selectA: Select a random embedding matrix

In mbinois/RRembo: Random EMbedding Bayesian Optimization for R

Select a random embedding matrix

Description

Select a random embedding matrix

Usage

selectA(
  d,
  D,
  type = "isotropic",
  control = list(n = 30, maxit = 100, maxit2 = 10)
)

Arguments

d	small dimension
D	high dimension
type	method of random sampling of coefficients or selection procedure, one of 'Gaussian' for standard Gaussian i.i.d. coefficients and orthonormalization 'isotropic' for a random matrix with equal row norms and orthonormal columns. It is obtained starting with a random Gaussian i.i.d. matrix, then alternating normalization of rows and orthonormalization of columns. 'optimized' for known optimal solutions (e.g., d = 2) or using a potential. Considering each row as a point on the d-hypersphere, try to maximize the minimum distance between any two points. 'standard' for the original REMBO iid random matrix.
control	list to be passed to optim in the optimized case (d > 2)

Value

randomly selected matrix with orthogonal columns and normalized rows (except for standard)

Author(s)

Mickael Binois

References

M. Binois (2015), Uncertainty quantification on Pareto fronts and high-dimensional strategies in Bayesian optimization, with applications in multi-objective automotive design, PhD thesis, Mines Saint-Etienne.

Examples

## Example of orthogonal projections
d <- 2; D <- 6
A1 <- selectA(d, D, type = 'Gaussian')
A2 <- selectA(d, D, type = 'isotropic')
A3 <- selectA(d, D, type = 'optimized')

n <- 10000
size <- 10
Y <- size * (2 * matrix(runif(n * d), n) - 1)

Z1 <- ortProj(randEmb(Y, A1), t(A1))
Z2 <- ortProj(randEmb(Y, A2), t(A2))
Z3 <- ortProj(randEmb(Y, A3), t(A3))

par(mfrow = c(1, 3))
plot(Z1, asp = 1)
plot(Z2, asp = 1)
plot(Z3, asp = 1)

par(mfrow = c(1, 1))

mbinois/RRembo documentation built on Sept. 16, 2023, 10:15 p.m.