Alternating Minimization Concept Question

$1$ point possible $($graded$)$

As in the video above, we now want to find and that minimize our new objective

In order to break a big optimization problem into smaller pieces that we know how to solve, we fix and find the best for that $.$ But a subtle and important point is that even if is best for $,$ then the that is best for might not be the original $!$ It's like how we might be some lonely person's best friend, even though they are not our best friend. In light of this, we repeat, like this: we fix and solve for $,$ then fix to be the result from the previous step and solve for $,$ and repeat this alternate process until we find the solution. This is an example of iterative optimization, where we greedily take steps in a good direction, but as we do so the context shifts so $$the good direction' evolves. Gradient descent, which we've already seen, has the same structure.

Consider the case $.$ The matrices and reduce to vectors and such that and $.$

When is fixed, finding that minimizes becomes equivalent to finding that minimizes $...$

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