C ¸agdas ¸bilen
email: cagdas.bilen@technicolor.com

Alexey Ozerov
email: alexey.ozerov@technicolor.com

Patrick Pérez
email: patrick.perez@technicolor.com

AUTOMATIC ALLOCATION OF NTF COMPONENTS FOR USER-GUIDED AUDIO SOURCE SEPARATION

Keywords: Nonnegative matrix factorization, Itakura-Saito divergence, generalized expectation-maximization, source separation

Nonnegative matrix or tensor factorization is a very popular approach for audio source separation. One important problem in nonnegative tensor factorization (NTF) in the context of user-guided audio source separation is the necessity to manually assign the NTF components to audio sources in order to be able to enforce prior information on the sources during the estimation process. In this paper, two new approaches to NTF based source separation are proposed, which do not require any manual component assignment to the sources, but estimate the underlying assignment automatically. Both algorithms use the prior information on the source samples in the estimation process along with either a limit on the minimum number of components each source uses or with a restriction that each component is used by sparse number of sources. The proposed methods are shown to outperform the classic approach with a manual distribution of the components equally among the sources.

INTRODUCTION

Audio source separation remains still very challenging, especially in the single-channel case and for reverberant mixtures [START_REF] Vincent | The signal separation evaluation campaign (2007-2010): Achievements and remaining challenges[END_REF]. Moreover, the resulting source separation performance depends greatly on the amount of prior information about the sources and/or the mixing process that can be incorporated within the corresponding source separation algorithm [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF]. One of new trends in audio source separation, referred as user-guided or user-assisted, consists in providing such prior information directly by a user [START_REF] Smaragdis | Separation by "humming": User-guided sound extraction from monophonic mixtures[END_REF][START_REF] Durrieu | Musical audio source separation based on user-selected f0 track[END_REF][START_REF] Rafii | A simple user interface system for recovering patterns repeating in time and frequency in mixtures of sounds[END_REF][START_REF] Fitzgerald | User assisted separation using tensor factorisation[END_REF][START_REF] Magoarou | Textinformed audio source separation using nonnegative matrix partial co-factorization[END_REF][START_REF] Badawy | On-the-fly audio source separation[END_REF][START_REF] Bryan | Isse: An interactive source separation editor[END_REF], and many of those approaches rely on time-frequency annotations of the spectrograms [START_REF] Laurberg | Structured non-negative matrix factorization with sparsity patterns[END_REF][START_REF] Wang | Musical audio stream separation[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF][START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF][START_REF] Duong | An interactive audio source separation framework based on non-negative matrix factorization[END_REF][START_REF] Lefèvre | A convex formulation for informed source separation in the single channel setting[END_REF][START_REF] Jeong | Informed source separation from monaural music with limited binary time-frequency annotation[END_REF].

Early approaches [START_REF] Laurberg | Structured non-negative matrix factorization with sparsity patterns[END_REF][START_REF] Wang | Musical audio stream separation[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF][START_REF] Duong | An interactive audio source separation framework based on non-negative matrix factorization[END_REF] are based on time annotations only, i.e., a user specifies (e.g., via a dedicated graphical user interface) which source is active at which moment. Then, the time annotations were extended to more general and flexible time-frequency annotations [START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Lefèvre | A convex formulation for informed source separation in the single channel setting[END_REF][START_REF] Jeong | Informed source separation from monaural music with limited binary time-frequency annotation[END_REF]. Finally, interactive frameworks [START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF], where user has a possibility of gradually completing and correcting timefrequency annotations, were proposed as well. In addition, within an interactive framework Duong et al. [START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF] proposed a method allowing dealing with early stage separation errors through uncertainty propagation principle and Jeong and Lee [START_REF] Jeong | Informed source separation from monaural music with limited binary time-frequency annotation[END_REF] proposed a method allowing dealing with user annotation errors through a sparsity-inducing ℓ1norm penalty.

