RFC: adopt new vector and matrix norm functions

[carlosgmartin]

Proposal:

Deprecate the following functions:

• norm
• vector_norm
• matrix_norm

Replace them with the following separate functions:

• p_norm(x, p): Computes the p-norm: $\|x\|_p = (\sum_{i < d} |x_i|^p)^{1/p}$.
• induced_norm(x, p, q): Computes the induced (p,q)-norm: $\|A\|_{p, q} = \sup_{\|x\|_p \leq 1} \|A x\|_q$.
  • $p = q = 2$ yields the spectral norm.
• entrywise_norm(x, p, q): Computes the entrywise (p,q)-norm: $\|A\|_{p, q} = \left( \sum_{j < n} \left( \sum_{i < m} |A_{ij}|^p \right)^{q/p} \right)^{1/q}$.
  • $p = q = 2$ yields the Frobenius norm.
• schatten_norm(x, p): Computes the Schatten p-norm: $\|A\|_p = \left( \sum_{i < \min\{m, n\}} \sigma_i(A)^p \right)^{1/p}$.
  • $p = 1$ yields the nuclear norm.
  • $p = 2$ yields the Frobenius norm.
  • $p = \infty$ yields the spectral norm.

Notice that the same value of p (or q) can yield different results, depending on which norm one is talking about.

Optionally, add the following convenience wrappers:

• nuclear_norm(x): Computes the nuclear norm.
• frobenius_norm(x): Computes the Frobenius norm.
• spectral_norm(x): Computes the spectral norm.
• spectral_radius(x): Computes the spectral radius (not actually a norm, but also useful).

Rationale:

Currently, norm and matrix_norm conflate different norms into a single function. This is apparently a design mistake inherited from MATLAB. Such a design (1) creates unnecessary confusion and (2) prevents access to alternative values of p and q for each kind of norm. It would be nice if the standard rectified this situation.

Also, for vectors, the ord=0 "0-norm", which counts the number of nonzero elements, is a misnomer: It is not actually the limit of the p-norm as $p \to 0$. To count the number of nonzero elements, count_nonzero should be used instead.

[kgryte]

@carlosgmartin Thank you for opening this proposal. I think the ship has sailed on this one, unfortunately.

While I personally agree that separate APIs would be better, I think it highly unlikely that we'd deprecate the existing norm APIs in the specification. It was already a battle to split norm into separate matrix_ and vector_ APIs (ref: #213 and earlier workgroup discussions), and array libraries are unlikely to be enthusiastic about adding even more functions to their namespaces.

As such, I'll go ahead and close this issue, as I don't believe there is much we can do atm to move this forward.