Exact CP-ALS block updates do not increase the objective

Fix a successful mode-$k$ update and keep all other factor matrices fixed. The residual objective as a function of the replaced factor $X$ is $\mathrm{\parallel }{A}_{(k)}-X{Z}_k^{\mathsf{T}}{\mathrm{\parallel }}_F^2$. Taking adjoints preserves the Frobenius norm and rewrites it as $\mathrm{\parallel }{A}_{(k)}^{\ast }-\overline{Z_k}{X}^{\ast }{\mathrm{\parallel }}_F^2$. Since $Z_k$ has full column rank, so does $\overline{Z_k}$; [F2] therefore makes the algorithm's update the unique minimizer of this block objective. The old factor is an admissible candidate, so the updated objective is at most the old one.

A rank-deficient-design branch stops before changing a factor, so the objective is identical before and after that branch. Every other executed update satisfies step 1.1; chaining these finitely many inequalities across any completed part of a sweep proves the objective is nonincreasing through all successful exact block updates. This asserts no convergence or global optimality.