Flexible Multi-layer Sparse Approximations of Matrices and Applications

The computational cost of many signal processing and machine learning techniques is often dominated by the cost of applying certain linear operators to high-dimensional vectors. This paper introduces an algorithm aimed at reducing the complexity of applying linear operators in high dimension by approximately factorizing the corresponding matrix into few sparse factors. The approach relies on recent advances in non-convex optimization. It is first explained and analyzed in details and then demonstrated experimentally on various problems including dictionary learning for image denoising, and the approximation of large matrices arising in inverse problems

Consensus Matching Pursuit of Multi-Trial Biosignals, with Application to Brain Signals

submittedTime-frequency representations are commonly used to analyze the oscillatory nature of bioelectromagnetic signals. There is a growing interest in sparse representations, where the data is described using few components. In this study, we adapt the Matching Pursuit of Mallat and Zhang for biosignals consisting of a series of variations around a similar pattern, with emphasis on multi-trial datasets encountered in MEG and EEG. The general principle of Matching Pursuit (MP) is to iteratively subtract from the signal its projection on the atom selected from a dictionary. The originality of our method is to select each atom using a voting technique that is robust to variability, and to subtract it by adapting the parameters to each trial. Because it is designed to handle inter-trial variability using a voting technique, the method is called Consensus Matching Pursuit (CMP). The method is validated on both simplified and realistic simulations, and on two real datasets (intracerebral EEG and scalp EEG ).We also compare our method to two other multi-trial MP algorithms: Multivariate MP (MMP) and Induced activity MP (IMP). CMP is shown to be able to sparsely reveal the structure present in the data, and to be robust to variability (jitter) across trials

Time-Frequency Mixed-Norm Estimates: Sparse M/EEG imaging with non-stationary source activations

International audienceMagnetoencephalography (MEG) and electroencephalography (EEG) allow functional brain imaging with high temporal resolution. While solving the inverse problem independently at every time point can give an image of the active brain at every millisecond, such a procedure does not capitalize on the temporal dynamics of the signal. Linear inverse methods (Minimum-norm, dSPM, sLORETA, beamformers) typically assume that the signal is stationary: regularization parameter and data covariance are independent of time and the time varying signal-to-noise ratio (SNR). Other recently proposed non-linear inverse solvers promoting focal activations estimate the sources in both space and time while also assuming stationary sources during a time interval. However such an hypothesis only holds for short time intervals. To overcome this limitation, we propose time-frequency mixed-norm estimates (TF-MxNE), which use time-frequency analysis to regularize the ill-posed inverse problem. This method makes use of structured sparse priors defined in the time-frequency domain, offering more accurate estimates by capturing the non-stationary and transient nature of brain signals. State-of-the-art convex optimization procedures based on proximal operators are employed, allowing the derivation of a fast estimation algorithm. The accuracy of the TF-MxNE is compared to recently proposed inverse solvers with help of simulations and by analyzing publicly available MEG datasets

Covariance-domain Dictionary Learning for Overcomplete EEG Source Identification

We propose an algorithm targeting the identification of more sources than channels for electroencephalography (EEG). Our overcomplete source identification algorithm, Cov-DL, leverages dictionary learning methods applied in the covariance-domain. Assuming that EEG sources are uncorrelated within moving time-windows and the scalp mixing is linear, the forward problem can be transferred to the covariance domain which has higher dimensionality than the original EEG channel domain. This allows for learning the overcomplete mixing matrix that generates the scalp EEG even when there may be more sources than sensors active at any time segment, i.e. when there are non-sparse sources. This is contrary to straight-forward dictionary learning methods that are based on the assumption of sparsity, which is not a satisfied condition in the case of low-density EEG systems. We present two different learning strategies for Cov-DL, determined by the size of the target mixing matrix. We demonstrate that Cov-DL outperforms existing overcomplete ICA algorithms under various scenarios of EEG simulations and real EEG experiments

