sGMCA

The sGMCA algorithm (Semi-blind Generalized Morphological Component Analysis) aims at solving Semi-Blind Source Separation (sBSS) problems, in which the spectra of the sought-after sources are constrained to belong to an unknown, physics-based, manifold. To that end, the Interpolatory AutoEncoder (IAE) framework is employed.

Introduction
Procedure
Dependencies
Parameters
Example
Authors
Reference
License

Introduction

Let us consider the forward model:

,

where:

are the multiwavelength data of size , stacked in a matrix,
is the mixing matrix,
are the sources, stacked in a matrix,
is a Gaussian, independent and identically distributed noise.

The sources are assumed to be sparse in the starlet representation . Moreover, in a semi-blind approach, we suppose that among the sources, have a spectrum modeled by a IAE and are fully unknown. Let be the indices of the modeled components. The sGMCA aims at minimizing the following objective function with respect to and :

where denotes the element-wise product, are the sparsity regularization parameters, is the m-dimensional Euclidean unit sphere, and is the manifold associated to the corresponding IAE model .

Procedure

The sGMCA algorithm is based on GMCA, which is a BSS procedure built upon a projected alternate least-squares (pALS) minimization scheme.

The sources and the mixing matrix are initialized with GMCA; at this point, the mixing matrix and the sources are likely to be contaminated by remnants of other components. and are then updated alternatively and iteratively until converge is reached. Each update comprises a least-squares estimate, so as to minimize the data-fidelity term, followed by the application of the proximal operator of the corresponding regularization term.

Dependencies

Required packages

Python (last tested with v3.7.6)
NumPy (last tested with v1.19.2)
JAX (last tested with v0.2.5)

Optional packages

matplotlib (last tested with v3.3.3)
tqdm (last tested with v4.52.0)

Parameters

Below are the three parameters of the sgmca function which must always be provided.

Parameter	Type	Information	Default value
`X`	(m,p) float numpy.ndarray	input data, each row corresponds to a channel	N/A
`n`	int	number of sources to be estimated	N/A
`models`	dict or str	IAE models of the spectra. Is either a dict of str: int (str is the model filename and int the nb of components following the model) or a str (same model applied to all components)	N/A

Below are the essential parameters of the sgmca function. They may be assigned their default value.

Parameter	Type	Information	Default value
`nnegA`	bool	non-negativity constraint on the spectra of which are not modeled	True
`nnegS`	bool	non-negativity constraint on	False
`nneg`	bool	non-negativity constraint on and , overrides nnegA and nnegS if not None	None
`nStd`	float	noise standard deviation contaminating . If None, MAD is used to calculate the thresholds .	None
`nscales`	int	number of starlet detail scales	2
`k`	float	parameter of the k-std thresholding	3
`K_max`	float	maximal L0 norm of the sources. Being a percentage, it should be between 0 and 1.	0.5
`stepSizeProj`	float	step size of the descent algorithm of the model constraint	0.1
`thrEnd`	bool	perform thresholding during the finale estimation of the sources	True
`eps`	(3,) float numpy.ndarray	stopping criteria of (1) the GMCA initialization, (2) sGMCA and (3) the descent algorithm of the model constraint	[1e-2, 1e-6, 1e-6]
`verb`	int	verbosity level, from 0 (mute) to 5 (most talkative)	0

Below are other parameters of the sgmca function, which can reasonably be assigned their default value.

Parameter	Type	Information	Default value
`AInit`	(m,n) float numpy.ndarray	initial value for the mixing matrix. If None, GMCA-based initialization.	None
`ARef`	(m,n_ref) or (m,) float numpy.ndarray	reference spectra of the mixing matrix, they are fixed during step #1 (0<n_ref<n)	None
`nbItMin1`	int	minimum number of iterations for GMCA initialization	100
`L1`	bool	if False, L0 rather than L1 penalization	True
`manualMapping`	bool	operate the model-to-source mapping manually	True
`doSemiBlind`	bool	perform semi-blind estimation	True
`nbItMax2`	int	maximum number of sGMCA iterations	50
`optimProj`	int	descent algorithm of the model constraint (0: Adam, 1: Momentum, 2: RMSProp, 3: AdaGrad, 4: Nesterov, 5: SGD)	3
`nbItProj`	int	maximum number of iterations of the descent algorithm of the model constraint	1000

Below are the values returned by the sgmca function.

Output	Type	Information
`A`	(m,n) float numpy.ndarray	Estimated mixing matrix
`S`	(n,p) float numpy.ndarray	Estimated sources

Example

Perform a sBSS on the data X with four sources, three of which having a spectrum modeled with a IAE.

models = {"model_1": 2,  # IAE model for component of type 1, two components are expected in A
          "model_2": 1}  # IAE model for component of type 2, one component is expected in A
A, S = sgmca(X=X, n=4, nStd=1e-7, models=models)

Authors

Rémi Carloni Gertosio
Jérôme Bobin
Fabio Acero

Reference

Rémi Carloni Gertosio, Jérôme Bobin, Fabio Acero, Semi-Blind Source Separation with Learned Constraints, Signal Processing, Volume 202, 2023, 108776

License

This project is licensed under the LGPL-3.0 License.

RCarloniGertosio / sGMCA