| # Optimal set of Least-Squares via SVD Theorem: optimal set of ordinary least-squares The optimal set of the OLS problem can be expressed as where is the pseudo-inverse of , and is the minimum-norm point in the optimal set. If is full column rank, the solution is unique, and equal to In general, the particular solution is the minimum-norm solution to the least-squares problem. Proof: The following proof relies on the SVD of , and the rotational invariance of the Euclidean norm. ## Optimal value of the problem Using the SVD we can find the optimal set to the lest-squares optimization problem Indeed, if is an SVD of , the problem can be written where we have exploited the fact that the Euclidean norm is invariant under the orthogonal transformation . With , and , and changing the variable to , we express the above as Expanding the terms, and using the partitioned notations , , we obtain Since is invertible, we can reduce the first term in the objective to zero with the choice . Hence the optimal value is We observe that the optimal value is zero if and only if , which is exactly the same as . ## Optimal set Let us detail the optimal set for the problem. The variable is partly determined, via its first components: . The remaining variables contained in are free, as does not appear in the objective function of the above problem. Thus, optimal points are of the form , with , , and free. To express this in terms of the original SVD of , we observe that means that where is partitioned as , with and . Similarly, the vector can be expressed as , with formed with the first columns of . Thus, any element in the optimal set is of the form where . (We will soon explain the acronym appearing in the subscript.) The free components correspond to the degrees of freedom allowed to by the nullspace of . ## Minimum-norm optimal point The particular solution to the problem, , is the minimum-norm solution, in the sense that it is the element of that has the smallest Euclidean norm. This is best understood in the space of -variables. Indeed,the particular choice corresponds to the element in the optimal set that has the smallest Euclidean norm. Indeed, the norm of is the same as that of its rotated version, . the first elements in are fixed, and since , we see that the minimal norm is obtained with . ### Optimal set via the pseudo-inverse The matrix , which appears in the expression of the particular solution mentioned above, is nothing else than the pseudo-inverse of , which is denoted . Indeed, we can express the pseudo-inverse in terms of the SVD as With this convention, the minimum-norm optimal point is . Recall that the last columns of form a basis for the nullspace of . Hence the optimal set of the problem is When is full column rank (, and ), the optimal set reduces to a singleton, as the nullspace is . The unique optimal point expresses as