percoordinate

Percoordinate is a concept describing operations that treat each coordinate of a vector independently. In mathematics and numerical analysis, a per-coordinate approach updates one component at a time, holding the others fixed, or applies a function independently to each coordinate.

In optimization, percoordinate methods include coordinate descent, where the objective is minimized with respect to a single coordinate at a time, cycling through coordinates until convergence. They are well suited for high-dimensional problems with separable structure and can exploit sparsity.

Variants include per-coordinate gradient updates, per-coordinate learning rates, and per-coordinate preconditioning, such as diagonal matrices that scale coordinates separately. These ideas also appear in numerical linear algebra when applying row- or column-wise transformations.

Advantages of percoordinate methods include low per-iteration cost, simplicity, and good performance on sparse or separable problems. Limitations include potentially slow convergence on ill-conditioned problems, sensitivity to coordinate ordering, and reduced effectiveness when coordinates interact strongly.

See also: coordinate descent; block coordinate descent; diagonal preconditioner.