I am a computer science research student working in application of Machine Learning to solve Computer Vision problems.
Since, lot of linear algebra(eigenvalues, SVD etc.) comes up when reading Machine Learning/Vision literature, I decided to take a linear algebra course this semester.
Much to my surprise, the course didn't look at all like Gilbert Strang's Applied Linear algebra(on OCW) I had started taking earlier. The course textbook is Linear Algebra by Hoffman and Kunze. We started with concepts of Abstract algebra like groups, fields, rings, isomorphism, quotient groups etc. And then moved on to study "theoretical" linear algebra over finite fields, where we cover proofs for important theorms/lemmas in the following topics:
Vector spaces, linear span, linear independence, existence of basis.
Linear transformations. Solutions of linear equations, row reduced
echelon form, complete echelon form,rank. Minimal polynomial of a
linear transformation. Jordan canonical form. Determinants.
Characteristic polynomial, eigenvalues and eigenvectors. Inner product
space. Gram Schmidt orthogonalization. Unitary and Hermitian
transformations. Diagonalization of Hermitian transformations.
I wanted to understand if there is any significance/application of understanding these proofs in machine learning/computer vision research or should I be better off focusing on the applied Linear Algebra?
Best Answer
If you want to do advanced computer vision, and not just implement algorithms, you will need to understand advanced algebraic concepts for linear transformations. You will also need to understand a bit of measure theory and analysis.
Why?
Because research level computer vision involves the development of algorithms. The development of these algorithms necessarily invokes the structural properties of the mathematical objects; properties such as measure, convergence, isometry, isomorphism, etc.
Furthermore, say you have the mechanical skills to develop a computational method. Any true research-level effort is also expected to demonstrate a proof of convergence, establish a domain in which the method is efficacious, compare the method to prior methods, and fundamentally compare the weaknesses and benefits.
This requires at least a solid understanding of graduate-level analysis and linear algebra.