As for the modeling, most of the approaches are based on popular nonnegative matrix factorization (NMF) or nonnegative tensor factorization (NTF) approximations [START_REF] Laurberg | Structured non-negative matrix factorization with sparsity patterns[END_REF][START_REF] Wang | Musical audio stream separation[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF][START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF][START_REF] Duong | An interactive audio source separation framework based on non-negative matrix factorization[END_REF] and many on NMF/NTF with Itakura-Saito (IS) divergence [START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF][START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF][START_REF] Duong | An interactive audio source separation framework based on non-negative matrix factorization[END_REF]. The use of NMF/NTF with Itakura-Saito (IS) divergence [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF] (as compared to other popular loss functions such as Euclidean distance or Kullback-Leibler divergence [START_REF] Lee | Algorithms for non-negative matrix factorization[END_REF]) is attractive for several reasons. First, its scale-invariance property [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF] makes it more suitable for modeling audio spectrograms. Second, its probabilistic Gaussian formulation [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF] facilitates many extensions such as multichannel formulations [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF][START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF] and separation errors management via uncertainty propagation [START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF]. As such, we consider in this work NMF/NTF modeling with IS divergence. Finally, there are also methods that are based on nuclear norm as a low-rank inducing penalty [START_REF] Lefèvre | A convex formulation for informed source separation in the single channel setting[END_REF][START_REF] Jeong | Informed source separation from monaural music with limited binary time-frequency annotation[END_REF], which usually leads to convex optimization problems.

All the above-mentioned NMF/NTF-based methods [START_REF] Laurberg | Structured non-negative matrix factorization with sparsity patterns[END_REF][START_REF] Wang | Musical audio stream separation[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF][START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF][START_REF] Duong | An interactive audio source separation framework based on non-negative matrix factorization[END_REF] suffer from the following important problem. The number of NMF/NTF components (i.e., rank-1 matrices or tensors) affected to each source must be defined in advance instead of being learned given a total budget of K components. For example, one needs specifying in advance: 4 components for "bass guitar", 10 components for "piano", and 6 components for "drums" (see, e.g., [START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF]). 1 This problem leads to the following potential disadvantages. First, the user spends more time by choosing a suitable number of components Kj (j = 1, . . . , J, and J is the total number of sources) instead of choosing just one total number of components K. Second, a suitable number of components chosen by user may be quite different from the one that would lead to the best source separation performance. This problem possibly arises from the fact that the modeling and the management of source activity constraints are not well separated. Indeed, a usual approach to manage time constraints is to set corresponding temporal activations to zero [START_REF] Laurberg | Structured non-negative matrix factorization with sparsity patterns[END_REF][START_REF] Wang | Musical audio stream separation[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF][START_REF] Duong | An interactive audio source separation framework based on non-negative matrix factorization[END_REF], and thus one needs to say in advance which component is affected to which source. However, in case of time-frequency annotations the latter trick does not work, and these constraints are usually managed via heuristic penalizations of the corresponding cost functions [START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF]. However, such penalizations need as well the information about allocation of the components among the sources.

In order to overcome the aforementioned limitations of the existing NMF/NTF-based methods we propose in this work a new method based on time-frequency annotations and the NTF model with IS divergence as in [START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF]. The main novelty of our proposal is that we change the way the time-frequency annotations are taken into account for model and sources estimation. Instead of using heuristic cost function penalizations as in [START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF] we propose estimating the NTF model from all available observations in the maximum likelihood (ML) or constrained ML sense under the corresponding Gaussian modeling (as in [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF]). We also slightly change the way the sources are estimated given the model. This is achieved via Wiener filtering as in [START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF] with the only difference that the filter is applied not only to the mixture, but to all available observations (mixture and partial observations of sources).

The main advantage of the proposed approach over the state of the art is that it does not require the number of NTF components per source Kj to be specified. One only needs specifying the total budget of components K that it then automatically allocated between sources during model estimation. Among the two variations of the proposed approach, the first one assigns a minimum number of components to each source and automatically estimates the assignment of the remaining components. The second variation assumes a sparse structure on the NTF coefficients so that the components contribute to a sparse number of sources to better model the independent behavior of the sources. There are also few secondary potential advantages to the proposed method. First, as it was already mentioned, the optimization criterion is not designed from some heuristic considerations, but it is the ML or a constraint ML criterion for the given model and observations. Second, as it will be explained more in detail below, the time-frequency annotation constraints are taken into account on both steps of estimation of the model and estimation of the sources, while in the state-of-the-art [START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF] these constraints are only taken into account in the model estimation stage. We evaluate our approach in case of temporal annotations on a dataset of music mixtures and compare it with the strategy of uniformly allocating the components.

