A regularized matrix factorization approach to induce structured sparselowrank solutions in the EEG inverse problem
 Jair MontoyaMartínez^{1}Email author,
 Antonio ArtésRodríguez^{1},
 Massimiliano Pontil^{2} and
 Lars Kai Hansen^{3}
https://doi.org/10.1186/16876180201497
© MontoyaMartínez et al.; licensee Springer. 2014
Received: 27 March 2014
Accepted: 16 June 2014
Published: 24 June 2014
Abstract
We consider the estimation of the Brain Electrical Sources (BES) matrix from noisy electroencephalographic (EEG) measurements, commonly named as the EEG inverse problem. We propose a new method to induce neurophysiological meaningful solutions, which takes into account the smoothness, structured sparsity, and low rank of the BES matrix. The method is based on the factorization of the BES matrix as a product of a sparse coding matrix and a dense latent source matrix. The structured sparselowrank structure is enforced by minimizing a regularized functional that includes the ℓ_{21}norm of the coding matrix and the squared Frobenius norm of the latent source matrix. We develop an alternating optimization algorithm to solve the resulting nonsmoothnonconvex minimization problem. We analyze the convergence of the optimization procedure, and we compare, under different synthetic scenarios, the performance of our method with respect to the Group Lasso and Trace Norm regularizers when they are applied directly to the target matrix.
Keywords
1 Introduction
where $\mathbf{Y}\in {\mathbb{R}}^{M\times T}$ and $\mathbf{A}\in {\mathbb{R}}^{M\times N}$ are known and represent, respectively, the EEG measurements matrix and the forward operator (a.k.a lead field matrix), $\mathbf{S}\in {\mathbb{R}}^{N\times T}$ denotes the BES matrix, and $\mathbf{E}\in {\mathbb{R}}^{M\times T}$ is a noise matrix. M denotes the number of EEG electrodes, N is the number of brain electrical sources, and T is the number of time instants.
This estimation problem is very challenging: N≫M, and the existence of silent BES (BES that produce nonmeasurable fields on the scalp surface) implies that the EEG inverse problem has infinite solutions: a silent BES can always be added to a solution of the inverse problem without affecting the EEG measurements. For all these reasons, the EEG inverse problem is an undetermined illposed problem [1–4].
A classical approach to solve an illposed problem is to use regularization theory, which involves the replacement of the original illposed problem with a ‘nearby’ wellposed problem whose solution approximates the required solution [5]. Solutions developed by this theory are stated in terms of a regularization function, which helps us to select, among the infinite solutions, the one that best fulfills a prescribed constrain (e.g., smoothness, sparsity, and low rank). To define the constrain, we can use mathematical restrictions (minimum norm estimates) or anatomical, physiological, and functional prior information. Some examples of useful neurophysiological information are [1, 6]: the irrotational character of the brain current sources, the (smooth) dynamic of the neural signals, the clusters formed by neighboring or functional related BES, and the smoothness and focality of the electromagnetic fields generated and propagated within the volume conductor media (brain cortex, skull, and scalp).
Several regularization functions have been proposed in the EEG community: Hämäläinen and Ilmoniemi in [7] proposed a squared Frobenius norm penalty ($\parallel \mathbf{S}{\parallel}_{\mathrm{F}}^{2}$), which they named Minimum Norm Estimate (MNE). This regularization function usually induces solutions that spread over a considerable part of the brain. Uutela et al. in [8] proposed an ℓ_{1}norm penalty (∥S∥_{1}). They named their approach Minimum Current Estimate (MCE). This penalty function promotes solutions that tend to be scattered around the true sources. Mixed ℓ_{1}ℓ_{2}norm penalties have also been proposed in the framework of the time basis, time frequency dictionaries, and spatial basis decomposition. These mixed norm approaches induce structured sparse solutions and depend on decomposing the BES signals as linear combinations of multiple basis functions, e.g., Ou et al. in [9] proposed the use of temporal basis functions obtained with singular value decomposition (SVD), Gramfort et al. in [10, 11] proposed the use of timefrequency Gabor dictionaries, and Haufe et al. in [12] proposed the use of spatial basis Gaussian functions. For a more detailed overview on inverse methods for EEG, see [2, 3, 13] and references therein. For a more detailed overview on regularization functions applied to structured sparsity problems, see [14–16] and references therein.
All of these regularizers try to induce neurophysiological meaningful solutions, which take into account the smoothness and structured sparsity of the BES matrix: during a particular cognitive task, only the BES related with the brain area involved in such a task will be active, and their corresponding time evolution will vary smoothly, that is, the BES matrix will have few nonzero rows, and in addition, the columns will vary smoothly. In this paper, we propose a regularizer that takes into account not only the smoothness and structured sparsity of the BES matrix but also its low rank, capturing this way the linear relation between the active sources and their corresponding neighbors. In order to do so, we propose a new method based on matrix factorization and regularization, with the aim of recovering the latent structure of the BES matrix. In the factorization, the first matrix, which acts as a coding matrix, is penalized using the ℓ_{21}norm, and the second one, which acts as a dense, full rank latent source matrix, is penalized using the squared Frobenius norm.
In our approach, the resulting optimization problem is nonsmooth and nonconvex. A standard approach to deal with the nonsmoothness introduced by the nonsmooth regularizers mentioned above is to reformulate the regularization problem as a secondorder cone programming (SOCP) problem [12] and use interior pointbased solvers. However, interior pointbased methods can not handle large scale problems, which is the case of large EEG inverse problems involving thousands of brain sources. Another approach is to try to solve the nonsmooth problem directly, using general nonsmooth optimization methods, for instance, the subgradient method [17]. This method can be used if a subgradient of the objective function can be computed efficiently [14]. However, its convergence rate is, in practice, slow ($O(1/\sqrt{k})$), where k is the iteration counter. In this paper, in order to tackle the nonsmoothness of the optimization problem, we depart from these optimization methods and use instead efficient firstorder nonsmooth optimization methods [5, 18, 19]: forwardbackward splitting methods. These methods are also called proximal splitting because the nonsmooth function is involved via its proximity operator. Forwardbackward splitting methods were first introduced in the EEG inverse problem by Gramfort et al. [10, 11, 13], where they used them to solve nonsmooth optimization problems resulting from the use of mixed ℓ_{1}ℓ_{2}norm penalties functions. These methods have drawn, increasing attention in the EEG, machine learning, and signal processing community, especially because of their convergence rates and their ability to deal with large problems [19–21].
On the other hand, in order to handle the nonconvexity of the optimization problem, we use an iterative alternating minimization approach: minimizing over the coding matrix while maintaining fixed the latent source matrix and viceversa. Both of these optimization problems are convex: the first one can be solved using proximal splitting methods, while the second one can be solve directly in terms of a matrix inversion.
The rest of the paper is organized as follows. In Section 2, we give an overview of the EEG inverse problem. In Section 3, we present the mathematical background related with the proximal splitting methods. The resulting nonsmooth and nonconvex optimization problem is formally described in Section 4. In Section 5, we propose an alternating minimization algorithm, and its convergence analysis is presented in Section 6. Section 7 is devoted to the numerical evaluation of the algorithm and its comparison with the Group Lasso and Trace Norm regularizers, which consider partially the characteristics of the matrix S: its structured sparsity by using the ℓ_{21}norm and its low rank by using the ℓ_{∗}norm, respectively. The advantages of considering both characteristics in a single method, like in the proposed one, become clear in comparison with the independent use of the Group Lasso and Trace Norm regularizers. Finally, conclusions are presented in Section 8.
2 EEG inverse problem background
The EEG signals represent the electrical activity of one or several assemblies of neurons [22]. The area of a neuron assembly is small compared to the distance to the observation point (the EEG sensors). Therefore, the electromagnetic fields produced by an active neuron assembly at the sensor level are very similar to the field produced by a current dipole [23]. This simplified model is known as the equivalent current dipole (ECD). These ECDs are also known by other names such as BES and current sources. Due to the uniform spatial organization of their dendrites (perpendicular to the brain cortex), the pyramidal neurons are the only neurons that can generate a net current dipole over a piece of cortical surface, whose field is detectable on the scalp [3]. According to [24], it is necessary to add the field of ∼10^{4} pyramidal neurons in order to produce a voltage that is detectable on the scalp. These voltages can be recorded by using different types of electrodes [22], such as disposable (gelless, and pregelled types), reusable disc electrodes (gold, silver, stainless steel, or tin), headbands and electrode caps, salinebased electrodes, and needle electrodes.
where $\mathbf{y}\left(t\right)\in {\mathbb{R}}^{M\times 1}$ is the EEG measurements vector, $\mathbf{s}\left(t\right)\in {\mathbb{R}}^{N\times 1}$ is the BES vector, $\mathbf{e}\left(t\right)\in {\mathbb{R}}^{M\times 1}$ is the noise vector, and $\mathbf{A}\in {\mathbb{R}}^{M\times N}$ is the lead field matrix. In a typical experimental setup, the number of electrodes (M) is ∼10^{2}, and the number of BES (N) is ∼10^{3}, 10^{4}. We can express the former model for all time instants {t_{1},t_{2},…,t_{ T }} (corresponding to some observation time window) by using the matrix formulation (1), where $\mathbf{Y}=\left[\mathbf{y}\left({t}_{1}\right),\mathbf{y}\left({t}_{2}\right),\dots ,\mathbf{y}\left({t}_{T}\right)\right]\in {\mathbb{R}}^{M\times T}$, $\mathbf{S}=\left[\mathbf{s}\left({t}_{1}\right),\mathbf{s}\left({t}_{2}\right),\dots ,\mathbf{s}\left({t}_{T}\right)\right]\in {\mathbb{R}}^{N\times T}$, and $\mathbf{E}=\left[\mathbf{e}\left({t}_{1}\right),\mathbf{e}\left({t}_{2}\right),\dots ,\mathbf{e}\left({t}_{T}\right)\right]\in {\mathbb{R}}^{M\times T}$. The i th row of the matrix Y represents the electrical activity recorded by the i th EEG electrode during the observation time window. In the BES matrix S, each row represents the time evolution of one brain electrical source, and each column represents the activity of all the corresponding sources in a particular time instant. Finally, the forward operator A summarizes the geometric and electric properties of the conducting media (brain, skull, and scalp) and establishes the link between the current sources and EEG sensors (A_{ i j } tells us how the j th BES influences the measure obtained by the i th electrode). Following this notation, the EEG inverse problem can be stated as follows: Given a set of EEG signals (Y) and a forward model (A), estimate the current sources within the brain (S) that produce these signals.
3 Mathematical background
3.1 Proximity operator
where ∥·∥ denotes the Euclidean norm. Note that the proximity operator is well defined, because the above minimum exists and is unique (the objective function if strongly convex).
3.2 Subdifferentialproximity operator relationship
where ∂ f(y) denotes the subdifferential of f at y.
3.3 Principles of proximal splitting methods
In optimization, (7) is known as forwardbackward splitting process [19]. It consists of two steps: first, it performs a forward gradient descend step ${\mathbf{S}}_{k}^{\ast}={\mathbf{S}}_{k}\gamma \nabla f\left({\mathbf{S}}_{k}\right)$ and then it performs a backward step ${\mathbf{S}}_{k+1}={\text{prox}}_{\mathrm{\gamma r}}\left({\mathbf{S}}_{k}^{\ast}\right)$.
From (7), we can see the importance of the proximity operator (associated to γ r(S)) with respect to the forwardbackward splitting methods, since their main step is to calculate it. If we would have a closedform expression for such proximity operator or if we could approximate it efficiently (with the approximation errors decreasing at appropriate rates [27]), then we could efficiently solve (7). Furthermore, when f has a Lipschitz continuous gradient, there are fast algorithms to solve (7). For instance, the Iterative Soft Thresholding Algorithm (ISTA) has a convergence rate of O(1/k), and the Fast Iterative ShrinkageThresholding Algorithm (FISTA) has a convergence rate of O(1/k^{2}) [5].
4 Problem formulation
where $\frac{1}{2}\left\right\mathbf{AS}\mathbf{Y}{}_{\mathrm{F}}^{2}$ is the square loss function (∥ ∥_{F} denotes the Frobenius norm), and λ Ω(S) is a nonsmooth penalty term that is used to encode the prior knowledge about the structure of the target matrix S.
where λ>0, ρ>0, $\mathbf{B}\in {\mathbb{R}}^{N\times K}$, $\mathbf{C}\in {\mathbb{R}}^{K\times T}$, and B(i,:), C(i,:) denote the i th row of B and C, respectively. K≪{N,T}, λ, and ρ are parameters of the model that must be adjusted.
Hence, the proposed regularization framework takes into account all the prior knowledge about the structure of the target matrix S.
5 Optimization algorithm
5.1 Matrix factorization approach
From (15), we can see that {Y_{ 1 },Y_{ 2 },…,Y_{ M }}⊂RowSpace(C), where Y_{ i } denotes the i th row of Y.
From the SVD theory [29], we know that {Y_{1},Y_{ 2 },…,Y_{ M }}⊂Row Space(V^{⊤}); therefore, we can choose C_{ 0 }=V^{⊤}. Then, given C_{ 0 }, we can start iterating using (13) and (14).
5.1.1 Minimization over B(fixed C)
 1.The gradient of the smooth function F _{ B }(B)$\begin{array}{l}\nabla {F}_{\mathbf{B}}\left(\mathbf{B}\right)=\frac{\partial {F}_{\mathbf{B}}\left(\mathbf{B}\right)}{\partial \mathbf{B}}={\mathbf{A}}^{\top}\left(\mathbf{A}\right(\mathbf{B}{\mathbf{C}}_{t1})\mathbf{Y}){\mathbf{C}}_{t1}^{\top}\end{array}$
where A^{⊤} denotes the transpose of the matrix A.
 2.An upper bound of the Lipschitz constant (L) of ∇F _{ B }(B) (it can also be estimated using a backtracking search routine [5])$\begin{array}{ll}\parallel \nabla {F}_{\mathbf{B}}\left({\mathbf{B}}_{1}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\nabla {F}_{\mathbf{B}}\left({\mathbf{B}}_{2}\right){\parallel}_{2}^{2}& \phantom{\rule{0.3em}{0ex}}=\parallel {\mathbf{A}}^{\top}\mathbf{A}{\mathbf{B}}_{1}{\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\mathbf{A}}^{\top}\mathbf{A}{\mathbf{B}}_{2}{\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}{\parallel}_{2}^{2}\\ \phantom{\rule{0.3em}{0ex}}=\parallel {\mathbf{A}}^{\top}\mathbf{A}({\mathbf{B}}_{1}{\mathbf{B}}_{2}){\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}{\parallel}_{2}^{2}\\ \phantom{\rule{0.3em}{0ex}}=\sum _{j=1}^{K}\parallel \left(\phantom{\rule{0.3em}{0ex}}{\mathbf{A}}^{\top}\mathbf{A}\left({\mathbf{B}}_{1}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\mathbf{B}}_{2}\right)\phantom{\rule{0.3em}{0ex}}\right){\left({\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}\phantom{\rule{0.3em}{0ex}}\right)}_{j}\phantom{\rule{0.3em}{0ex}}{\parallel}_{2}^{2}\end{array}$where ${\left({\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}\right)}_{j}$ denotes the jth column of the matrix ${\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}$. Taking into account that $\parallel \mathbf{Q}x{\parallel}_{2}\le \parallel \mathbf{Q}\parallel {}_{2}\parallel x{\parallel}_{2},\phantom{\rule{2.77626pt}{0ex}}\forall x\in {\mathbb{R}}^{N},\forall \mathbf{Q}\in {\mathbb{R}}^{M\times N}$[29], where ∥·∥_{2} denotes the spectral norm, we get:$\begin{array}{ll}\hfill \parallel \nabla {F}_{\mathbf{B}}\left({\mathbf{B}}_{1}\right)\nabla {F}_{\mathbf{B}}\left({\mathbf{B}}_{2}\right){\parallel}_{2}^{2}& =\sum _{j=1}^{K}\parallel \left({\mathbf{A}}^{\top}\mathbf{A}\left({\mathbf{B}}_{1}{\mathbf{B}}_{2}\right)\right)\\ \phantom{\rule{1em}{0ex}}\times {\left({\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}\right)}_{j}{\parallel}_{2}^{2}\\ \le \sum _{j=1}^{K}\parallel {\mathbf{A}}^{\top}\mathbf{A}\left({\mathbf{B}}_{1}{\mathbf{B}}_{2}\right)\parallel {}_{2}^{2}\phantom{\rule{2.77626pt}{0ex}}\\ \phantom{\rule{1em}{0ex}}\times \parallel {\left({\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}\right)}_{j}{\parallel}_{2}^{2}\\ \le \parallel {\mathbf{A}}^{\top}\mathbf{A}\left({\mathbf{B}}_{1}{\mathbf{B}}_{2}\right)\parallel {}_{2}^{2}\\ \phantom{\rule{1em}{0ex}}\times \sum _{j=1}^{K}\parallel {\left({\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}\right)}_{j}{\parallel}_{2}^{2}\\ \le \parallel {\mathbf{A}}^{\top}\mathbf{A}\left({\mathbf{B}}_{1}{\mathbf{B}}_{2}\right)\parallel {}_{2}^{2}\phantom{\rule{2.77626pt}{0ex}}\\ \phantom{\rule{1em}{0ex}}\times \parallel {\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}{\parallel}_{2}^{2}\end{array}$(18)From (18), taking into account that the spectral norm is submultiplicative ($\parallel \mathbf{\text{PQ}}\parallel {}_{2}\le \parallel \mathbf{P}\parallel {}_{2}\phantom{\rule{2.77626pt}{0ex}}\parallel \mathbf{Q}\parallel {}_{2},\phantom{\rule{2.77626pt}{0ex}}\forall \mathbf{P}\in {\mathbb{R}}^{M\times N},\phantom{\rule{2.77626pt}{0ex}}\forall \mathbf{Q}\in {\mathbb{R}}^{N\times T}$), it follows that:$\begin{array}{rcl}\parallel \nabla {F}_{\mathbf{B}}\left({\mathbf{B}}_{1}\right)& & \nabla {F}_{\mathbf{B}}\left({\mathbf{B}}_{2}\right){\parallel}_{2}^{2}\le \parallel {\mathbf{A}}^{\top}\mathbf{A}\parallel {}_{2}^{2}\phantom{\rule{2.77626pt}{0ex}}\\ \parallel {\mathbf{B}}_{1}& & {\mathbf{B}}_{2}\parallel {}_{2}^{2}\phantom{\rule{2.77626pt}{0ex}}\parallel {\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}{\parallel}_{2}^{2}\end{array}$and, using the fact that $\parallel \mathbf{P}\parallel {}_{2}^{2}\le \parallel \mathbf{P}{\parallel}_{2}^{2},\phantom{\rule{2.77626pt}{0ex}}\forall \mathbf{P}\in {\mathbb{R}}^{M\times N}$, we obtain:$\begin{array}{ll}\parallel \nabla {F}_{\mathbf{B}}\left({\mathbf{B}}_{1}\right)\nabla {F}_{\mathbf{B}}\left({\mathbf{B}}_{2}\right){\parallel}_{2}^{2}& \le \parallel {\mathbf{A}}^{\top}\mathbf{A}\parallel {}_{2}^{2}\phantom{\rule{2.77626pt}{0ex}}\parallel {\mathbf{B}}_{1}{\mathbf{B}}_{2}{\parallel}_{2}^{2}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\times \parallel {\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}{\parallel}_{2}^{2}\phantom{\rule{2em}{0ex}}\\ \le L\phantom{\rule{2.77626pt}{0ex}}\parallel {\mathbf{B}}_{1}{\mathbf{B}}_{2}{\parallel}_{2}^{2}\phantom{\rule{2em}{0ex}}\end{array}$(19)
where $L=\parallel {\mathbf{A}}^{\top}\mathbf{A}\parallel {}_{2}\phantom{\rule{2.77626pt}{0ex}}\parallel {\mathbf{C}}_{t1}{\mathbf{C}}_{t1}^{\top}{\parallel}_{2}$.
 3.Proximal operator associated to the nonsmooth function λ∥·∥_{2,1}$\begin{array}{ll}\phantom{\rule{12.0pt}{0ex}}{\text{prox}}_{\lambda \parallel \xb7{\parallel}_{2,1}}\left(\mathbf{B}\right)& =\underset{\mathbf{X}}{\text{argmin}}\left\{\lambda \parallel \mathbf{X}{\parallel}_{2,1}+\frac{1}{2}\parallel \mathbf{X}\mathbf{B}{\parallel}_{2}^{2}\right\}\phantom{\rule{2em}{0ex}}\\ ={\left[{\left({\text{prox}}_{\lambda \parallel \xb7{\parallel}_{2,1}}\left(\mathbf{B}\right)\right)}_{i,:}\right]}_{i=1}^{i=N}\phantom{\rule{2em}{0ex}}\\ ={\left[\frac{{\mathbf{B}}_{i,:}}{\parallel {\mathbf{B}}_{i,:}{\parallel}_{2}}{(\parallel {\mathbf{B}}_{i,:}{\parallel}_{2}\lambda )}_{+}\right]}_{i=1}^{i=N}\phantom{\rule{2em}{0ex}}\end{array}$(20)
where (·)_{+}= max(·,0), and by convention $\frac{0}{0}=0$.
5.1.2 Minimization over C(fixed B)
The matrix $\left[{\mathbf{B}}_{\mathbf{t}}^{\top}{\mathbf{A}}^{\top}\mathbf{A}{\mathbf{B}}_{\mathbf{t}}+{\mathbf{I}}_{K}\right]\in {\mathbb{R}}^{K\times K}$, and K is supposed to be small; therefore, calculating its corresponding inverse matrix is quite cheap.
6 Convergence analysis
We are going to analyze the convergence behavior of Algorithm 1 by using the global convergence theory of iterative algorithms developed by Zangwill [30]. Note that in this theory, the term ‘global convergence’ do not imply convergence to a global optimum for all initial points. The property of global convergence expresses, in a sense, the certainty that the algorithm converges to the solution set. Formally, an iterative algorithm ξ, on the set X, is said to be globally convergent provided, for any starting point x_{0}∈X, the sequence {x_{ n }} generated by ξ has a limit point [31].
In order to use the global convergence theory of iterative algorithms, we need a formal definition of iterative algorithm, as well as the definition of a setvalued mapping (a.k.a pointtoset mapping) [30]:
Definition 6.1
Definition 6.2
which generates a sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ via the rule x_{n+1}∈ξ(x_{ n }), n=0,1,…
Now that we know the main building blocks of the global convergence theory of iterative algorithms, we are in a position to state the convergence theorem related to Algorithm 1:
Theorem 6.1
Then, the limit of any convergent subsequence of ${\left\{{\mathbf{B}}_{\mathbf{t}},{\mathbf{C}}_{\mathbf{t}}\right\}}_{t=1}^{\infty}$ is in Ω_{ B } and Ω_{ C }.
This convergence theorem is a direct application of Zangwill’s global convergence theorem [30]. Before going in this assertion, let us show some definitions and theorems used in the proof.
Definition 6.3
with x_{ ∞ }∈X (we write convergence of subsequences as x_{ n }→x_{ ∞ }, which is equivalent to $\underset{n\to \infty}{\text{lim}}{x}_{n}={x}_{\infty}$).
Definition 6.4
Definition 6.5
 1.
x _{ n }→x _{0} a s n→∞, x _{ n }∈X
 2.
y _{ n }→y _{0} a s n→∞, y _{ n }, y _{0}∈Y
 3.
y _{ n }∈Π(x _{ n })
implies y_{0}∈Π(x_{0}). The map Π is called closed on S⊂X provided is closed at each x∈S.
Theorem 6.2
 1.
Π _{ A } is closed at x _{0}
 2.
Π _{ B } is closed on Π _{ A }(x _{0})
 3.
if x _{ n }→x _{0} and y _{ n }∈Π _{ A }(x _{ n }), then there exists y _{0}∈Y, such that for some sequence $\left\{{y}_{{n}_{j}}\right\}$, ${y}_{{n}_{j}}\to {y}_{0}$ as j→∞.
Then, the composite map Π_{ C }=Π_{ B }∘Π_{ A } is closed at x_{0}.
Lemma 6.1
then, Ψ is closed at x if Ψ(x) is nonempty.
Theorem 6.3
has an optimal solution x^{∗}∈S.
Theorem 6.4
 1.
All point x _{ n } are in a compact set S⊂X.
 2.There is a continuous function $\alpha :X\to \mathbb{R}$ such that
 (a)
if x∉Γ, then α(x ^{′})<α(x)∀x ^{′}∈M _{ x }(x).
 (b)
if x∈Γ, then α(x ^{′})≤α(x)∀x ^{′}∈M _{ x }(x).
 (a)
 3.
The map M _{ x }(x) is closed at x if x∉Γ.
Then, the limit of any convergent subsequence of ${\left\{{x}_{n}\right\}}_{n=0}^{\infty}$ is in Γ. That is, accumulation points x^{∗} of the sequence x_{ n } lie in Γ. Furthermore, α(x_{ n }) converges to α^{∗}, and α(x^{∗})= α^{∗} for all accumulation points x^{∗}.
Proof
where $Z(\mathbf{C},t)=\frac{1}{2}\left\right{\mathbf{\text{A(B}}}_{\mathbf{t}}\mathbf{\text{C)}}\mathbf{Y}{}_{\mathrm{F}}^{2}+\lambda \left{\mathbf{B}}_{\mathbf{t}}\right{}_{2,1}+\frac{1}{2}\left\right\mathbf{C}{}_{\mathrm{F}}^{2}$.
To prove this theorem by using Zangwill’s global convergence theorem, we need to prove that all its corresponding assumptions are fulfilled. In order to prove assumption 1, let us analyze the sequences ${\left\{{\mathbf{B}}_{\mathbf{t}}\right\}}_{t=1}^{\infty}$ and ${\left\{{\mathbf{C}}_{\mathbf{t}}\right\}}_{t=1}^{\infty}$. The sequence ${\left\{{\mathbf{B}}_{\mathbf{t}}\right\}}_{t=1}^{\infty}$ is generated by using FISTA, which is a convergent algorithm (B_{ t }→B_{ ∞ }) that guarantees that B_{ t }∈Ω_{ B }[5, 18]. Hence, using Definition 6.3, we can see that the sequence ${\left\{{\mathbf{B}}_{\mathbf{t}}\right\}}_{t=1}^{\infty}$ generated by (23) lies in a compact set. On the other hand, the sequence ${\left\{{\mathbf{C}}_{\mathbf{t}}\right\}}_{t=1}^{\infty}$ is generated by (22), which guarantees that C_{ t }∈Ω_{ C }. This sequence always converges to a point inside Ω_{ C }, which implies that Ω_{ C } also lies in a compact set. This concludes the proof of assumption 1.
 (a)
if C _{ t }∉Γ, then Z(C _{ t+1 },t+1)<Z(C _{ t },t) ∀C _{ t+1 }∈Φ(C _{ t })
 (b)
if C _{ t }∈Γ, then Z(C _{ t+1 },t+1)≤Z(C _{ t },t) ∀C _{ t+1 }∈Φ(C _{ t })
From (25), we can see that the sequence ${\left\{{\mathbf{C}}_{\mathbf{t}}\right\}}_{t=1}^{\infty}$ will always lie in Γ (because C_{ t } is generated by (22)); therefore, we only need to prove (b).
In order to prove assumption 3, we need to prove that Φ is closed at C if C∉Γ. To do so, we are going to use Theorem 6.2; therefore, we need to prove that Φ_{ B } and Φ_{ C } are both closed maps: from (23) and (24), we can see that their corresponding objective functions are both continuous $\forall \mathbf{B}\in {\mathbb{R}}^{N\times K}$ and $\forall \mathbf{C}\in \phantom{\rule{1em}{0ex}}{\mathbb{R}}^{K\times T}$, respectively; hence, by using Weierstrass Theorem and Lemma 6.1, we can conclude that Φ_{ B } and Φ_{ C } are both closed maps for any C_{ t1 } and B_{ t }, respectively, and by using Theorem 6.2, we can conclude that Φ is closed on any C_{ t1 }.
Finally, from all the previous proofs and Zangwill’s global convergence theorem, it follows that the limit of any convergent subsequence of ${\left\{{\mathbf{B}}_{\mathbf{t}},{\mathbf{C}}_{\mathbf{t}}\right\}}_{t=1}^{\infty}$ is in Ω_{ B } and Ω_{ C }.
7 Numerical experiments
where q=min{N,T} and σ_{ i }(S) denotes the i th singular value of S. Both problems (28) and (29) were solved using the FISTA implementation of the SPArse Modeling Software (SPAMS) [33, 34].
In order to have a reproducible comparison of the different regularization approaches, we generated two synthetic scenarios:

M=128 EEG electrodes, T=161 time instants, N=413 current sources within the brain, but only 12 of them are active: 4 main active sources with their corresponding 2 nearest neighbor sources are also active. The other 401 sources are not active (zero electrical activity). Therefore, in this scenario, the synthetic matrix S is a structured sparse matrix with only 12 nonzero rows (the rows associated to the active sources).

M=128 EEG electrodes, T=161 time instants, N=2,052 current sources within the brain, but only 40 of them are active: 4 main active sources with their corresponding 9 nearest neighbor sources are also active. The other 2012 sources are not active (zero electrical activity). Therefore, in this scenario, the synthetic matrix S is a structured sparse matrix with only 40 nonzero rows (the rows associated to the active sources).
In both scenarios, the simulated electrical activity (simulated waveforms) associated to the four Main Active Sources (MAS) was obtained from a face perceptionevoked potential study [35, 36]. To obtain the simulated electrical activity associated to each one of the active neighbor sources, we simply set it as a scaled version of the electrical activity of its corresponding nearest MAS (with a scaled factor equal to 0.5). Hence, there is a linear relation between the four MAS and their corresponding nearest neighbor sources; therefore, in both scenarios, the rank of the synthetic matrix S is equal to 4.
As forward model (A), we used a threeshell concentric spherical head model. In this model, the inner sphere represents the brain, the intermediate layer represents the skull, and the outer layer represents the scalp [37]. To obtain the values of each one of the components of the matrix A, we need to solve the EEG forward problem [38]: Given the electrical activity of the current sources within the brain and a model for the geometry of the conducting media (brain, skull and scalp, with its corresponding electric properties), compute the resulting EEG signals. This problem was solved by using the SPM software [39]. Taking into account the comments mentioned in Section 2, the N simulated current sources were positioned on a mesh located on the brain cortex, with an orientation fixed perpendicular to it.
where ${\mathbf{Y}}_{{\text{TS}}_{i}}\in {\mathbb{R}}^{{M}_{{\text{TS}}_{i}}\times T}$, and ${\mathbf{A}}_{{\text{TS}}_{i}}\in {\mathbb{R}}^{{M}_{{\text{TS}}_{i}}\times N}$ (TS_{ i } denotes the index set of the rows that belongs to the i th test set). Once the estimated matrix $\widehat{\mathbf{S}}$ has been found, we apply a threshold to remove spurious sources with almost zero activity. We have set this threshold equal to the 1% of the mean energy of all the sources.
7.1 Performance evaluation
In order to evaluate the performance of the regularizers, we compare the waveform and localization of the four MAS present in the synthetic BES matrix against the four MAS estimated by each one of the regularizers. We also compare the sparsity pattern of the estimated BES matrix $\widehat{\mathbf{S}}$ against the sparsity pattern of the synthetic BES matrix S, as well as the synthetic and predicted EEG measurements.
From these results, we can see that the proposed Matrix Factorization approach outperforms both the Group Lasso and Trace Norm regularizers. The main reason for this is because it combines their two main features: it combines the structured sparsity (from Group Lasso) and the low rank (from Trace Norm) into one unified framework, which implies that it is able to induce structured sparselowrank solutions which are biologically plausible: few active sources, with linear relations between them.
8 Conclusions
We have presented a novel approach to solve the EEG inverse problem, which is based on matrix factorization and regularization. Our method combines the ideas behind the Group Lasso (structured sparsity) and Trace Norm (low rank) regularizers into one unified framework. We have also developed and analyzed the convergence of an alternating minimization algorithm to solve the resulting nonsmoothnonconvex regularization problem. Finally, using simulation studies, we have compared our method with the Group Lasso and Trace Norm regularizers when they are applied directly to the target matrix, and we have shown the gain in performance obtained by our method, hence proving the effectiveness and efficiency of the proposed algorithm.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. They are also grateful to Dr. Carsten Stahlhut, from DTU Informatics, for valuable discussions about EEG brain imaging and also to Dr. Alexandre Gramfort, from Telecom ParisTech, for his advices related with EEG signal processing and nonsmooth convex programming theory applied to the EEG inverse problems. This work has been partly supported by Ministerio de Economía of Spain (projects ‘DEIPRO’ (id. TEC200914504C0201) and ‘COMONSENS’ (id. CSD200800010), ‘ALCIT’ (id. TEC201238800C0301), and ‘COMPREHENSION’ (id. TEC201238883C0201)). Authors LKH and MP were funded by Banco Santander and Universidad Carlos III de Madrid’s Excellence Chair programme.
Authors’ Affiliations
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