Why estimated covariance matrix by glasso is always zero?

Question

I am using glasso function from glasso package, as follow:

obj <- glasso(var(X), rho = 0.09, zero = info, approx=TRUE)

Regardless of rho value, all of entries in obj$w, estimated covariance matrix, are zero. Do you have any idea why this happens?

For your information, the dimension of var(X) is 1990 x 1990 and the number of rows in info is 1959841.

EDIT: You can download X and info variables as RData file from this link: https://www.dropbox.com/s/t9s4iw6ulbys72o/varX.RData?dl=0

Answer 1

I was having the same problem and discovered it was related to the parameter approx=TRUE, which lets glasso use an approximation method used to speed up the inversion of the covariance matrix, so the problem may be related to the original covariance matrix s not fulfilling the assumptions used for this method.

In my case my matrix wasn't that big (800x800), so I was able to fix the problem just by setting the approx parameter to FALSE and being a little patient, but I know this may not be feasible for larger matrices, so I hope someone could share a better solution for this problem.

Answer 2

I did some experimentation with the glasso function and I find that it does not return the covariance matrix w, and the inverse covariance matrix wi lacks the diagonal terms and is not symmetric.

Setting up ...

library(glasso)

n <- 20
k <- 3
x <- matrix(rnorm(n*k),ncol=k)

Since n > k, the covariance matrix can be inverted directly.

> var(x)
            [,1]        [,2]       [,3]
[1,]  1.37818760 -0.03676559 -0.0446064
[2,] -0.03676559  1.15692561 -0.4592093
[3,] -0.04460640 -0.45920931  1.1981595
> solve(var(x))
           [,1]       [,2]       [,3]
[1,] 0.72803016 0.03997524 0.04242491
[2,] 0.03997524 1.02163778 0.39304343
[3,] 0.04242491 0.39304343 0.98683155

With approx=FALSE the glasso function uses the lasso method and gets the right answer.

> glasso(var(x),rho=0,approx=FALSE)[c('w','wi')]
$w
            [,1]        [,2]        [,3]
[1,]  1.37818760 -0.03676557 -0.04460639
[2,] -0.03676557  1.15692561 -0.45920931
[3,] -0.04460639 -0.45920931  1.19815952

$wi
           [,1]       [,2]       [,3]
[1,] 0.72803010 0.03997522 0.04242489
[2,] 0.03997386 1.02163778 0.39304343
[3,] 0.04242438 0.39304343 0.98683155

Setting approx=TRUE causes the function to use the Meinshausen-Buhlmann method, and that seems to be poorly implemented.

> glasso(var(x),rho=0,approx=TRUE)[c('w','wi')]
$w
     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0
[3,]    0    0    0

$wi
            [,1]        [,2]        [,3]
[1,]  0.00000000 -0.03912856 -0.04299102
[2,] -0.05490688  0.00000000 -0.39828827
[3,] -0.05827283 -0.38471896  0.00000000

Now that I look at it, the wi returned by setting approx=TRUE looks like the covariance, not inverse covariance, matrix. This is a serious error in glasso function. Or maybe it's an error in the documentation. Either way, it's serious.