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NmfBetaLoss

What Nmf.Fit is asked to minimise, and therefore what "a good factorization" means for the data in hand.

The beta divergence is one family with a parameter; this package ships the two members of it that solver="mu" computes in closed form. They are not two ways of reaching one answer. Each is the maximum-likelihood fit under a different noise model, so they disagree about which errors are worth trading against which — and a matrix of counts and a matrix of measurements do not want the same trade.

Members

Member	Value	What it does
Frobenius	0	Half the squared Frobenius norm of X − W H, β = 2. The Gaussian noise model, and what a matrix of continuous measurements wants. scikit-learn's beta_loss="frobenius".
KullbackLeibler	1	The generalised Kullback–Leibler divergence, β = 1. The Poisson noise model, and what counts want — a term-document matrix included, which is why it is here. scikit-learn's beta_loss="kullback-leibler".

The choice changes the arithmetic and not only the objective. Frobenius updates W through two dense products; Kullback–Leibler divides X by W H where X is non-zero and broadcasts a row of sums as the denominator, which is why it is the slower of the two on a wide matrix and why it snaps H below machine epsilon to zero where Frobenius does not.

ReconstructionError is reported in the loss that produced it, so numbers from two different members are not comparable — a Kullback–Leibler fit of a corpus is routinely the "larger" of the two and is not the worse one. Compare a loss against itself, across ranks or across initialisations, and never across this enum.

Other β values — Itakura–Saito at β = 0, or anything between — are not offered. They cost a general power in the inner loop, and neither of the two data shapes this package targets asks for one.