SPCAvRP_subspace {SPCAvRP}

Computes the leading eigenspace using the SPCAvRP algorithm for the eigenspace estimation

Description

Computes m leading eigenvectors of the sample covariance matrix which are sparse and orthogonal, using A x B random axis-aligned projections of dimension d.

Usage

SPCAvRP_subspace(data, cov = FALSE, m, l, d = 20, 
A = 600, B = 200, center_data = TRUE)

Arguments

data

Either the data matrix (p x n) or the sample covariance matrix (p x p).

cov

TRUE if data is given as a sample covariance matrix.

m

The dimension of the eigenspace, i.e the number of principal components to compute.

l

Desired sparsity level of the eigenspace (i.e. the number of non-zero rows in output$vector) (see Details).

d

The dimension of the random projections (see Details).

A

Number of projections over which to aggregate (see Details).

B

Number of projections in a group from which to select (see Details).

center_data

TRUE if the data matrix should be centered (see Details).

Details

This function implements the SPCAvRP algorithm for the eigenspace estimation (Algorithm 3 in the reference given below).

If the true sparsity level k of the eigenspace is known, use l = k and d = k.

If the true sparsity level k of the eigenspace is unknown, the appropriate choice of l can be done, for example, by running the algorithm (for any l) and inspecting the computed output importance_scores. The default choice for d is 20, but we suggest choosing d equal to or slightly larger than l.

It is desirable to choose A (and B = ceiling(A/3)) as big as possible subject to the computational budget. In general, we suggest using A = 300 and B = 100 when the dimension of data is a few hundreds, while A = 600 and B = 200 when the dimension is on order of 1000.

If center_data == TRUE and data is given as a data matrix, the first step is to center it by executing scale(data, center_data, FALSE), which subtracts the column means of data from their corresponding columns.

Value

Returns a list of two elements:

vector

A matrix whose m columns are the estimated eigenvectors.

value

An array with m estimated eigenvalues.

importance_scores

An array of length p with importance scores for each variable 1 to p.

Author(s)

Milana Gataric, Tengyao Wang and Richard J. Samworth

References

Milana Gataric, Tengyao Wang and Richard J. Samworth (2018) Sparse principal component analysis via random projections https://arxiv.org/abs/1712.05630

See Also

SPCAvRP, SPCAvRP_deflation

Examples

p <- 50 # data dimension
k1 <- 8 # sparsity of each induvidual component
v1 <- 1/sqrt(k1)*c(rep(1, k1), rep(0, p-k1)) # first principal compnent (PC)
v2 <- 1/sqrt(k1)*c(rep(0,4), 1, -1, 1, -1, rep(1,4), rep(0,p-12)) # 2nd PC
v3 <- 1/sqrt(k1)*c(rep(0,6), 1, -rep(1,4), rep(1,3), rep(0,p-14)) # 3rd PC
Sigma <- diag(p) + 40*tcrossprod(v1) + 20*tcrossprod(v2) + 5*tcrossprod(v3) # population covariance 
mu <- rep(0, p) # pupulation mean
n <- 2000 # number of observations
loss = function(u,v){
  sqrt(abs(1-sum(v*u)^2))
}
loss_sub = function(U,V){
  U<-qr.Q(qr(U)); V<-qr.Q(qr(V))
  norm(tcrossprod(U)-tcrossprod(V),"2")
}
set.seed(1)
X <- mvrnorm(n, mu, Sigma) # data matrix


spcavrp.sub <- SPCAvRP_subspace(data = X, cov = FALSE, m = 2, l = 12, d = 12,
                             A = 200, B = 70, center_data = FALSE)

subspace_estimation<-data.frame(
  loss_sub(matrix(c(v1,v2),ncol=2),spcavrp.sub$vector),
  loss(spcavrp.sub$vector[,1],v1),
  loss(spcavrp.sub$vector[,2],v2),
  crossprod(spcavrp.sub$vector[,1],spcavrp.sub$vector[,2]))
colnames(subspace_estimation)<-c("loss_sub","loss_v1","loss_v2","inner_prod")
rownames(subspace_estimation)<-c("")
print(subspace_estimation)

plot(1:p,spcavrp.sub$importance_scores,xlab='variable',ylab='w',
     main='importance scores w \n (may use to choose l when k unknown)')