On minimizing matrix norm (AB-C)
In general, you can't. You can find a matrix that will minimize $||\tilde{A}^TB-C||$, and another that minimizes $||\tilde{A}-A||$, but, in general, the two matrices you find won't be the same. In fact, the matrix that solves the second problem is always $\tilde{A} = A$, and, if $||{A}^TB-C||$ is not the minimum of the first function, then your problem doesn't have a solution.

However, what would a machine learning practicioner do? Instead of finding a matrix that solves both of the problems (that might not exist), you can aim to solve a combination of the two problems. You can aim to find the matrix that minimizes
$$||\tilde{A}^TB-C||^2 + ||\tilde{A}-A||^2
$$
or, in general, the matrix that minimizes 
$$||\tilde{A}^TB-C||^2 + \lambda ||\tilde{A}-A||^2
$$
for some $\lambda &gt;0$. If $\lambda \approx 0$, then you are just solving the first problem, and if $\lambda &gt;&gt;0$, then you are just solving the second one.

And, this is just a quadratic optimization problem, that can be solved using gradient descent and will provide a unique global maximum.