Hey! Out of curiosity, what is the limitation that prevents imate from working with products of rectangular matrices?
For any real-valued matrix $A \in \mathbb{R}^{n \times m}$ and positive semi-definite matrix $P \in \mathbb{R}^{n \times n}$ given by $P = A A^T$ (or $P = A^T A)$, I would like to estimate various Schatten norms of both $A$ and $P$, e.g.
from math import prod
import numpy as np
from scipy.sparse import random
from scipy.sparse.linalg import svds
from imate import schatten, trace
## Generate random semi-sparse matrix
n = 500
A = random(n, 40, density=0.050)
P = A @ A.T
## Ground truth
sv = np.sort(np.abs(svds(A, k = min(A.shape)-1)[1]))
ev = np.sort(np.linalg.eigh(P.todense())[0])
print(f"Density of P: {P.nnz/prod(P.shape)}")
print(f"(A) Schatten-1 / nuclear norm: {np.sum(sv[sv > 1e-16])}")
print(f"(A) Schatten-2 / frobenius norm: {np.sum(sv[sv > 1e-16]**2)**(1/2)}")
print(f"(P) Schatten-1 / nuclear norm: {np.sum(ev[ev > 1e-16])}")
print(f"(P) Schatten-2 / frobenius norm: {np.sum(ev[ev > 1e-16])**(1/2)}")
# Density of P: 0.096224
# (A) Schatten-1 / nuclear norm: 110.72781488837865
# (A) Schatten-2 / frobenius norm: 18.074377305479523
# (P) Schatten-1 / nuclear norm: 330.2851529166273
# (P) Schatten-2 / frobenius norm: 56.81765769772352I can get the Schatten-norms out of $P$ with imate:
schatten(P, p=1.0, method="exact")*P.shape[0]
schatten(P, p=2.0, method="exact")*P.shape[0]**(1/2)
# 330.28515291662717
# 56.81765769772352s
But I also want the norms for $A$. Initially I thought to use that fact that the singular values of $A$ are the square roots of the eigenvalues of $P$. This works well enough using numpy:
print(f"(A) Schatten-1 / nuclear norm, derived from P: {np.sum(np.sqrt(ev[ev > 1e-16]))}")
print(f"(A) Schatten-2 / frobenius norm, derived from P: {np.sum(ev[ev > 1e-16])**(1/2)}")
# (A) Schatten-1 / nuclear norm, derived from P: 112.62572173011456
# (A) Schatten-2 / frobenius norm, derived from P: 18.173749005547183But when I try this for A via the gramian approach I get an error:
schatten(A, gram=True, p=1.0, method="slq")
# ValueError: Input matrix should be a square matrix.
EDIT: Actually the Schatten-p norm of a general matrix expressed as a trace quantity is given by 3rd equation in section 2.3 of Ubaru's paper:
$$ \lVert X \rVert_p = \left(\sum\limits_{i=1}^r \sigma_i^p \right)^{1/p} = \left(\sum\limits_{i=1}^r \lambda_i^{p/2} \right)^{1/p}$$
This doesn't seem capturable by imate's definition of the Schatten norm. Am I missing something? Or is this particular matrix function just yet to be implemented in imate?
Hey! Out of curiosity, what is the limitation that prevents
imatefrom working with products of rectangular matrices?For any real-valued matrix$A \in \mathbb{R}^{n \times m}$ and positive semi-definite matrix $P \in \mathbb{R}^{n \times n}$ given by $P = A A^T$ (or $P = A^T A)$ , I would like to estimate various Schatten norms of both $A$ and $P$ , e.g.
I can get the Schatten-norms out of$P$ with
imate:But I also want the norms for$A$ . Initially I thought to use that fact that the singular values of $A$ are the square roots of the eigenvalues of $P$ . This works well enough using
numpy:But when I try this for
Avia the gramian approach I get an error:EDIT: Actually the Schatten-p norm of a general matrix expressed as a trace quantity is given by 3rd equation in section 2.3 of Ubaru's paper:
This doesn't seem capturable by imate's definition of the Schatten norm. Am I missing something? Or is this particular matrix function just yet to be implemented in
imate?