SCoT: a Python toolbox for EEG source connectivity

Analysis of brain connectivity has become an important research tool in neuroscience. Connectivity can be estimated between cortical sources reconstructed from the electroencephalogram (EEG). Such analysis often relies on trial averaging to obtain reliable results. However, some applications such as brain-computer interfaces (BCIs) require single-trial estimation methods. In this paper, we present SCoT—a source connectivity toolbox for Python. This toolbox implements routines for blind source decomposition and connectivity estimation with the MVARICA approach. Additionally, a novel extension called CSPVARICA is available for labeled data. SCoT estimates connectivity from various spectral measures relying on vector autoregressive (VAR) models. Optionally, these VAR models can be regularized to facilitate ill posed applications such as single-trial fitting. We demonstrate basic usage of SCoT on motor imagery (MI) data. Furthermore, we show simulation results of utilizing SCoT for feature extraction in a BCI application. These results indicate that CSPVARICA and correct regularization can significantly improve MI classification. While SCoT was mainly designed for application in BCIs, it contains useful tools for other areas of neuroscience. SCoT is a software package that (1) brings combined source decomposition and connectivtiy estimation to the open Python platform, and (2) offers tools for single-trial connectivity estimation. The source code is released under the MIT license and is available online at github.com/SCoT-dev/SCoT

Blind Multilinear Identification

We discuss a technique that allows blind recovery of signals or blind identification of mixtures in instances where such recovery or identification were previously thought to be impossible: (i) closely located or highly correlated sources in antenna array processing, (ii) highly correlated spreading codes in CDMA radio communication, (iii) nearly dependent spectra in fluorescent spectroscopy. This has important implications --- in the case of antenna array processing, it allows for joint localization and extraction of multiple sources from the measurement of a noisy mixture recorded on multiple sensors in an entirely deterministic manner. In the case of CDMA, it allows the possibility of having a number of users larger than the spreading gain. In the case of fluorescent spectroscopy, it allows for detection of nearly identical chemical constituents. The proposed technique involves the solution of a bounded coherence low-rank multilinear approximation problem. We show that bounded coherence allows us to establish existence and uniqueness of the recovered solution. We will provide some statistical motivation for the approximation problem and discuss greedy approximation bounds. To provide the theoretical underpinnings for this technique, we develop a corresponding theory of sparse separable decompositions of functions, including notions of rank and nuclear norm that specialize to the usual ones for matrices and operators but apply to also hypermatrices and tensors.Comment: 20 pages, to appear in IEEE Transactions on Information Theor

MEG and EEG data analysis with MNE-Python

Magnetoencephalography and electroencephalography (M/EEG) measure the weak electromagnetic signals generated by neuronal activity in the brain. Using these signals to characterize and locate neural activation in the brain is a challenge that requires expertise in physics, signal processing, statistics, and numerical methods. As part of the MNE software suite, MNE-Python is an open-source software package that addresses this challenge by providing state-of-the-art algorithms implemented in Python that cover multiple methods of data preprocessing, source localization, statistical analysis, and estimation of functional connectivity between distributed brain regions. All algorithms and utility functions are implemented in a consistent manner with well-documented interfaces, enabling users to create M/EEG data analysis pipelines by writing Python scripts. Moreover, MNE-Python is tightly integrated with the core Python libraries for scientific comptutation (NumPy, SciPy) and visualization (matplotlib and Mayavi), as well as the greater neuroimaging ecosystem in Python via the Nibabel package. The code is provided under the new BSD license allowing code reuse, even in commercial products. Although MNE-Python has only been under heavy development for a couple of years, it has rapidly evolved with expanded analysis capabilities and pedagogical tutorials because multiple labs have collaborated during code development to help share best practices. MNE-Python also gives easy access to preprocessed datasets, helping users to get started quickly and facilitating reproducibility of methods by other researchers. Full documentation, including dozens of examples, is available at http://martinos.org/mne