The rest of the paper is organized as follows. The problem to be solved is introduced and the proposed approach is described in Section 2. Section 3 is devoted to experiments and some conclusions are drawn in Section 4.

PROPOSED APPROACH

Problem Definition

Let us consider a single channel mixture composed of J sources with the mixing equation

x f n = J j=1 s jf n , (1)
in which x f n is the Short-Time Fourier Transform (STFT) coefficient of the mixture at frequency bin f ∈ 1, F and time window n ∈ 1, N and s jf n is the STFT coefficient of the source j for the same time-frequency coordinate. A subset, Ω ⊂ 1, J × 1, F × 1, N , of the time-frequency samples of the sources are assumed to be known up to an additive noise factor b jf n such that

y jf n = s jf n + b jf n , for all (j, f, n) ∈ Ω. (2)
The problem that is considered in this paper is the problem of estimating the source signals, {s jf n } j,f,n , given the mixture, {x f n } f,n and a subset of the measured source samples, {y jf n } (j,f,n)∈Ω . In practice the observed samples can be set as annotated silent periods of the sources. 2 Hence for the rest of this 2 The algorithms presented in this paper are formulated directly in STFT domain assuming the partial source observations are formulated in this domain, e.g., via some user annotations as in [START_REF] Lefèvre | Semi-supervised NMF with time-frequency annotations for singlechannel source separation[END_REF][START_REF] Bryan | An efficient posterior regularized latent variable model for interactive source separation[END_REF][START_REF] Duong | Temporal annotation-based audio source separation using weighted nonnegative matrix factorization[END_REF]. However a formulation with respect to time domain samples is also possible as presented in [START_REF] Bilen | Compressive samplingbased informed source separation[END_REF].

paper we shall assume that the support of measurements, Ω, indicates the time frequency points for which the corresponding source is known to be silent, i.e. y jf n = 0, ∀(j, f, n) ∈ Ω.

Model Assumptions

The measurement noise, b jf n , and the source samples, s jf n , are assumed to be independently Gaussian distributed with variances σ 2 b,jf n and v jf n respectively such that

b jf n ∼ Nc 0, σ 2 b,jf n , s jf n ∼ Nc(0, v jf n ). (3)
The tensor of the source power spectrum is modeled to be low rank such that

v jf n = K k=1 q jk w f k h nk , (4)
where the number of components, K, is sufficiently small. This low rank tensor model is defined by the nonnegative matrices

Q = [q jk ] j,k , W = [w f k ] f,k and H = [h nk ] n,k being, respectively, J × K, F × K and N × K.
The NTF model parameters are represented by θ = {Q, W, H}. Similar to NMF, the matrix H and W represent the power distribution of the K components among the time and frequency bins respectively. The matrix Q represents the power distribution of the K components among the sources.

NTF Model Constraints

Assuming the tensor P = [p jf n ] j,f,n of the power spectra is known (with p jf n = |s jf n | 2 ), NTF model parameters can be estimated using the multiplicative update (MU) rules minimizing the Itakura-Saito (IS) divergence [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF] between the given 3-valence tensor of source power spectra, P, and the 3-valence tensor of the NTF model approximation, V = [v jf n ] j,f,n , defined as

DIS(P V) = j,f,n dIS(p jf n v jf n ), (5)
where dIS(x y) = x/ylog(x/y) -1 is the IS divergence. The motivation to use the IS divergence instead of some other possible divergences (such as Kullback-Leibler divergence) is that the solution minimizing the IS divergence is shown to be equivalent to estimating in the ML sense [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF]. The following simple multiplicative update (MU) rules are derived (as in [START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF]) to estimate the model parameters, Q, W, H, to minimize the IS divergence:

q jk ← q jk f,n w f k h nk p jf n v -2 jf n f,n w f k h nk v -1 jf n , (6)
w f k ← w f k j,n h nk q jk p jf n v -2 jf n j,n h nk q jk v -1 jf n , (7)
h nk ← h nk j,f w f k q jk p jf n v -2 jf n j,f w f k q jk v -1 jf n . ( 8
)
These MU rules must be repeated several times till some convergence criteria are met.

