Sparsity of representations of signals has been shown to be a key concept of fundamental importance in fields such as blind source separation, compression, sampling and signal analysis. The aim of this paper is to compare several commonly-used sparsity measures based on intuitive attributes. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper, six properties are discussed: (Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates, and Babies), each of which a sparsity measure should have. The main contributions of this paper are the proofs and the associated summary table which classify commonly-used sparsity measures based on whether or not they satisfy these six propositions. Only two of these measures satisfy all six: the pq-mean with p les 1, q > 1 and the Gini index.

Comparing Measures of Sparsity

Recently, sparse representation has attracted a lot of interest in various areas. However, the standard sparse representation does not consider the intrinsic structure, i.e., the nonzero elements occur in clusters, called group sparsity. Furthermore, there is no dictionary learning method for group sparse representation considering the geometrical structure of space spanned by atoms. In this paper, we propose a novel dictionary learning method, called Dictionary Learning with Group Sparsity and Graph Regularization (DL-GSGR). First, the geometrical structure of atoms is modeled as the graph regularization. Then, combining group sparsity and graph regularization, the DL-GSGR is presented, which is solved by alternating the group sparse coding and dictionary updating. In this way, the group coherence of learned dictionary can be enforced small enough such that any signal can be group sparse coded effectively. Finally, group sparse representation with DL-GSGR is applied to 3-D medical image denoising and image fusion. Specifically, in 3-D medical image denoising, a 3-D processing mechanism (using the similarity among nearby slices) and temporal regularization (to perverse the correlations across nearby slices) are exploited. The experimental results on 3-D image denoising and image fusion demonstrate the superiority of our proposed denoising and fusion approaches.

Group-Sparse Representation With Dictionary Learning for Medical Image Denoising and Fusion

Effective common spatial pattern (CSP) feature extraction for motor imagery (MI) electroencephalogram (EEG) recordings usually depends on the filter band selection to a large extent. Subband optimization has been suggested to enhance classification accuracy of MI. Accordingly, this study introduces a new method that implements sparse Bayesian learning of frequency bands (named SBLFB) from EEG for MI classification. CSP features are extracted on a set of signals that are generated by a filter bank with multiple overlapping subbands from raw EEG data. Sparse Bayesian learning is then exploited to implement selection of significant features with a linear discriminant criterion for classification. The effectiveness of SBLFB is demonstrated on the BCI Competition IV IIb dataset, in comparison with several other competing methods. Experimental results indicate that the SBLFB method is promising for development of an effective classifier to improve MI classification.

Sparse Bayesian Learning for Obtaining Sparsity of EEG Frequency Bands Based Feature Vectors in Motor Imagery Classification.

Sparsity is a recurrent theme in machine learning and is used to improve performance of algorithms such as non-negative matrix factorization and the LOST algorithm. Our aim in this paper is to compare several commonly-used sparsity measures according to intuitive attributes that a sparsity measure should have. Sparsity of representations of signals in fields such as blind source separation, compression, sampling and signal analysis has proved not just to be useful but a key factor in the success of algorithms in these areas. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper we discuss six properties (robin hood, scaling, rising tide, cloning, bill gates and babies) that we believe a sparsity measure should have. The main contribution of this paper is a table which classifies commonly-used sparsity measures based on whether or not they satisfy these six propositions. Only one of these measures satisfies all six: the Gini index.

Comparing measures of sparsity

https://hal.archives-ouvertes.fr/docs/01/01/18/56/PDF/BeckACHBWGBM14-NIvf.pdf

EEG extended source localization: Tensor-based vs. conventional methods.

How to localize the neural electric activities effectively and precisely from the scalp EEG recordings is a critical issue for clinical neurology and cognitive neuroscience. In this paper, based on the spatial sparse assumption of brain activities, proposed is a novel iterative EEG source imaging algorithm, Lp norm iterative sparse solution (LPISS). In LPISS, the lp(ples1) norm constraint for sparse solution is integrated into the iterative weighted minimum norm solution of the underdetermined EEG inverse problem, and it is the constraint and the iteratively renewed weight that forces the inverse problem to converge to a sparse solution effectively. The conducted simulation studies with comparison to LORETA and FOCUSS for various dipoles configurations confirmed the validation of LPISS for sparse EEG source localization. Finally, LPISS was applied to a real evoked potential collected in a study of inhibition of return (IOR), and the result was consistent with the previously suggested activated areas involved in an IOR process

Y. Tian

Papers

Lp Norm Iterative Sparse Solution for EEG Source Localization