Select the bandwidth of random-feature covariates by cross-validation

Chooses the Gaussian-kernel bandwidth \(\beta\) (and optionally the number of random features \(D\)) for Random Fourier Feature covariates by cross-validation, in the same way that nmfkc.kernel.beta.cv does for exact kernel covariates. For every candidate the random features are regenerated on the columns of U and the model is cross-validated column-wise:

• type = "signed": cosine features from nmfkc.signed.rff (signed), fitted with nmfkc.signed through nmfkc.signed.cv;

• type = "positive": positive random features from nmfkc.rff.positive (non-negative), fitted with nmfkc through nmfkc.cv.

The candidate with the smallest cross-validated prediction error is returned. Since the same random map cannot be shared across folds without leaking (the map is data-independent, so it can: only the fit is cross-validated), one draw per candidate is used and its seed is fixed so the comparison across \(\beta\) is paired.

For large \(N\) set sample.size: the selection then runs on a random subsample of the columns (default: all columns), after which the chosen \(\beta\) is used with nmfkc.signed.rff.gram / nmfkc.rff.positive.gram on the full data.

Usage

nmfkc.rff.beta.cv(
  Y,
  rank = 2,
  U,
  beta = NULL,
  D = 500L,
  type = c("signed", "positive"),
  plot = TRUE,
  ...
)

Arguments

Y

Response matrix \(P \times N\) (non-negative for type = "positive").

rank

Number of bases \(Q\).

U

Input matrix \(p \times N\); columns are samples.

beta

Numeric vector of candidate bandwidths. Default NULL: the seven-point grid of nmfkc.kernel.beta.nearest.med around the median nearest-neighbour heuristic (computed on the subsample when sample.size is set).

D

Integer vector of candidate numbers of random features. A single value (default 500) selects \(\beta\) only; several values select over the \((\beta, D)\) grid.

type

"signed" (cosine RFF, default) or "positive".

plot

Logical. If TRUE (default), plot the CV objective against beta (one line per D).

...

Passed to the CV function (nmfkc.signed.cv or nmfkc.cv: e.g. nfolds, epsilon, maxit). Also accepts:

• sample.size: number of columns used for the selection (default all).

• seed: integer seed for the subsample and the random maps (default 123; the caller's stream is restored).

• cores: evaluate the candidates in parallel with .nmfkc.parlapply (default getOption("mc.cores", 1L)). Each candidate is deterministic given its index, so the result is identical for any cores.

• hyperbolic: passed to nmfkc.rff.positive.

Value

A list with

beta, D

The selected bandwidth and feature count.

objfunc

Matrix of CV objective values, length(beta) rows by length(D) columns.

beta.candidates, D.candidates

The grids.

type, sample.size, n.used

Settings used.

Lifecycle

This function is experimental. The interface may change in future versions.

See also

nmfkc.kernel.beta.cv (exact kernel; also works for Nystr\"om covariates with U = landmarks, V = data), nmfkc.signed.rff.gram, nmfkc.rff.positive.gram

Examples

# \donttest{
data(iris)
set.seed(1)
idx <- sample(nrow(iris), 100)
U <- t(scale(iris[idx, 1:4]))
levs <- levels(iris$Species)
Y <- sapply(iris$Species[idx], function(s) as.integer(levs == s)); rownames(Y) <- levs

## signed RFF: choose beta over the median-heuristic grid at D = 50
sel <- nmfkc.rff.beta.cv(Y, rank = 3, U, D = 50, plot = FALSE)
sel$beta
#> [1] 0.2681876
g   <- nmfkc.signed.rff.gram(Y, U, beta = sel$beta, D = 50, seed = 1)
fit <- nmfkc.signed(Y, A = g, rank = 3, verbose = FALSE)

## positive RFF: select over a (beta, D) grid
sel2 <- nmfkc.rff.beta.cv(Y, rank = 3, U, D = c(100, 400), type = "positive",
                          plot = FALSE, verbose = FALSE)
#> Warning: candidate beta = 577.8, D = 100 failed (missing value where TRUE/FALSE needed); treated as Inf.
#> Warning: candidate beta = 124.5, D = 100 failed (missing value where TRUE/FALSE needed); treated as Inf.
#> Warning: candidate beta = 26.82, D = 100 failed (missing value where TRUE/FALSE needed); treated as Inf.
#> Warning: candidate beta = 577.8, D = 400 failed (missing value where TRUE/FALSE needed); treated as Inf.
#> Warning: candidate beta = 124.5, D = 400 failed (missing value where TRUE/FALSE needed); treated as Inf.
#> Warning: candidate beta = 26.82, D = 400 failed (missing value where TRUE/FALSE needed); treated as Inf.
sel2$objfunc
#>                D=100      D=400
#> 577.79268        Inf        Inf
#> 124.48166        Inf        Inf
#>  26.81876        Inf        Inf
#>   5.77793 0.40283493 0.60412928
#>   1.24482 0.24717367 0.55493823
#>   0.26819 0.09430055 0.06380387
#>   0.05778 0.07752821 0.07436292
# }