Squared Exponential Kernel Matrix is not Positive Definite

Question

Related to this question, but adding a nugget isn't helping. Possibly I'm just making a silly coding mistake:

x <- cbind(runif(5),runif(5))
d <- dist(x)
k <- exp(-d^2)
chol(k)
# Error in chol.default(k) :
#    the leading minor of order 1 is not positive definite

Adding a nugget doesn't help:

k <- as.matrix(k)
diag(k) <- diag(k) + 0.1
chol(k)
# Error in chol.default(k) :
#   the leading minor of order 1 is not positive definite

What am I doing wrong here with my squared exponential kernel so that it isn't positive definite?

Answer 1

> as.matrix(k)
          1         2         3         4         5
1 0.0000000 0.8305698 0.6825535 0.8021312 0.7282046
2 0.8305698 0.0000000 0.8834873 0.6476372 0.6340222
3 0.6825535 0.8834873 0.0000000 0.3842710 0.3584702
4 0.8021312 0.6476372 0.3842710 0.0000000 0.9839804
5 0.7282046 0.6340222 0.3584702 0.9839804 0.0000000

The dist function gives a triangular structure with no diagonal and converting it to a matrix inside chol.default gives zeroes on the diagonal. You need to fix up the diagonal.