We are going to follow a top-down approach here and discuss the applications first. I have explained the math behind SVD after the applications for those interested in how it works underneath.
You just need to know four things to understand the applications:
SVD is the decomposition of a matrix A into 3 matrices – U, S, and V
S is the diagonal matrix of singular values. Think of singular values as the importance values of different features in the matrix
The rank of a matrix is a measure of the unique information stored in a matrix. Higher the rank, more the information
Eigenvectors of a matrix are directions of maximum spread or variance of data
How many times have we faced this issue? We love clicking images with our smartphone cameras and saving random photos off the web. And then one day – no space! Image compression helps deal with that headache.
It minimizes the size of an image in bytes to an acceptable level of quality. This means that you are able to store more images in the same disk space as compared to before.
Image compression takes advantage of the fact that only a few of the singular values obtained after SVD are large. You can trim the three matrices based on the first few singular values and obtain a compressed approximation of the original image. Some of the compressed images are nearly indistinguishable from the original by the human eye.