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Generalized PLS-SVD via Implicit Operator (memory-safe)

Source: R/gplssvd_op.R
gplssvd_op.Rd

Compute the top-k singular triplets of \(S = Xe' Ye\) without materializing the whitened matrices \(Xe = Mx^{1/2} X Wx^{1/2}\), \(Ye = My^{1/2} Y Wy^{1/2}\) when doing so would densify sparse data. When the whitening is sparsity-preserving (identity/diagonal metrics) or the data are dense, the whitened blocks are precomputed once so each matrix-vector product in the iterative SVD costs two multiplies.

Usage

gplssvd_op(
  X,
  Y,
  XLW = NULL,
  YLW = NULL,
  XRW = NULL,
  YRW = NULL,
  k = 2,
  center = FALSE,
  scale = FALSE,
  svd_backend = c("eigencore", "irlba", "RSpectra"),
  svd_opts = list(tol = 1e-07, maxitr = 1000),
  constraints_remedy = c("error", "ridge", "clip", "identity")
)

Arguments

X

n x I matrix (numeric or Matrix)

Y

n x J matrix (numeric or Matrix)

XLW

Row metric for X (M_X): NULL/identity, numeric length-n, diagonalMatrix, or PSD Matrix

YLW

Row metric for Y (M_Y)

XRW

Column metric for X (W_X)

YRW

Column metric for Y (W_Y)

k

Number of components. If k exceeds min(ncol(X), ncol(Y)), a warning is issued and k is silently truncated to that maximum.

center, scale

Logical; pre-center/scale columns of X, Y before metrics

svd_backend

One of "eigencore" (default) or "irlba"; "RSpectra" is accepted as a deprecated alias of "eigencore". Ignored whenever both ncol(X) <= 64 and ncol(Y) <= 64, in which case S is materialized densely and solved with base::svd().

svd_opts

List of options for the backend: tol (both backends) and maxitr (irlba only; the eigencore partial SVD has no iteration cap). An incomplete eigencore solve raises genpca_solver_nonconvergence; try a less stringent tol if the requested accuracy cannot be reached.

constraints_remedy

What to do with a metric that is not positive semi-definite: "error" (default), "ridge", "clip" or "identity"; repairs emit a genpca_metric_repaired warning. See genpca().

Value

A list with elements:

d

Length-k numeric vector of singular values of \(S = Xe' Ye\).

u

I x k matrix; left singular vectors of S (orthonormal in the Euclidean metric).

v

J x k matrix; right singular vectors of S (orthonormal in the Euclidean metric).

p

I x k matrix of generalized X-weights, \(p = W_X^{-1/2} u\).

q

J x k matrix of generalized Y-weights, \(q = W_Y^{-1/2} v\).

fi

I x k matrix of X-variable scores, \(F_i = W_X p D\) (columns of p rescaled by the singular values).

fj

J x k matrix of Y-variable scores, \(F_j = W_Y q D\).

lx

N x k matrix of X row latent variables, \(L_x = M_X^{1/2} X W_X p\).

ly

N x k matrix of Y row latent variables, \(L_y = M_Y^{1/2} Y W_Y q\).

k

Integer; number of components actually returned (may be less than the requested k if it exceeded min(I, J)).

dims

A list list(N, I, J) with the row count N and column counts I = ncol(X), J = ncol(Y).

center

A list list(X, Y) of the length-I / length-J column means subtracted from X/Y (all zero when center = FALSE).

scale

A list list(X, Y) of the length-I / length-J column scale factors divided out of X/Y (all one when scale = FALSE).

Details

Naming map: this function names its metrics XLW/YLW (left/row weights) and XRW/YRW (right/column weights); these correspond to Mx/My (row metrics) and Ax/Ay (column metrics) in genpca()'s and genpls()'s M/A convention.

References

Beaton, D. (2020). Generalized eigen, singular value, and partial least squares decompositions: The GSVD package. arXiv:2010.14734.

Abdi, H. (2007). Partial least square regression PLS-Regression. In N. Salkind (Ed.), Encyclopedia of Measurement and Statistics. Thousand Oaks, CA: Sage.

Examples

set.seed(1)
X <- matrix(rnorm(40 * 6), 40, 6)
Y <- matrix(rnorm(40 * 4), 40, 4)
op <- gplssvd_op(X, Y, k = 2, center = TRUE)
round(op$d, 3)
#> [1] 20.428 14.384