nmfkc.ard: Automatic relevance determination for NMF rank (experimental)

In nmfkc: Non-Negative Matrix Factorization with Kernel Covariates

Automatic relevance determination for NMF rank (experimental)

Description

Prototype of Tan & Fevotte's (2013) ARD-NMF (Euclidean / \beta = 2). Unlike the cross-validation (nmfkc.ecv, nmfkc.bicv) and stability (nmfkc.consensus) engines that scan a range of ranks, ARD fits NMF once at an over-complete rank and prunes automatically: each component k carries a relevance weight \lambda_k with an inverse-gamma prior; the multiplicative updates gain a penalty + w_{fk}/\lambda_k (L2 / half-normal prior) or + 1/\lambda_k (L1 / exponential), which drives unsupported components to zero. The number of surviving components is the estimated rank. Covariates are ignored (plain NMF).

This is a model-based point estimate: the result depends on the prior, the starting rank and the random initialization, and can vary run to run. Use it as a complement to the CV / consensus engines, not a sole criterion.

Usage

nmfkc.ard(Y, rank = NULL, nrun = 10, plot = FALSE, ...)

Arguments

Y	Observation matrix (F \times N), non-negative.
rank	Over-complete starting rank K (must exceed the true rank). NULL (default) uses min(F, N, 20).
nrun	Number of random-initialization restarts (default 10). ARD is a sensitive point estimate, so several restarts are advisable; the reported rank is the mode of the per-run estimates (rank.runs), with a representative modal fit kept for plot/W/H.
plot	Logical; draw the relevance bar plot.
...	Advanced options, rarely needed (defaults in parentheses): prior ("L2": half-normal / squared-energy group shrinkage, "L1": exponential / sparser); seed (123, random-initialization seed); a (1) and b ((F + N)/K * mean(Y)), the inverse-gamma prior (a smaller b over-prunes, a larger one prunes nothing); maxit (3000) and epsilon (1e-6) for optimisation control; tol (1e-3), the relevance threshold below which a component is counted as pruned.

Details

Relation to Tan & Fevotte (2013). The update equations reproduce the paper's \ell_2 / \ell_1 ARD-NMF for the Euclidean (\beta = 2) case exactly: the multiplicative penalties, the closed-form \lambda_k update (f(w_k) + f(h_k) + b)/c and the constant c (their Eq. 33). Two deliberate simplifications keep it practical: only \beta = 2 is implemented (the paper covers the general \beta-divergence), and the default b is an empirical per-component energy scale (F + N)/K \cdot \bar{Y} that reliably avoids the winner-take-all collapse, rather than the paper's method-of-moments value (their Eq. 38).

Value

An object of class "nmfkc.ard": a list with rank (estimated = mode over restarts), rank.runs (the per-run estimates), relevance (representative run, descending), lambda, W, H (ordered by relevance), rank.init, prior, nrun, objfunc (final objective value of the representative run) and objfunc.iter (its per-iteration trajectory).

References

V. Y. F. Tan and C. Fevotte (2013). Automatic relevance determination in nonnegative matrix factorization with the beta-divergence. IEEE TPAMI 35(7):1592–1605. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1109/TPAMI.2012.240")}.

See Also

nmfkc.ecv, nmfkc.bicv, nmfkc.consensus, nmfkc.rank

Examples

set.seed(1)
X <- matrix(abs(rnorm(40 * 3)), 40, 3)
B <- matrix(abs(rnorm(3 * 60)), 3, 60)
ar <- nmfkc.ard(X %*% B, rank = 10)   # over-complete start
ar$rank                                # ~ 3 surviving components
plot(ar)

nmfkc documentation built on July 14, 2026, 1:07 a.m.