Cost Functions

class triqs_maxent.cost_functions.cost_function.CostFunction(chi2=None, S=None, H_of_v=None, A_of_H=None, chi2_factor=1.0)

Bases: DoublyDerivableFunction

The base class for the function to be minimized

Parameters:

chi2Chi2

the expression for chi2 (e.g. NormalChi2)

SEntropy

the expression for S (e.g. NormalEntropy)

H_of_vH_of_v

the expression for \(H(v)\) (e.g. NormalH_of_v)

A_of_HA_of_H

the expression for \(A(H)\) (e.g. IdentityA_of_H)

chi2_factorfloat

an additional factor for the \(\chi^2\) term of the cost function (default: 1.0)

set_alpha(alpha)

Set the hyper-parameter.

f(v)

Calculate the function to be minimized, \(Q_{\alpha}(v)\).

Parameters:

varray

vector in singular space giving the solution; if None, the last supplied value should be reused

d(v)

Calculate the derivative of the function to be minimized

Parameters:

varray

vector in singular space giving the solution if None, the last supplied value should be reused

dd(v)

Calculate the 2nd derivative of the function to be minimized

Parameters:

varray

vector in singular space giving the solution if None, the last supplied value should be reused

parameter_change()

Notify the function that parameters have changed

This allows to reprocess certain values.

MaxEnt cost function

class triqs_maxent.cost_functions.maxent_cost_function.MaxEntCostFunction(d_dv=False, dA_projection=2, **kwargs)

Bases: CostFunction

The usual MaxEnt cost function

The expression for the cost function is

\[Q_\alpha(v) = \frac12 \chi^2(H(v)) \eta - \alpha S(H(v)),\]

where \(\eta\) is an additional factor for the \(\chi^2\) term, which can be given as chi2_factor (default: 1.0).

This function implements the usual derivatives d and dd, where by default (i.e., with d_dv = False) d is \(\frac{\partial Q_\alpha}{\partial H}\) and dd is \(\frac{\partial^2 Q_\alpha}{\partial H \partial v}\). If dA_projection>0, the derivatives are projected into singular space; either by multiplying from the left by \(V^\dagger\) (i.e., the matrix with the right singular vectors, if dA_projection=1) or by \(\partial H/\partial v\) (if dA_projection=2).

If d_dv = True, we have d giving \(\frac{\partial Q_\alpha}{\partial v}\); dd is again the derivative of d with respect to v.

This cost function should be used for general \(\chi^2\) and \(S\). If you use the normal values, it is possible to use the BryanCostFunction instead.

Parameters:

d_dvbool

see explanation above

dA_projectionint (0, 1, 2)

see explanation above; it is ignored if d_dv = True

**kwargs

gets passed to CostFunction

f(v)

Calculate the function to be minimized, \(Q_{\alpha}(v)\).

Parameters:

varray

vector in singular space giving the solution

dH(v)

Calculate the derivative of the function with respect to H

d(v)

Calculate the derivative of the function to be minimized

Parameters:

varray

vector in singular space giving the solution

ddH(v)

Calculate the 2nd derivative of the function with respect to H

dd(v)

Calculate the 2nd derivative of the function to be minimized

Parameters:

varray

vector in singular space giving the solution

Bryan Cost Function

class triqs_maxent.cost_functions.bryan_cost_function.BryanCostFunction(chi2_factor=1.0)

Bases: CostFunction

The usual MaxEnt cost function

This cost function contains some simplifications that are only valid if the normal \(\chi^2\) and \(S\) are used, i.e., it does only work for diagonal entries of the spectral function. For a general cost function, use MaxEntCostFunction.

The expression for the cost function is

\[Q_\alpha(v) = \frac12 \chi^2(H(v)) \eta - \alpha S(H(v)),\]

where \(\eta\) is an additional factor for the \(\chi^2\) term, which can be given as chi2_factor (default: 1.0).

The derivatives d and dd do not actually give the derivatives of the cost function, but a simplified version that is obtained when multiplying from the left by \(V^\dagger\).

Parameters:

chi2_factorfloat

an additional factor for the \(\chi^2\) term of the cost function (default: 1.0)

f(v)

Calculate the function to be minimized, \(Q_{\alpha}(v)\).

Parameters:

varray

vector in singular space giving the solution

dH(v)

Calculate the derivative of the function with respect to H

d(v)

Calculate the derivative of the function to be minimized

Parameters:

varray

vector in singular space giving the solution

ddH(v)

Calculate the 2nd derivative of the function with respect to H

dd(v)

Calculate the 2nd derivative of the function to be minimized

Parameters:

varray

vector in singular space giving the solution