The NTF model parameters can be initialized and constrained in various ways to enable an accurate representation. In this paper we will define and compare 3 schemes:

1. Pre-Assigned Estimation: Given the inactive time-frequency samples, y jf n , (j, f, n) ∈ Ω, one can set h nk = 0, ∀(j, f, n) ∈ Ω and k ∈ Kj , as well as q jk = 0, ∀j ∈ 1, J and k ∈ Kj where Kj represents the set of components belonging to source j and J j=1 #Kj = K. This is the classical method that has been used in various approaches [START_REF] Laurberg | Structured non-negative matrix factorization with sparsity patterns[END_REF][START_REF] Wang | Musical audio stream separation[END_REF][START_REF] Ozerov | Multichannel nonnegative tensor factorization with structured constraints for user-guided audio source separation[END_REF][START_REF] Duong | An interactive audio source separation framework based on non-negative matrix factorization[END_REF]. In this case Q is set such that each component is assigned to only one source and the temporal activation of each component in H is assigned with respect to the temporal activation of the corresponding source. Hence it is required to assign the components to the sources via the sets Kj manually. Once the Q and H matrices are initialized, the silent temporal samples of the sources are no longer used in the signal estimation step unlike the proposed methods. 2. Relaxed Estimation: In this approach a minimum number of components, Kmin, are assigned to each source through Q at the initialization and the remainder of the matrix Q is estimated automatically. When Kmin = 0, this approach is simply estimating all the model parameters, θ, relying on the likelihood maximization. 3. Sparse Estimation: It is often the case that the sources exhibit independent characteristics, hence the representation can be more accurate when the components are assigned only to a single source. The Relaxed Estimation does not enforce each component to exclusively contribute to one source. In Sparse Estimation, the coefficients in Q and H are constrained to be sparse so that a structure of zeros similar to the initialized patterns in the Pre-Assigned Estimation can be attained without the need to manually specify how many components are assigned to each source. It should also be noted that unlike Pre-Assigned and Relaxed Estimation, Sparse Estimation cannot be performed by specific initialization, instead the multiplicative update rules for the model parameters must be modified to perform sparse NTF decomposition as described in [START_REF] Le Roux | Sparse NMF? half-baked or well done?[END_REF].

Signal Estimation Criterion

Let Ω f n ⊂ 1, J be the set of source indices defined as

Ω f n {j|(j, f, n) ∈ Ω} , (9)
and let s f n [s 1f n , . . . , s J f n ] T and #Ω f n -length column vector y f n [y jf n ] T j∈Ω f n . Now we can define an observation vector for each time-

frequency point (f, n) as o f n y T f n , x f n T . Let O = {o f n } f,n
the set of all observed data. In our approach we estimate the model in the maximum likelihood sense, i.e., maximizing the likelihood of the observed data given the model parameters, which writes: 3

p(O|θ) = F f =1 N n=1 1 πΣo f n o f n exp -o H f n Σ -1 o f n o f n o f n , (10)
since o f n may be shown zero-mean complex Gaussian vector with covariance matrix Σo f n o f n that can be expressed as follows:

Σo f n o f n = Σy f n y f n Σ H x f n y f n Σx f n y f n Σx f n x f n , (11)
where until convergence criteria met 7: end procedure and I(Ω f n ) is the J × #Ω f n matrix consisting of the columns of J × J identity matrix IJ that belong to Ω f n .

Σy f n y f n = I H (Ω f n )diag [v jf n ] j I(Ω f n ) (12) + diag [σ 2 b,jf n ]j∈Ω f n , ( 13
)
Σx f n y f n = [v jf n ]j I(Ω f n ), Σx f n x f n = j v jf n , (14)

Algorithm Summary

The general flow of the proposed generalized expectation maximization (GEM) algorithm is the same for all three approaches, with the difference of initialization and update of the NTF model parameters as described in Section 2.3. After the initialization, the algorithm iteratively alternates between updating the posterior power spectra, P, given the model parameters and updating the NTF model parameters by the multiplicative update rules given the posterior power spectra, as described in Section 2.3. The posterior power spectra can be computed as

P = E [|s jf n | 2 ] j,f,n |O; θ = |E {[s jf n ] j,f,n |O; θ}| 2 (15)
+ Σs f n s f n f,n
where

E {[s jf n ] j,f,n |O; θ} = Σ H o f n s f n Σ -1 o f n o f n o f n f,n , (16)
Σs f n s f n = Σs f n s f n -Σ H o f n s f n Σ -1 o f n o f n Σo f n s f n , (17)
Σo f n s f n =      v 1f n • • • 0 . . . . . . . . . 0 • • • v J f n v 1f n • • • v J f n      , Σs f n s f n =    v 1f n • • • 0 . . . . . . . . . 0 • • • v J f n    . (18)
The overall GEM algorithm steps are summarized in Algorithm 1.

