Reference

• https://convex-optimization.github.io/
• http://www.cs.yale.edu/homes/vishnoi/Publications_files/approx-survey.pdf
• passwd:dohwAj-serkiw-4gamny
• http://www.ifp.illinois.edu/~angelia/
• https://sites.google.com/view/philpare
• https://cauribe.rice.edu/elec570/

Convex Factorization Machine

• https://mblondel.org/
• https://mblondel.org/publications/mblondel-ecmlpkdd2015.pdf

Convex Matrix Factorization

• https://www.quora.com/Why-is-the-matrix-factorization-optimization-function-in-recommender-systems-not-convex

The reason is that it is a non-constant function with more than one global minima. 
A function like this is highly likely to be nonconvex.

Say you observe a m*n matrix A and you wanna factorize it into UV^t, where U is m*r and V is n*r. 
Suppose there is some U_0 and V_0 that attains the minimum of our objective \|A - UV^t\|_F^2, then U_0*Q and V_0*Q is also a minimum of the same objective for any r*r orthogonal matrix Q, since (U_0Q)(V_0Q)^t = U_0(QQ^t)V_0 = U_0V_0. 
If the objective function is convex, then it must be constant on the line segment connecting these two points, where it actually isn’t.

In principle, a function that is permutation / rotation invariant is usually nonconvex. 
However, the above objective is bi-convex in (U,V) so people often use alternating minimization algorithm to solve it.