Singular value Decomposition (SVD) versus Schmidt Decomposition (SD) :

sinbad989 (52)in #steemstem • 8 years ago

Image Compression with SVD

The singular value decomposition (SVD) is not only a classical theory in matrix computation and analysis but also is a powerful tool in machine learning and modern data analysis. The image above is an example of the application of an SVD in image compression.

What is a singular value decomposition? Its definition comes in many forms depending on your domain of application.

Singular value Decomposition:

A factorization of a real or a complex matrix (quantum mechanics)
A factorization of A into the product of three matrices A = UDV* where the columns of U and V are orthonormal and the matrix D is diagonal with real positive entries. (computer science)
The generalization of the eigendecomposition of a positive semidefinite normal matrix (mathematics)

Derivation of reduced singular value decomposition

Application:

Signal processing
Statistics
Image Compression
Cross-references in Google search engines

The most amazing fact about all matrices is the following theorem.

Every matrix A has singular value decomposition.

It has the following characteristics:

Singular values are uniquely determined, if A is square singular values are distinct
Basis components of U and V are also unique up to a complex sign

Application of Schmidt Decomposition in Quantum Information

Schmidt Decomposition:

It is just a singular value decomposition in disguise
A particular way of expressing a vector in the tensor product of two inner product spaces
A restatement of the singular value decomposition in a different context

Most properties of Schmidt decomposition are of physical interest. It's no surprise to see that most of its application are in physical science, mostly in quantum information.

Application: