Create biglasso.m

Sparse-inverse covariance estimation with the bigraphical lasso.

Estimates a sparse precision (inverse-covariance) matrix through a series of LASSO regressions on partitions of the sample covariance matrix. The main difference from the graphical lasso algorithm is the parametrisation of the precision matrix as a Kronecker sum of two sparse precision matrices, Psi and Theta. This is also known in algebraic graph theory as the Cartesian product of two graphs (since the precisions individually inscribe connectivities of a GMRF). The Kronecker sum of Psi and Theta is defined as:

    kron(Psi, eye(size(Theta))) + kron(eye(size(Psi)), Theta).

From this, is it clear that the Kronecker sum is decomposed into the precisions involved in a product of Gaussian distributions. As such, the bigraphical lasso can learn a model for any design matrix with dependencies across columns (Theta) as well as across rows (Psi).

ARG S : the sample covariance matrix (the outer product matrix X'X/n). We assume S comes from data with standardised predictors (columns with zero means, unit
        norms). It has size d x d.

ARG T : the feature covariance matrix (the inner product matrix XX'/d). We assume T comes from data with standardised datapoints (rows with zero means, unit 
        norms). It has size n x n.

ARG lambda : the L1 shrinkage (regularisation) parameter.

The function accepts a variable length input argument list, in the format ['arg1name', arg1value , 'arg2name', arg2value, ...].
Available options are:

    'warnInit':     Initialisation for W (estimated covariance).
    'maxIter' :     Maximum number of iterations.
    'thresh'  :     Convergence threshold for coordinate descent. Each inner coordinate-descent loop continues until the relative change in any coefficient is
                    less than thresh. Defaults value is 1E-4.

RETURN W : the estimated covariance matrix, inv(kronSum(Psi, Theta)).

RETURN W_dual : the estimated dual covariance matrix, inv(kronSum(Theta, Psi)).

RETURN Psi : the estimated row-precision matrix.

RETURN Theta : the estimated column-precision matrix.

SEEALSO : glmnet, glasso

COPYRIGHT : Alfredo Kalaitzis, 2012 - 2014