Enhancing the SVD compression losslessly

Abstract

Orthonormality is the foundation of a number of matrix decomposition methods. For example, Singular Value Decomposition (SVD) implements the compression by factoring a matrix with orthonormal parts and is pervasively utilized in various fields. Orthonormality, however, inherently includes constraints that would induce redundant information, preventing SVD from deeper compression and even making it frustrated as the data fidelity is strictly required. In this paper, we theoretically prove that these redundances resulted by orthonormality can be completely eliminated in a lossless manner. An enhanced version of SVD, namely E-SVD, is accordingly established to losslessly and quickly release constraints and recover the orthonormal parts in SVD. According to our theory, advantages of E-SVD over SVD become increasingly evident with the rising requirement of data fidelity. In particular, E-SVD will reduce 25% storage units as SVD reaches its limitation and fails to compress data. Empirical evidences from remote sensing and internet of things justify our theory by demonstrating the consistent compression superiority of E-SVD over SVD, and the additional calculation cost of E-SVD is of the same magnitude with SVD. Digital image compression experiment with typical size 375 × 500 also shows that E-SVD needs the least storage space as compared to alternative compression solutions when the rank is sufficient to maintain a good visual quality. The presented theory sheds insightful lights on the constraint solution in orthonormal matrices and E-SVD, guaranteed by which will profoundly enhance the SVD-based compression in the context of explosive growth in both data acquisition and fidelity levels.