Sparse Reverberant Audio Source Separation via Reweighted Analysis Simon Arberet, Pierre Vandergheynst, Rafael E. Carrillo, Jean-Philippe Thiran and Yves Wiaux

Abstract— We propose a novel algorithm for source signals estimation from an underdetermined convolutive mixture assuming known mixing filters. Most of the state-of-the-art methods are dealing with anechoic or short reverberant mixture, assuming a synthesis sparse prior in the time-frequency domain and a narrowband approximation of the convolutive mixing process. In this paper, we address the source estimation of convolutive mixtures with a new algorithm based on i) an analysis sparse prior, ii) a reweighting scheme so as to increase the sparsity, iii) a wideband data-fidelity term in a constrained form. We show, through theoretical discussions and simulations, that this algorithm is particularly well suited for source separation of realistic reverberation mixtures. Particularly, the proposed algorithm outperforms state-of-the-art methods on reverberant mixtures of audio sources by more than 2 dB of signal-to-distortion ratio on the BSS Oracle dataset.

I. I NTRODUCTION Most audio recordings can be viewed as mixtures of several audio signals (e.g., musical instruments or speech), called source signals or sources, that are usually active simultaneously. The sources may have been mixed synthetically with a mixing console or by recording a real audio scene using microphones. The mixing of N audio sources on M channels is often formulated as the following convolutive mixing model: xm (t) =


(amn ? sn )(t) + em (t),

1 ≤ m ≤ M,



where sn (t) ∈ R and xm (t) ∈ R denote sampled time signals of respectively the n-th source and the m-th mixture (t being a discrete time index), amn (t) ∈ R denote the finite (sampled) impulse response of some causal filter, and ? denotes convolution. The goal of the convolutive Blind Source Separation (BSS) problem is to estimate the N source signals sn (t) (1 ≤ n ≤ N ), given the M mixture signals xm (t) (1 ≤ m ≤ M ). When the number of sources is larger than the number of mixture channels (N > M ), the BSS problem is said to be underdetermined and is often addressed by sparsity-based approaches [1]–[4] consisting in the following two steps: i) at the first step the mixing parameters are estimated as in [3], [5]–[7], and ii) at the second step, the source are estimated e.g. using a minimum mean squared error (MMSE) or a Maximum Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected] This work was supported by the European Commissions FP7-FET program, SMALL project (grant agreement no. 225913). The authors are with the Signal Processing Laboratory, Electrical Engineering Department, École Polytechnique Fédérale de Lausanne (EPFL), Station 11, CH-1015 Lausanne, Switzerland. (e-mail: [email protected])

A Posteriori (MAP) estimator given a sparse source prior and the mixing parameters. Audio signals are usually not sparse in the time domain. However, the source coefficients can be estimated in some time-frequency (TF) domain, where they are sparse, by using for example the short time Fourier transform (STFT). The mixing equation (1) is then approximated by the so-called narrowband approximation [8], as follows [9]: ˆ )ˆs(t, f ) + e ˆ (t, f ) ≈ A(f ˆ(t, f ) x


ˆ (t, f ) ∈ CM , ˆs(t, f ) ∈ CN and e ˆ(t, f ) ∈ CM are the where x vectors of mixture, source and noise STFT coefficients in TF ˆ ) = [ˆ bin (t, f ), and A(f amn (f )]M,N m,n=1 is an M ×N complexvalued mixing matrix with elements a ˆmn (f ) being the discrete Fourier transforms of the m × n filters amn (t), ∀t. The narrowband approximation (2) is valid when the length of the mixing filters is short compared to that of the TF analysis window. In reverberant environments, the mixing filters are long and thus the approximation error of the narrowband model is important. This error has been characterised both numerically and theoretically in [10]. Recently, some approaches have been proposed to overcome the limitation of the narrowband approximation, for the mixing filter estimation [7] and the sources estimation [10]–[12]. In [11] and [12], the narrowband approximation has been circumvented via a statistical modeling of the mixing process, while in [10], Kowalski et al. proposed a convex optimization framework based on a wideband `2 mixture fitting cost. There are different ways to model the sparsity and estimate the sources in the TF domain. Binary masking techniques [3], [13]–[16] assume that, in each TF bin, there is at most one active source, i.e. the source TF representations are disjoint. First a mask is estimated for each source, and then the sources are reconstructed from the TF coefficients given by these masks. Another approach is to minimise a convex cost function by penalising the source TF coefficients with a `1 cost [2], [10], [17], which is a sparsity promoting norm. Basis pursuit denoising (BPDN) [18] is a very classical approach that use the `1 cost to promote sparsity and which has been shown to be particularly efficient. First because there are proofs of stable recovery of the sparsest solution, i.e. having the smallest `0 cost, secondly because the `1 cost function is convex, and thus the optimization can be solved in a polynomial time as opposed to the `0 cost which leads to NP-hard problems. The classical BPDN formulation [18] is an unconstrained convex optimization problem, which has an alternative constrained formulation. As for the unconstraint formulation, this constraint formulation has a regularisation parameter. However, when the noise level is known, the parameter of the constrained formulation is far more easy to set than the unconstrained formulation. Moreover, with this constrained


formulation, it is possible to use a reweighing scheme which consists in solving a sequence of (constrained) weighted `1 problems, and which usually give a sparser solution than the one of the simple `1 problem. While synthesis sparse priors have been widely used for the source modeling including `0 cost (i.e. binary masking), `1 cost and mixed norm `1,2 cost [10], analysis sparse priors have been used only in some specific applications such as image restoration where it has shown very promising results. The `1 analysis problem have been studied only recently [19]– [21], and its understanding is today still shallow and partial. Moreover it has to our knowledge never been used in audio source separation. In this article, we focus on addressing the source estimation task, i.e. the second stage of a typical BSS approach, assuming that the mixing filters amn are known. We propose a novel algorithm, for convolutive source separation which is based on a reweighted scheme of an analysis sparse prior. This algorithm introduces three important contributions with respect to the state-of-the-art, and which will be carefully evaluated in the experimental section: i) The algorithm is based on an analysis sparsity prior, which is fundamentally different than the synthesis prior when the analysis operator is a redundant frame (such as a redundant STFT), ii) the algorithm is based on a reweighting scheme that mimics the `0 minimization behaviour and thus promotes a stronger sparsity assumption than the `1 cost, and iii) the algorithm is based on a wideband mixture fitting constraint, and thus first avoid the narrowband approximation, and secondly offers a strong fidelity term (in a new constrained formulation) without the need to fix a regularization parameter (in a standard regularized formulation). The organization of the remainder of the paper is the following. In section II, we introduce our notations and the state-of-the-art approaches. In section III, we discuss convex optimization approaches for sparse inverse problems. In section IV, we introduce our algorithm for audio source separation, and in section V, we provide numerical results of our algorithm compared to the state-of-the-art methods. II. S TATE OF THE A RT A. The convolutive mixture model in operator and matrix form The mixture model (1) can be written as: x = A(s) + e.


where x ∈ RM ×T is the matrix of the mixture composed of xm (t) entries, with 1 ≤ t ≤ T , T being the number of samples. Similarly s ∈ RN ×T is the matrix of sources composed of the sn (t) entries, e ∈ RM ×T is the matrix of the noise composed of em (t) entries, and A : RN ×T → RM ×T is the discrete linear operator defined by [A(s)]m,t =


where xvec ∈ RM T , svec ∈ RN T and evec ∈ RM T are the unfolded vectors of the matrices x, s and e, respectively, and A is a matrix of size M T ×N T composed of M ×N Toeplitz blocks Amn of size T × T . B. Time-frequency transform Underdetermined source separation is an ill-posed inverse problem, which needs additional assumptions to be solved. As stated in the introduction, a powerful assumption is the sparsity of the sources in some representation. Audio signals are known to be sparse in the TF domain, and a popular TF representation is obtained via the STFT. The STFT operator Ψ ∈ CT ×B transforms a multichannel signal s of length T , into a matrix ˜s ∈ CN ×B populated by the B = QF time-frequency column vectors ˆs(t, f ), with t = qL/R, L being the window size, R the redundancy ratio, and q = 1, . . . , Q, and f = 1, . . . , F the time frame index and frequency index, respectively: ˜s = s Ψ,

(amn ? sn )(t).

The adjoint operator A : RM ×T → RN ×T of A is obtained by applying the convolution mixing process with the adjoint filters P a∗nm (t) , amn (−t), ∀t instead of amn , that is: M ∗ [A (x)]n,t = m=1 (a∗nm ? xm )(t).


and the ISTFT is obtained by applying the adjoint operator Ψ∗ ∈ CB×T on the STFT coefficients ˜s: s = ˜s Ψ∗ .


C. Narrowband methods a) Binary masking: The DUET method [3], as most of the convolutive source separation methods [2], [17], [22] is based on the narrowband approximation (2). DUET is a binary masking method [13]–[16] which exploits the assumption that at each TF point, there is approximately one active source. In other words, the number of nonzero entries of the source vector ˆs(t, f ) (defined just after (2)) is equal to one. This can be expressed using the `0 quasi-norm as kˆs(t, f )k0 = 1. DUET, as many BSS methods is a two-step approach composed of a localization-based clustering step followed by a TF (binary) masking step. A lot of efforts has been made in the literature on the clustering step [13]–[16]. However when the mixing filters are known, there is no need to perform such a step. Once the filters (or en estimation of these filters) are known, the DUET method minimizes, for each TF point, a narrowband data fidelity term under a `0 constraint on the source vector: argmin

n=1 ∗

Note that Eq. (3) can be written in the following matrix form: xvec = Asvec + evec ,

ˆ s(t,f )∈CN s.t. kˆ s(t,f )k0 =1

ˆ )ˆs(t, f )k2 . kˆ x(t, f ) − A(f 2


As there is only one possible active source, this problem can be efficiently minimized with a combinatorial optimization strategy, where the data fidelity cost associated with each possible active source is computed, and the selected source is the one leading to the lowest cost.


b) `p norm minimization: Other methods relax the `0 cost with an `p norm, with p > 0, which is defined for a PI vector or a matrix z with I entries as kzkp = ( i=1 |zi |p )1/p . The `p norm promotes sparsity when p ≤ 1 but is not convex when p < 1. As a consequence the `1 norm is often preferred. Experiments [10] show that the DUET method is more robust than the `1 norm minimization in reverberant situations. D. Wideband Lasso Kowalski et al. [10] proposed a variational formulation of the source estimation problem where the data fidelity term used in (6) which is based on the narrowband model (2) is replaced with a wideband data fidelity term according to the time mixing model (1) (equivalently (3)). This leads to the following Wideband Lasso (WB Lasso): 1 argmin kx − A(˜s Ψ∗ )k22 + λP(˜s). ˜ s∈CN ×B 2


The first term is the wideband data fidelity term measuring the fit between the observed mixture x and the mixing model (3) obtained with the source STFT coefficients ˜s given the mixing system A, and the second term P(˜s) is a sparse synthesis regularization term. The parameter λ ∈ R+ governs the balance between the data term and the regularization term. Kowalski et al. [10] proposed to minimize problem (7) for an `1 cost P(˜s) = k˜sk1 and a mixed norm cost P(˜s) = k˜sk21,2 (in order to promote disjointness of the source in the TF domain without constraining the number of active sources per TF bin) using a forward backward scheme (ISTA) [23], [24] and its accelerated versions [25] (FISTA and the Nesterov scheme). Experimental results in [10] showed that, in reverberant situations, the Wideband Lasso (WB Lasso) was significantly better than the Narrowband Lasso (i.e. problem (7) with the narrowband data fidelity term of (6)) and that the `1,2 mixed norm regularization on the sources was not performing better than the `1 norm regularization. III. O PTIMIZATION A. Constrained vs unconstrained problems The Lasso (also called Basis pursuit denoising (BPDN)) problem (7) with P(˜s) = k˜sk1 has an alternative constrained formulation: argmin k˜sk1

noise is very small or null. Indeed, for small λ, FISTA requires a larger number of iterations to reach convergence, and secondly, its convergence speed strongly depends on the chosen initialization. One can refer to [26] for more details on the convergence of this algorithm. So as to let the algorithm converge in practice, it is possible to use a continuation trick [10], [26] also known as warm start, which consists in running the algorithm multiple times, first with a large value of λ, and then iteratively decrease the value of λ and initialize the algorithm with the result of the previous run. This continuation trick is quite efficient in practice, but requires a significant computational effort specially when λ is small (i.e. when the noise is small). B. Reweighted `1 vs `1 minimization The `1 minimization problem is equivalent to `0 minimization when the forward operator A (in Eq. (3)) satisfies certain conditions defined in the context of compressed sensing. The main difference between these two problems, is that the `0 minimization does not depend on the magnitudes of the coefficients, while the `1 does. One way to mimic the minimization behavior of the `0 cost is to replace the `1 norm in (8) by a weighted `1 norm [27], whichP is defined, for a vector or a I matrix z with I entries zi , as i=1 wi |zi |. The idea behind the weighted `1 norm, is that large weights will encourage small components while small weights will encourage larger components. Moreover, if the non-zero components have very small weights, the weighted norm will be independent of the precise value of the non-zero components. As the original signal and weights are not known in advance, the weights are first initialised to have unit value, i.e. the weighted norm corresponds to the `1 norm in that case. From the solution of this convex problem, the weights are then re-estimated in a way that penalizes more the small coefficients and less the large coefficients. The appropriate weights are then computed via a sequence of weighted `1 problems. Each of these weights is essentially the inverse of the value of the solution of the previous problem. This procedure has been observed to be very effective in reducing the number of measurements needed for recovery, and to outperform standard `1 -minimization in many situations, see e.g. [20], [27]. C. Analysis vs synthesis problems

˜ s∈CN ×B

subject to kx − A(˜s Ψ∗ )k2 ≤ ,


which is equivalent to the unconstrained formulation (7) for some (unknown) value of λ and . It follows that determining the proper value of λ in (7) is akin to determining the power limit of the noise [18]. However, there is no optimal strategy to fix the regularization parameter λ even if the noise level is known, therefore constrained problems, such as (8), offer a stronger fidelity term when the noise power is known, or can be estimated a priori. Moreover, as stated in [10], [26], algorithms to optimize (8) such as FISTA, have some convergence issue for small values of λ, which is the case in our problem where the

The BPDN (constrained (8) or unconstrained (7)) defines the optimization in the sparse representation domain finding the optimal representation vector ˜s and then recovering the true signal through the synthesis relation s = ˜s Ψ∗ . These methods are known as synthesis based methods in the literature. Synthesis-based problems may also be substituted by analysis based problems, where instead of estimating a sparse representation of the signal, the methods recover the signal s itself [19]: argmin ks Ψ k1 s∈RN ×T

subject to kx − A(s)k2 ≤ ,



In the case of orthonormal bases, the two approaches are equivalent. However, when Ψ is a redundant frame or an overcomplete dictionary, the two problems are no longer equivalent. The analysis of the geometry of the two problems, studied in [19], [21], show that there is a large gap between the two formulations. One remark to make is that the analysis problem does not increase the dimensionality of the problem (relative to the signal dimension) when overcomplete dictionaries are used. Empirical studies have shown very promising results for the analysis approach [19]. [20] provides a theoretical analysis of the `1 analysis problem coupled with redundant dictionaries in the context of compressed sensing.

D. Source recovery guarantees The literature on Compressed Sensing (CS) and sparse recovery gives some insight about the theoretical guarantees to recover the sources from a mixture by solving problem (9). The sufficient recovery condition for the analysis problem (9) depends [20] both on the analysis sparsity of the sources (k = ks Ψ k0 ) and on properties of the forward operator A, more precisely its matrix form A. The analysis dictionary Ψ is supposed to be a tight frame and can be highly coherent. A sufficient condition for accurate recovery, is that the matrix A satisfies the so-called D-RIP and that the signal has a sparse representation in Ψ. The D-RIP [20] is an extension of the restricted isometry property (RIP) for signals spare in a dictionary, which is a generalisation of a basis, while the RIP is a property that characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. We do not have any proof of the validity of the D-RIP for matrix A, but there are some clues about the conditions of its validity. Indeed, A is a matrix composed of M × N Toeplitz blocks, each of them corresponding to the convolution with the filter amn . It has been shown [28] that Toeplitz matrices satisfy, with an overwhelming probability, the RIP, for filters of length P having i.i.d. Gaussian (or zero mean bounded distribution) entries, when the p sparsity k of the signal of length T is such that k < c P/ log(T ), for a constant c. Other results [20] show that a lot of matrices that satisfy the RIP (e.g. matrices with Gaussian, subgaussian, or Bernoulli entries) also satisfy the D-RIP. If we assume it is also the case for random Toeplitz matrices, we see that, in the case of a singlechannel, single-source “mixture” (M = N = 1), the longer the filters, the larger the sparsity of the sources can be. This discussion suggests that the source estimation, solving problem (9), should be better in reverberant conditions, where the filters are long (large P ) and where the coefficients of the filters can be relatively well modeled by an i.i.d. Gaussian distribution or zero mean bounded distribution. This trend will be confirmed in the experimental section V-D.

IV. R EWEIGHTED A NALYSIS A LGORITHM Our proposed algorithm is based on a reweighted `1 analysis method.

Let us define the weighted `1 problem argmin ks Ψ kW,1 s∈RN ×T

subject to kx − A(s)k2 ≤ ,


N ×B where W ∈P R+ is a matrix with positive entries wij , and kzkW,1 , i,j wij |zij | is the weighted `1 norm and  is a bound on the `2 norm of the noise e. Assuming i.i.d. real Gaussian noise with variance σe2 , the `2 norm term in (10) follows a χ2 distribution√ with M T degrees of freedom. Thus we can set 2 = (M T + 2 2M T )σe2 , where σe2 is the variance of the noise. This choice provides a likely bound for kek2 , since the probability that kek22 exceeds 2 is the probability that a χ2 with M T degrees of freedom exceeds its √ mean, M T , by at least two times the standard deviation 2M T , which is very small. In the noise-free case, we can choose a very small value of  ( → 0), or replace the `2 constraint by the linear equality constraint x = A(s). The solution to (10) is denoted as ∆(x, A, W, ), which is a function of the data vector x, the mixing operator A, the weights matrice W, and the bound  on the noise level estimator. Recall that in the reweighting approach a sequence of weighted `1 problems is solved, and each of these weights is essentially the inverse of the value of the solution of the previous problem. In practice, we update the weights at each iteration, i.e. after solving a complete weighted `1 problem, by the function δ f (δ, ·) = (11) δ+|·|

applied entrywise on the weights wij , ∀i, j. So as to approximate the `0 norm, we used the reweighted `1 algorithm with a homotopy strategy [29, p. 296] which consists in solving a sequence of weighted `1 problems with a decreasing sequence {δ (k) } (k denoting the iteration variable) and warm start initialization. This process is then repeated until a stationary solution is reached. A. The SSRA algorithm The resulting algorithm defined in Algorithm 1 is similar to the Sparsity Averaging Reweighted Analysis (SARA) algorithm recently proposed by part of the authors for compressive imaging [30], [31]. The main difference is that our redundant sparsity operator Ψ is not built as concatenation of orthonormal bases and that the forward operator A involved in (10) to compute ∆(x, A, W, ) is different from the classical CS matrices which are usually populated with i.i.d. random entries. A rate parameter β, with 0 < β < 1, controls the decrease of the sequence δ (k) = βδ (k−1) = β k δ0 such that δ (k) → 0 as k → ∞. However, if there is noise, we set a lower bound as δ (k) > σe˜, where σe˜ is the standard deviation of the noisep in the representation domain and is computed as σe˜ = σe M T /2N B, which gives a rough estimate for a baseline above which significant signal components could be identified.


Algorithm 1: SSRA algorithm for source estimation Input: x, A, Ψ, . Initialize: k := 1, W(0) := 1N ×B , ρ := 1. Compute the solution of Problem (10): s(0) := ∆(x, A, W(0) , ), δ (0) := std(s(0) Ψ). while ρ > η and k < Kmax do Update the weight matrix:  (k) (k−1) Wij := f δ (k−1) , s˜ij , for i = 1, . . . , N, j = 1, . . . , B, with ˜s(k−1) = s(k−1) Ψ. Compute the solution of Problem (10): s(k) := ∆(x, A, W(k) , ). Update δ (k) := max(βδ (k−1) , σe˜). Update ρ := ks(k) − s(k−1) k2 /ks(k−1) k2 k := k + 1 end return s(k−1)

Algorithm 2: Douglas-Rachford algorithm Initialize: k = 0, z(0) ∈ RN ×T , αk ∈ (0, 2), γ > 0. repeat s(k) = proxγf2 (z(k) ) z(k+1) = z(k) + αk (proxγf1 (2s(k) − z(k) ) − s(k) ) k = k + 1. until convergence; return s(k)

As a starting point we set s(0) as the solution of the `1 problem and δ (0) = std(s(0) Ψ), where std(·) stands for the empirical standard deviation of the signal, fixing the signal scale. The reweighting process ideally stops when the relative variation between successive solutions ks(k) − s(k−1) k2 /ks(k−1) k2 is smaller than some bound η, with 0 < η < 1, or after the maximum number of iterations allowed, Kmax , is reached. In our implementation, we fixed η = 10−3 and β = 10−1 . B. Convex optimization algorithms At each iteration of Algorithm 1, the solution ∆(x, A, W, ) of problem (10) has to be computed. Problem (10) consists of minimizing a non-smooth convex function under an `2 -ball constraint. Hence, it is not possible to use conventional smooth optimization techniques based on the gradient. However we can use proximal optimization methods [32] that are efficient convex optimization algorithms that can deal with non-smooth functions and which are particularly well suited for large scale problems. We first introduce the general framework of proximal splitting methods for solving convex problems. We then derive the proximity operators involved in our optimization problem (10), which defined the elementary operations that are required to fit problem (10) into the general proximal splitting framework, and we finally describe the Douglas-Rachford (DR) algorithm which is a well adapted algorithm to solve convex optimization problems involving two non-smooth functions. 1) Proximal splitting methods: Proximal splitting methods solve optimization problems of the form: argmin f1 (z) + f2 (z)



where f1 (z), f2 (z), are convex functions from RI to R. Note that any convex constrained problem can be formulated as an unconstrained problem by using the indicator function iC (·) of the convex constraint set C as one of the functions in

(12), e.g. f2 (z) = iC (z) where C represents the constraint set, and iC (z) = 0 if z ∈ C, and +∞ otherwise. Problem (10) can be seen as a particular instance of problem (12), with f1 (s) = ks Ψ kW,1 and f2 (s) = iB` (s), where 2 B`2 = {s ∈ RN ×T | k A(s) − xk2 ≤ } is the set of matrices s that are satisfying the fidelity constraint kx − A(s)k2 ≤ . The key concept in proximal splitting methods is the use of the proximity operator of a convex function, which is a natural extension of the notion of a projection operator onto a convex set. For example, the proximal operator of the `1 norm is the soft-thresolding operator, and the proximal operator of the indicator function of a constraint is simply the projection operator onto the constraint set. Solution to (12) is reached iteratively by successive application of the proximity operator associated with each function f1 and f2 . See [32] for a review of proximal splitting methods and their applications in signal and image processing. We give in the appendix, the definition of the proximity operator and then derive these operators for the functions f1 (s) = ks Ψ kW,1 and f2 (s) = iB` (s) of our optimization 2 problem (10). 2) Douglas-Rachford Algorithm: The Douglas-Rachford (DR) algorithm [33] solves problem (12) by splitting, i.e. by performing a sequence of calculations involving separately the individual proximity operators proxγf1 and proxγf2 . Moreover it does not require Lipschitz-differentiability of any of the functions fi . The general form of the DR algorithm to solve problem (12) is given in Algorithm 2. This algorithm has been proved to converge to a solution of Problem (12). In practice, we used the value αk = 1, ∀k, and γ = 0.1. While the DR algorithm converges when the number of iterations tends to infinity, we have to choose a stopping criterion. We chose to stop the algorithm when the relative change of the objective value between is less than two successive estimates a given value ηdr , i.e. f1 (s(k) ) − f1 (s(k−1) ) /f1 (s(k) ) < ηdr , or when the number of iterations is greater than a given value Miter . In our experiments, we fixed ηdr = 0.01, and Miter = 200.

V. E XPERIMENTS We evaluated our algorithm with state-of-the-art methods over convolutive mixtures of speech sources in different mixing conditions. For all the experiments, the test signals are sampled at 11 kHz and we use a STFT with cosine windows.



Reweighting X 5 5 5

Analysis X X 5 5

Constrained X X X 5

A. Experimental protocol We used the same experimental protocol as in [10]. The mixing filters were room impulse responses simulated via the Roomsim toolbox [34], with a room size of dimension 3.55 m × 4.45 m × 2.5 m (l × w × h), and with the same microphones and sources configuration as in [10] and in the SASSEC and SISEC evaluation campaigns [12], [35], [36]. The number of microphones was M = 2, and the number of sources was varied in the range 3 ≤ N ≤ 6. The distances of the sources from the center of the microphone pairs was varied between 80 cm and 1.2 m. The different sets of mixing filters were generated corresponding to three different reverberation times RT60 (anechoïc, RT60 = 50 ms, RT60 = 250 ms) and two different microphone spacings d (d = 5 cm and d = 1 m). We also provide an experiment using Binaural Room Impulse Responses (BRIRs) [37] captured in a large room of dimension 8.72 m × 8.02 m × 4.25 m with a RT60 of 890 ms (room D in [37]). A Cortex (MK.2) Head and Torso Simulator (HATS), positioned at (4.36 m, 3.73 m, 1.7 m), and Genelec 8020A loudspeakers, were used to capture the responses. The loudspeakers were placed around the HATS on an arc in the median plane with a 1.5 m radius and angles 45◦ , 15◦ , −10◦ , −5◦ . Each set of mixing filters was convolved with the 10 different sets of speech sources of the BSS Oracle dataset1 [38] composed of 30 male and female speech signals, yielding 10 mixtures per mixing condition, i.e. with a similar protocol as in [10], [11], [38]–[40]. We choose to not add additional noise to the mixture in order to only evaluate the source separation performance of the algorithms. In order to evaluate each feature of our SSRA algorithm (Algorithm 1), we evaluate different variations of it: (i) with and without reweighting, (ii) with synthesis instead of analysis, (iii) using the Lagrangian unconstrained formulation instead of the constrained one. These different variations are summarized in table I. For all the constrained methods, we set  = 10−4 . Note that ideally we would have set  = 0 or used the proximity operator (24) in the noise-free case, but both approaches take an infinite number of iterations to reach convergence, and thus we need anyway to specify a tolerance. Note that the unconstrained synthesis approach corresponds to the WB Lasso method of [10], and that the reweighting approach cannot be easily used in the unconstrained case, because of the difficulty to adjust the λ parameter. Indeed, changing the weights of the weighted `1 norm would automatically change the balance between the data-fidelity term and the regularisation term and thus a new value of λ should be set to compensate this unbalance. 1 available

at http://bass-db.gforge.inria.fr/bss_oracle

As stated before, the WB Lasso method needs the continuation trick to converge but at the cost of additional computation. In the following experiment, we used the continuation trick (CT) with the sequence λ(k) = 10−k , k = 1, . . . , 8. We also give the results without the continuation trick, that is when λ is set directly to λ = 10−8 . The WB Lasso method with the continuation trick is denoted “WB Lasso CT” in the following experiments, while the WB Lasso method without the continuation trick is simply denoted “WB Lasso”. We also compared our algorithm with the classical DUET method [3] for source estimation (i.e. the clustering step of DUET for mixing filters estimation is skipped and the source estimation step of DUET is initialised with the known mixing system A), as well as the results of the near-optimal 2 binary mask [38] which give an upper bound on the performance we can expect with any binary masking method such as DUET or others [14]–[16]. The performance is evaluated for each source using the signal-to-Distortion Ratio (SDR), signal-to-interference ratio (SIR) and signal-to-artefact ratio (SAR), as defined in [42], which indicates the overall quality of each estimated source compared to the target. We then average this measure over all the sources and all the mixtures for each mixing condition. B. Performance analysis as a function of the window size In order to setup the good STFT window size L for each of the method, we made a first experiment where, for a given mixing configuration where N = 4 sources, RT60 = 250 ms and d = 1 m, we compute the source separation performance in term of SDR as a function of L. Figure 1 illustrates the results of this experiment. We can 12 SSRA BPDN A BPDN S WB Lasso CT WB Lasso DUET

10 8 SDR (dB)


6 4 2 0 −2 128



1024 Window Size




Fig. 1. Variation of the average SDR as a function of the STFT window length L over speech mixtures with N = 4 sources, RT60 = 250 ms and d = 1 m.

first notice that the proposed SSRA approach outperforms significantly the state of the art (WB Lasso CT, DUET) 2 Due to the non-orthogonality of the STFT, oracle masks as defined in [38] (there are other definitions of the binary mask as in [41]) have to be determined jointly in all TF points. As it is infeasible for realistic signals involving millions of samples, we rather obtained near-optimal masks by minimizing the distortion on the target estimate in each TF point separately.


C. Performance analysis as a function of redundancy of the STFT It is known [43] that increasing the redundancy of the STFT of synthesis-based methods can improve the source separation performance by reducing the musical noise. However, it also increases the calculation cost. On the other hand, one of the main advantages of the analysis approach compared to the synthesis approach, is that adding redundancy in the sparse transform (i.e. here the STFT) does not increase the size of the solution. As mentioned by Candès et al. [20], incoherence of the columns of Ψ is not necessary to guarantee the source recovery of the analysis problem. What matters is that the columns of the Gram matrix Ψ∗ Ψ are reasonably sparse, which is the case of a redundant STFT. Thus, it is interesting to check if adding redundancy in the STFT, by increasing the overlap ratio between successive windows, can improve the source separation performance. In this experiment we vary the redundancy ratio R by powers of 2 in such a way that Ψ remains a tight frame, which is important, in an algorithm point of view, so as to be able to use Proposition 2 in order to have a fast proximal operator for f1 , and from a theoretical point of view [20] (see section III-D). Results depicted in Fig. 2 show that the synthesis approaches improve their performance when the redundancy increases but it stabilises quickly around R=4 or R=8. Also, as predicted, the time of computation increases quickly with the redundancy. The computation time of the synthesis methods was growing so fast with respect to the redundancy that, for some of these methods, we decided to stop the computation before the end, hence the incomplete curves in Figure 2. Unfortunately, the source separation performance of the analysis approach (BPDN A) decreases when the redundancy increases. We do not have a theoretical explanation of this behaviour and we let this question for future work as it is not the main focus of this study. However, for the reweighting analysis approach (SSRA), we can improve the performance by about 0.5 dB with a redundancy of 4 (we called this variant SSRA 4) instead

11 10 9

SDR (dB)

8 7 6 5 4 3 2 1 2



16 Redundancy


(a) Source separation performance


30 CPU Time per mixture (hours)

whatever the window size. It is interesting to notice that the window length of L = 512 samples is the optimal size for all the methods except DUET, for which the optimal window size is L = 2048. This is to be expected since the narrowband approximation is better when the window size is large compared to the filter length. Other trends can be observed: the analysis approach (i.e. BPDN A) improves slightly the performance (about 0.5 dB in SDR) with respect to the synthesis approach (i.e. BPDN S) and the reweighting with analysis approach (i.e. SSRA) improves the performance more significantly. The performance of the constrained (BPDN S) and the unconstrained (i.e. WB Lasso CT) synthesis approaches, is very similar, as predicted by the theory (when  → 0 and λ → 0). Moreover, we will see in section V-C that the computational time of BPDN S is more than 6 time lower than the one of WB Lasso CT. On the other hand, the performance of WB Lasso without the continuation trick, is significantly worse (more than 2 dB of difference with WB Lasso CT at the optimum window size L = 512).

25 20 15 10 5 0 2



16 Redundancy


(b) Computational performance Fig. 2. Average SDR in Decibel and computational time in hours, as a function of the redundancy R over speech mixtures with N = 4 sources, RT60 = 250 ms and d = 1 m.

of 2 (called SSRA 2, or simply SSRA as before) without significantly increasing the time of computation. D. Performance analysis as a function of reverberation time and microphone spacing We evaluate in Table II the proposed approach and its variants, with respect to the filter length (RT60 ), and microphone spacing. As a comparison, we show the results of state-of-the-art methods, such as DUET [3] and WB Lasso CT [10], as well as the results of the near-optimal binary mask [38]. We also applied an ANOVA at 1%, as in [10], so as to test if a method 3 performs statistically better than another one. Algorithms which perform statistically best are in bold. According to the performance results in term of SDR shown in Table II, the analysis methods seem to work better with 3 The near-optimal binary mask is not taken into account in this ANOVA analysis because it is not a competitive method but rather an oracle estimation.






WB Lasso CT [10]

DUET [3]

Near-optimal binary mask [38]

SDR (dB) SIR (dB) SAR (dB) SDR (dB) SIR (dB) SAR (dB) SDR (dB) SIR (dB) SAR (dB) SDR (dB) SIR (dB) SAR (dB) SDR (dB) SIR (dB) SAR (dB) SDR (dB) SIR (dB) SAR (dB)

simulated RIR via Roomsim toolbox [10], [34] anechoic 50 ms 250 ms 5 cm 1m 5 cm 1m 5 cm 1m 1.6 8.0 3.9 8.8 6.6 10.2 6 14.4 9.7 16.2 14 18.2 9 9.5 7.6 9.9 8.7 11.2 3.4 5.5 4.2 8.0 5.8 8.7 5.2 9.7 7.4 13.4 10.1 14.7 12.1 9.3 9.6 9.7 9.5 10.3 3.9 7.7 4.3 7.2 6.0 7.9 6.9 13.8 9.1 13.4 11.7 14.4 9.6 9.1 7.8 8.7 8.5 9.3 4.9 7.7 4.4 7.3 6.1 8.0 10.1 13.8 9.2 13.4 11.7 14.5 8.1 9.2 7.8 8.7 8.6 9.3 5.9 7.3 5.5 6.4 2.6 3.4 14.2 15.4 13.4 13.9 8.9 9.9 7 8.2 6.7 7.5 4.8 5 9.5 10.8 9.3 10.7 8.1 9.3 18.9 21.0 18.9 20.9 18.5 20.3 10.1 11.3 9.9 11.1 8.6 9.7

realistic (long) RT60 while synthesis methods perform better when the RT60 is small. The reweighting is working well with realistic (long) RT60 , and with shorter RT60 when the microphone spacing is large (1 m), but not when the spacing is small (5 cm). The proposed SSRA method drastically improves performance over the other methods (about 2 dB better than WB Lasso CT) in environments with realistic reverberation (RT60 ≥ 250 ms). Note that it even beats the near-optimal binary mask by more than 10 dB in SDR, SIR and SAR when the reverberation is long (RT60 = 890 ms). The weak performance of the near-optimal binary mask in highly reverberant situation, producing negative SDR values, has already been observed in other articles, e.g. [38]. This is due to the fact that the narrowband approximation is very bad in highly reverberant situations and also that the binary assumption that sources have disjoint TF supports, has limited validity. As for the WB Lasso, the performance of SSRA is less good than DUET in anechoic and low reverberation environment (RT60 = 50 ms) when the microphone spacing is low (d = 5 cm). As shown experimentally in [10], for low reverberant or anechoic environments, it is possible to improve the performance of WB Lasso (and also probably SSRA), by replacing the wideband data fidelity term with the narrowband one, but as discussed in [10], for these mixing conditions, there are a lot of methods based on the narrowband approximation that already work well, and some of them like DUET are moreover very fast. As a consequence, it is not justified to use complex methods like WB Lasso or SSRA for these conditions. For all the sparse recovery methods based on the wideband formulation, the performance is better when the filter is longer. This trend has a theoretical explanation as discussed in section III-D. An analysis of the SIR and SAR is also provided in Table II. We can observe that the analysis approach (BPDN A) tends to have few artefacts (high SAR) while the synthesis approach (BPDN S) tends to have little interferences (high SIR), while

BRIR [37] 890 ms 17.2 cm 13.2 23.7 13.8 10.6 19.3 11.4 10.1 19.1 10.8 10.1 18.9 10.8 −10.9 −2.1 −5.7 −2.9 12.4 −2.5

the reweighed scheme cumulates these two qualities: i.e. high SAR and high SIR. E. Performance with respect to the number of sources In addition to the previous experiments where we evaluated the different methods in the case of a mixture of 4 sources, we compare now the different methods for mixtures with 3 ≤ N ≤ 6 sources and with RT60 = 250 ms and d = 1 m. The results are depicted in Fig. 3. 16 SSRA 4 SSRA 2 BPDN A BPDN S WB Lasso CT WB Lasso DUET

14 12 10 SDR (dB)

RT60 d

8 6 4 2 0 −2 3




Number of sources

Fig. 3. Variation of the average SDR as a function of N over speech mixtures with RT60 = 250 ms and d = 1 m.

Whatever the number of sources, the results depicted in Fig. 3 show that the gap of performance between our methods and the state-of-the-art (WB Lasso CT) is nearly constant with respect to the number of sources and that: i) The analysis approach improves the separation performance compared to the synthesis one by between 0.5 dB and 1 dB of SDR. ii) The reweighting combined with analysis, improves the performance by between 2 and 3 dB of SDR. iii) The constrained approach leads to similar results as the non-constrained one,


which is predicted by the theory, however, a) it has been necessary to use the costly continuation trick so as to make the non-constrained approach converge at a cost of slowing down the computation by a factor of 8 (i.e. the number of λ(k) steps of the continuation trick), b) we cannot simply use (because of the λ setting) the very efficient reweighting scheme, with this approach, iii) In the noise-free case we know that the best setting for λ is zero, however in the noisy case, there is no obvious way to setup λ as opposed to  in the constraint approach.

F. Robustness to error in the filter estimation In the previous experiment, we assumed that we knew the mixing system A. However, in practice, one would usually only have an estimation of it. In order to evaluate the robustness of our approach, we made the following experiment where instead of using the true filters as our input mixing system A, we used a corrupted mixing system Ab where a gaussian noise has been added to the true filters for different amounts of noise. To setup the , we used the methodology described in section IV, with σe being estimated by assuming that the SNR of the mixing system is equal to the SNR of the mixture, i.e. we used σ be := std(x) · 10−SN R/20 , where SN R is the signal-to-noise ratio of the mixing system. The results of the experiment, depicted in figure 4 show that the performance of the tested algorithms 4 are stable with respect to the noise level. We can also note that the performance of SSRA with a corrupted mixing system is better than the one of DUET with a clean mixing system (3.4 dB) when the level of corruption is less than 10 dB (i.e. filter SNR higher or equal to 10 dB).

VI. C ONCLUSION We proposed a novel algorithm based on a reweighted analysis sparse prior for reverberant audio source separation, assuming that the mixing filters are known. This algorithm, based on i) an analysis sparse prior, ii) a reweighting scheme so as to increase the sparsity, iii) a wideband data-fidelity term in a constrained form, has been evaluated on convolutive mixtures of sources and compared with state-of-theart methods. We also evaluated the analysis versus synthesis prior, as well as the reweighted versus non-reweighted scheme, and the constrained versus unconstrained data-fidelity term, on mixtures with different levels of reverberation, different numbers of sources, and when the filters are corrupted with different levels of noise. Our conclusion is that the reweighted analysis sparse prior with a constrained wideband data-fidelity term works better than any of the tested methods for realistic reverberant mixtures and that the gain of performance is by between 2 and 3 dB of SDR with respect to the state-of-the-art. Another advantage of our algorithm, is that we can easily increase the redundancy of the analysis operator, for example by increasing the redundancy of the STFT, without significantly increasing the complexity. A possible extension of this work will be to model the sources with a structured sparsity prior instead of an `1 cost. Another extension would be, in addition to the sources estimation, to estimate the mixing filters, possibly with an alternating optimization approach. It would also be interesting to study more formally the D-RIP for the narrowband and wideband linear operators so as to have a deeper understanding of the source recovery conditions depending on the mixing conditions. A PPENDIX A. Proximity operator We give the definition of the proximity operator and derive these operators for the functions f1 (s) = ks Ψ kW,1 and f2 (s) = iB` (s).


10 8


SDR (dB)


Definition 1. (Proximity operator) Let fi be a lower semicontinuous convex function from CI to ]−∞, +∞]. The proximity operator of fi , denoted proxfi is given by:

4 2

1 proxfi (z) , argmin fi (u) + kz − uk22 . 2 I u∈C

0 −2 −4 −6 0



20 30 Filter SNR (dB)


This definition extends naturally for some matrices z and u, by replacing the `2 norm with the Frobenius norm. We recall that L is a frame if its adjoint satisfies the generalized Parseval relation with bounds ν1 and ν2 : ν1 kzk2 ≤ k L∗ zk2 ≤ ν2 kzk2 ,

Fig. 4. Variation of the average SDR as a function of the filter SNR over speech mixtures with RT60 = 250 ms and d = 1 m.

4 Note that, we did not evaluate the WB Lasso method in this experiment, because first, there is not a proper strategy to setup λ in the noisy case, and also it is known that there is a value of λ for which the results are the same as BPDN S.



with 0 < ν1 ≤ ν2 < ∞. The frame is tight when ν1 = ν2 = ν and L L∗ = ν I. So as to derive the proximity operators of f1 and f2 , we need the following lemma: Lemma 1. If L is a tight frame, i.e. L L∗ = ν I, ν > 0, then proxf (L ·) (z) = z + ν −1 L∗ (proxνf − I)(L z)



Lemma 2. If L is a general frame with bounds ν1 and ν2 , Let µk ∈ (0, 2/ν2 ), Define (k) u(k+1) = µk (I − proxµ−1 f )(µ−1 + L p(k) − y) k u


p(k+1) = z − L∗ u(k+1)



Then p(k) → proxf (L ·−y) (z) linearly. The proof Lemma 1 can be found in [33] the one of Lemma 2 can be found in [44]. Proposition 1. (Prox of λk · k1 ) Let z ∈ CI . Then u = proxλk·k1 (z) = (proxλ|·| (zi ))1≤i≤I is given entrywise by soft thresholding: zi (|zi | − λ)+ (18) ui = proxλ|·| (zi ) = |zi | where (·)+ = max(0, ·). The proof of this proposition can be found in [24]. Applying Lemma 1, we get a closed form solution of the proximal operator of f1 (s) = ks Ψ kW,1 : ˜ ∈ CN ×B Proposition 2. (Prox of f1 (·) = k · Ψ kW,1 ) Let z N ×T T ×B and z ∈ R . If Ψ ∈ C is a tight frame, i.e. Ψ Ψ∗ = ν I, ×B and W ∈ RN is a matrix of positive weights wij , then + proxk·Ψ kW,1 (z) = z + ν −1 (proxνk·kW,1 − I)(z Ψ) Ψ∗ (19) with proxνk·kW,1 (˜ z) = (proxνwij |·| (˜ zij ))1≤i≤N,1≤j≤B


where proxνwij |·| is the soft thresholding operator given in (18) with λ = νwij . Applying Lemma 2, we get an iterative solution of the proximal operator of f2 = iB` : 2

Proposition 3. (Prox of f2 (·) = iB` (·)) If A is a general 2 frame with bounds ν1 and ν2 , Let µk ∈ (0, 2/ν2 ), Define u(k+1) = µk (I − proxik·k

2 ≤

(k) )(µ−1 + A(p(k) ) − x) k u (21)

p(k+1) = z − A∗ (u(k+1) )


where : proxik·k

2 ≤

(u) = min(1, /kuk2 )u.


Then p(k) → proxiB (z) linearly. `2

The tightest possible frame bound ν2 is the largest spectral value of the frame operator A A∗ , which can be computed using the power iteration algorithm as in [10]. Recursion (21)(22) is a forward-backward splitting scheme applied to the dual problem [44] which we accelerated with a Nesterov-type update [25]. In the noise-free case, we can also replace the `2 -ball constraint set B`2 with the affine constraint set Ceq = {s ∈ RN ×T | A(s) = x} and derive a closed form solution of the proximal operator of f2 (·) = iCeq (·) as in [44]: Proposition 4. (Prox of f2 (·) = iCeq (·)) proxiCeq (z) = z + A∗ (A A∗ )−1 (x − A(z)).


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Simon Arberet was born in France in 1980. He graduated from École Supérieure d’Électricité (Supélec), Paris, France in 2003. He received the BA degree in music from University of Paris 8, France, in 2004 and the M.S. degree in computer science from the Pierre and Marie Curie University (Paris 6), France in 2005. He received the Ph.D degree in signal processing from University of Rennes 1, France in 2008. He was with the National Center for Scientific Research (CNRS) from 2005 to 2008 while working towards his Ph.D degree. He worked with the National Institute for Research in Computer and Control Science (INRIA) in 2009. He was scientific researcher at the Signal Processing Laboratory (LTS2), Ecole Polytechnique Fédérale de Lausanne (EPFL) from 2009 to 2012, and he is now with the Swiss Center for Electronics and Microtechnology (CSEM). His research interests include statistical signal processing, sparse models and algorithms in signal/image processing, inverse problems and blind audio source separation.

Pierre Vandergheynst received the M.S. degree in physics and the Ph.D. degree in mathematical physics from the Université catholique de Louvain, Louvain-la-Neuve, Belgium, in 1995 and 1998, respectively. From 1998 to 2001, he was a Postdoctoral Researcher with the Signal Processing Laboratory, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland. He was Assistant Professor at EPFL (2002-2007), where he is now an Associate Professor. His research focuses on harmonic analysis, sparse approximations and mathematical data processing in general with applications covering signal, image and high dimensional data processing, sensor networks, computer vision. He was co-Editor-in-Chief of Signal Processing (2002-2006) and Associate Editor of the IEEE Transactions on Signal Processing (2007-2011), the flagship journal of the signal processing community. He has been on the Technical Committee of various conferences, serves on the steering committee of the SPARS workshop and was co-General Chairman of the EUSIPCO 2008 conference. Pierre Vandergheynst is the author or co-author of more than 70 journal papers, one monograph and several book chapters. He has received two IEEE best paper awards. Professor Vandergheynst is a laureate of the Apple 2007 ARTS award and of the 2009-2010 De Boelpaepe prize of the Royal Academy of Sciences of Belgium.

Rafael E. Carrillo (S’07–M’12) received the B.S. and the M.S. degrees (with honors) in Electronics Engineering from the Pontificia Universidad Javeriana, Bogotá, Colombia, in 2003 and 2006 respectively and the Ph.D. degree in Electrical Engineering from the University of Delaware, Newark, DE, in 2011. He was a visiting lecturer from 2003 to 2006 at the Pontificia Universidad Javeriana and a Teaching Fellow at the University of Delaware from 2009 to 2011. Currently, he is a postdoctoral researcher at the Institute of Electrical Engineering, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. His research interests include signal and image processing, compressive sensing, inverse problems, and robust, nonlinear and statistical signal processing. Dr. Carrillo was the recipient of the “Mejor trabajo de grado” award, given to outstanding master thesis at the Pontificia Universidad Javeriana, in 2006, the University of Delaware Graduate Student Fellowship in 2007 and the Signal Processing and Communications Graduate Faculty Award from the University of Delaware in 2010.


Jean-Philippe Thiran (S’91–M’98–SM’03) received the Elect. Eng. and Ph.D. degrees from the Université catholique de Louvain, Louvain-laNeuve, Belgium, in 1993 and 1997, respectively. Since 2004 he has been a Professor at the Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, and director of the Signal Processing Laboratory (LTS5) at EPFL. Since 2011 he also holds a part time Associate Professor position with the University Hospital Center (CHUV) and University of Lausanne (UNIL). His current scientific interests include image segmentation, prior knowledge integration in image analysis, partial differential equations and variational methods in image analysis and multimodal signal processing, with many applications including remote sensing, medical imaging and human-computer interactions. Dr. Thiran was Co-editor-in-Chief of the “Signal Processing” international journal from 2001 to 2005. He is currently an Associate Editor of IEEE Transactions on Image Processing. Among many other scientific duties, he was the General Chairman of the 2008 European Signal Processing Conference (EUSIPCO 2008), a tutorial co-chair of the IEEE Int. Conf. on Image Processing in 2011 (ICIP-2011) and will be the technical co-chair of ICIP-2015. He is a Senior Member of the IEEE, member of the IVMSP technical committees of the IEEE Signal Processing Society.

Yves Wiaux received the M.S. degree in physics and the Ph.D. degree in theoretical physics from the Université catholique de Louvain (UCL), Louvainla-Neuve, Belgium, in 1999 and 2002, respectively. He was a Postdoctoral Researcher at the Signal Processing Laboratories of the Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland, from 2003 to 2008. He was also a Postdoctoral Researcher of the Belgian National Science Foundation (F.R.S.FNRS) at the Physics Department of UCL from 2005 to 2009. He is now a Senior researcher at the Departments of radiology of the University of Geneva and the Lausanne University Hospital, Switzerland, with joint affiliation to EPFL. His research lies at the intersection between signal processing (inverse problems, sparsity, compressed sensing) and applications to biomedical imaging (magnetic resonance) and astronomical imaging (interferometry) and data analysis (cosmic microwave background).

Sparse Reverberant Audio Source Separation via ...

accelerated versions [25] (FISTA and the Nesterov scheme). Experimental .... norm term in (10) follows a χ2 distribution with MT degrees of freedom. Thus we can ...... and results,” Lecture Notes in Computer Science, vol. 4666, p. 552,. 2007.

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