what do we mean by variational model?

Question

I encountered the term in a paper where the author constrcuted a new model (that he called a variational model): $$\min_{D,W,C}\|Y-DC\|_F^2 + \gamma J(C,W,L)$$ based on a classic one: $$\min_{D,C}\|Y-DC\|_F^2 $$

Answer 1

Variational model is one, where we optimize over functions as opposed to over values. Lets take the classic case from your question $$\min_{D,C}\|Y-DC\|_F^2 $$ in which we optimize over $D$ and $C$, which are (probably) meant to be functions. I say probably, because you did not supply the article you are reading.

You can look into Euler-Lagrange equations as an introdution to variational calculus, that deals with such questions. The classic case is finding the Brachistochrone curve.

Another example of this is the expectation minimization algorithm. In contemporary deep learning, variational autoencoders work with this idea, where a neural network is used to approximate the function (as opposed to minimizing over the entire function space).