Kron2SumCov

class glimix_core.cov.Kron2SumCov(G, dim, rank)

Implements K = C₀ ⊗ GGᵀ + C₁ ⊗ I.

C₀ and C₁ are d×d symmetric matrices. C₀ is a semi-definite positive matrix while C₁ is a positive definite one. G is a n×m matrix and I is a n×n identity matrix. Let M = Uₘ Sₘ Uₘᵀ be the eigen decomposition for any matrix M. The documentation and implementation of this class make use of the following definitions:

• X = GGᵀ = Uₓ Sₓ Uₓᵀ

• C₁ = U₁ S₁ U₁ᵀ

• Cₕ = S₁⁻½ U₁ᵀ C₀ U₁ S₁⁻½

• Cₕ = Uₕ Sₕ Uₕᵀ

• D = (Sₕ ⊗ Sₓ + Iₕₓ)⁻¹

• Lₓ = Uₓᵀ

• Lₕ = Uₕᵀ S₁⁻½ U₁ᵀ

• L = Lₕ ⊗ Lₓ

The above definitions allows us to write the inverse of the covariance matrix as:

K⁻¹ = LᵀDL,

where D is a diagonal matrix.

Example

>>> from numpy import array
>>> from glimix_core.cov import Kron2SumCov
>>>
>>> G = array([[-1.5, 1.0], [-1.5, 1.0], [-1.5, 1.0]])
>>> Lr = array([[3], [2]], float)
>>> Ln = array([[1, 0], [2, 1]], float)
>>>
>>> cov = Kron2SumCov(G, 2, 1)
>>> cov.C0.L = Lr
>>> cov.C1.L = Ln
>>> print(cov)
Kron2SumCov(G=..., dim=2, rank=1): Kron2SumCov
  LRFreeFormCov(n=2, m=1): C₀
    L: [[3.]
        [2.]]
  FreeFormCov(dim=2): C₁
    L: [[1. 0.]
        [2. 1.]]

__init__(G, dim, rank)

Constructor.

Parameters

• dim (int) – Dimension d for the square matrices C₀ and C₁.

• rank (int) – Maximum rank of the C₁ matrix.

Methods

LdKL_dot(v)	Implements L(∂K)Lᵀv.
__init__(G, dim, rank)	Constructor.
gradient()	Gradient of K.
gradient_dot(v)	Implements ∂K⋅v.
listen(func)	Listen to parameters change.
logdet()	Implements log|K| = - log|D| + n⋅log|C₁|.
logdet_gradient()	Implements ∂log|K| = Tr[K⁻¹∂K].
solve(v)	Implements the product K⁻¹⋅v.
value()	Covariance matrix K = C₀ ⊗ GGᵀ + C₁ ⊗ I.

Attributes

C0	Semi-definite positive matrix C₀.
C1	Definite positive matrix C₁.
D	(Sₕ ⊗ Sₓ + Iₕₓ)⁻¹.
G	User-provided matrix G, n×m.
Ge	Result of US from the SVD decomposition G = USVᵀ.
Lh	Lₕ.
Lx	Lₓ.
name	Name of this function.
nparams	Number of parameters.