. This problem can be modeled with a matrix fill, assuming that each row of the matrix represents the same user rating for different movies, and each column represents a different user's rating for the same movie. The number of users is huge, the number of movies is huge, so the dimension of this matrix is very large. Due to the limited number of films scored by users, only a small fraction of the element values in this matrix are known and may contain noise or errors. So the problem with Netflix is how to infer the values of unknown elements from this incomplete matrix. The more accurate The matrix fills, the more the film is recommended for users to suit their preferences. Due to the limited number of factors affecting the user's interest in the film, such as film themes, actors, eras, directors, etc., this matrix is essentially a low-rank matrix.

For details, please refer to the article: the Theory and application of low rank matrix recovery _ from compression transmission. pdf

Review of low-rank recovery algorithms. pdf

Research progress and prospect of compression perception and its application in image processing. pdf

Data dimensionality Reduction--low rank recovery