sporco.pgm.cmod

PGM algorithm for the CMOD problem

Functions

getPcn(zm, nnc)	Construct constraint set projection function.
zeromean(v)	Subtract mean of each column of matrix.
normalise(v)	Normalise columns of matrix.

Classes

CnstrMOD(*args, **kwargs)	PGM algorithm for a constrained variant of the Method of Optimal Directions (MOD) [20] problem, referred to here as Constrained MOD (CMOD).
WeightedCnstrMOD(*args, **kwargs)	PGM algorithm for a weighted variant of Constrained MOD (CMOD) problem.

Function Descriptions

sporco.pgm.cmod.getPcn(zm, nnc)

Construct constraint set projection function.

Parameters

zmbool

Flag indicating whether the projection function should include column mean subtraction

nncbool

Flag indicating whether the projection function should include clipping to positive values

Returns

fnfunction

Constraint set projection function

sporco.pgm.cmod.zeromean(v)

Subtract mean of each column of matrix.

Parameters

varray_like

Input dictionary array

Returns

vzndarray

Dictionary array with column means subtracted

sporco.pgm.cmod.normalise(v)

Normalise columns of matrix.

Parameters

varray_like

Array with columns to be normalised

Returns

vnrmndarray

Normalised array

Class Descriptions

class sporco.pgm.cmod.CnstrMOD(*args, **kwargs)

Bases: sporco.pgm.pgm.PGM

PGM algorithm for a constrained variant of the Method of Optimal Directions (MOD) [20] problem, referred to here as Constrained MOD (CMOD).

Solve the optimisation problem

\[\mathrm{argmin}_D \| D X - S \|_2^2 + \iota_C(D)\]

where \(\iota_C(\cdot)\) is the indicator function of feasible set \(C\) consisting of matrices with unit-norm columns.

After termination of the solve method, attribute itstat is a list of tuples representing statistics of each iteration. The fields of the named tuple IterationStats are:

Iter : Iteration number

DFid : Value of data fidelity term \((1/2) \| D X - S \|_2^2\)

Cnstr : Constraint violation measure

Rsdl : Residual

L : Inverse of gradient step parameter

Time : Cumulative run time

Parameters

Zarray_like, shape (M, K)

Sparse representation coefficient matrix

Sarray_like, shape (N, K)

Signal vector or matrix

dsztuple

Dictionary size

optCnstrMOD.Options object

Algorithm options

class Options(opt=None)

Bases: sporco.pgm.pgm.PGM.Options

CnstrMOD algorithm options

Options include all of those defined in pgm.PGM.Options, together with additional options:

ZeroMeanFlag indicating whether the solution

dictionary \(D\) should have zero-mean components.

NonNegCoefFlag indicating whether the solution

dictionary \(D\) should have non-negative coefficients.

Note that ZeroMean and NonNegCoef may not both be True.

Parameters

optdict or None, optional (default None)

CnstrMOD algorithm options

setcoef(Z)

Set coefficient array.

getdict()

Get final coefficient array.

grad_f(V=None)

Compute gradient of data fidelity for variable V or self.Y.

prox_g(V)

Compute proximal operator of \(g\).

hessian_f(V)

Compute Hessian of \(f\) applied to V.

eval_objfn()

Compute components of objective function as well as total contribution to objective function.

obfn_f(X=None)

Compute data fidelity term \((1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2\).

obfn_cns()

Compute constraint violation measure \(\| P(\mathbf{y}) - \mathbf{y}\|_2\).

reconstruct(X=None)

Reconstruct representation.

class sporco.pgm.cmod.WeightedCnstrMOD(*args, **kwargs)

Bases: sporco.pgm.cmod.CnstrMOD

PGM algorithm for a weighted variant of Constrained MOD (CMOD) problem.

Solve the optimisation problem

\[\mathrm{argmin}_D \| D X - S \|_W^2 + \iota_C(D)\]

where \(\iota_C(\cdot)\) is the indicator function of feasible set \(C\) consisting of matrices with unit-norm columns and \(\| \cdot \|_W\) denotes the weighted Frobenius norm defined as (note that this is not the standard definition)

\[\| X \|_W^2 = \| W^{1/2} \odot X \|_F\]

so that

\[\| X \|_W^2 = \sum_i \| W_i^{1/2} \mathbf{x}_i \|_2^2 = \sum_i \| \mathbf{x}_i \|_{W_i}^2 \;,\]

where \(\mathbf{x}_i\) and \(\mathbf{w}_i\) are the \(i^{\text{th}}\) columns of \(X\) and \(W\) respectively, and \(W_i = \mathrm{diag}(\mathbf{w}_i)\).

After termination of the solve method, attribute itstat is a list of tuples representing statistics of each iteration. The fields of the named tuple IterationStats are:

Iter : Iteration number

DFid : Value of data fidelity term \((1/2) \| D X - S \|_W^2\)

Cnstr : Constraint violation measure

Rsdl : Residual

L : Inverse of gradient step parameter

Time : Cumulative run time

Parameters

Zarray_like, shape (M, K)

Sparse representation coefficient matrix

Sarray_like, shape (N, K)

Signal matrix

Warray_like, shape (N, K)

Weight matrix

dsztuple

Dictionary size

optWeightedCnstrMOD.Options object

Algorithm options

grad_f(V=None)

Compute gradient of data fidelity for variable V or self.Y.

obfn_f(X=None)

Compute data fidelity term \((1/2) \| D \mathbf{x} - \mathbf{s} \|_W^2\).