EXPERIMENTS

In order to assess the performance of the proposed algorithm, 6 different music mixtures are selected, 4 each composed of 3 sources (bass or drums, guitar and vocals). Source separation is performed on each mixture with the total number of components fixed to 15 5 , using 3 different approaches: using equal number of components pre-assigned for each source (Pre-assigned Estimation), assigning 2 components per source and estimating the rest of the parameters using the proposed estimation method (Relaxed Estimation) and estimating the signals by assuming distribution of components among the sources are sparse (Sparse Estimation). The STFT is computed using a half-overlapping sine window of 1024 samples (64 ms) and the proposed GEM algorithm is run for 500 iterations. The sources in the mixtures are artificially silenced during a percentage of the total time, an example of which is shown in Figure 1.

The overall performance of the compared algorithms can be seen in Figure 2. It can be noticed that the proposed Sparse Estimation method often outperforms blindly assigning equal number of components for each source. This can be attributed to the fact that the sources in the test mixtures often have varying complexity, for example the vocals or guitar are known to have greater spectral patterns variability than the bass and drums. Since the proposed method allows uneven distribution of the components among the sources, the performance is often better. The Relaxed Estimation method can also be seen to have better or close performance to Pre-assigned Estimation for most of the mixtures. This approach also allows automatic assignment of components hence it can benefit from nonuniform distribution of components among the sources. However it does not enforce the components to be exclusively assigned to any of the sources, therefore its performance is usually worse than Sparse Estimation which induces exclusivity by enforcing sparsity in the matrix Q.

It should be noted that, even though not displayed in Figure 2, the use of the initialization in Relaxed Estimation and the sparsity constraint in Sparse Estimation are also tested without the use of silent samples from the sources in the signal estimation step. However the performance was significantly worse than results shown in Figure 2. This demonstrates that the inclusion of the known (silent) samples in the signal estimation is essential for the success of the proposed automatic component allocation algorithms. Another important remark is that, when the known samples of the sources are zero, i.e. the sources are known to be silent, it is necessary to apply the proposed algorithms with non-zero noise variance, σ 2 b,jf n , to avoid numerical errors in matrix inversion during the signal estimation step. Fortunately, this necessity does not create a drawback since the noise variance, σ 2 b,jf n , can be chosen sufficiently small to avoid inaccurate signal reconstructions.

CONCLUSION

In this paper, two new methods are presented which enable automatic allocation of NTF components among multiple sources when performing source separation with low-rank NTF models and user guided annotations. The Relaxed Estimation method does not impose any structure on the NTF model parameters except assigning a small number of components exclusively to each source and relies on the maximum likelihood estimation using the prior information on the annotations to determine the unknown NTF model. The Sparse Estimation method imposes a sparsity prior on the distribution of the components among the sources so that each component is forced to be assigned to a single source. The proposed methods are tested against the classic approach of manually assigning equal number of components to each source on artificially silenced music mixtures, sources of which have variable degrees of complexity. As a result it has been shown that Sparse Estimation outperforms the other approaches in most of the cases, and even the Relaxed Estimation performs better than the classic approach for many test mixtures.

The proposed algorithms are still not perfect and can perform worse than the classic method in rare cases for which strictly constraining the NTF model is more useful than having larger degrees of freedom in the model.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Source 1 (top left), source 2 (top right), source 3 (bottom left) and the mixture for the mixture (bottom right) 5 in the experiment setup.

  GEM algorithm for Source Separation using NTF model 1: procedure SSEPARATION-NTF({x f n } f,n , {y jf n } (j,f,n)∈Ω )

	2:	Initialize non-negative Q, W, H with respect to Pre-
		Assigned, Relaxed or Sparse Estimation
	3:	repeat
	4:	Estimate posterior power spectra,
		P = E [|s jf n | 2 ] j,f,n |O; θ
	5:	Update Q, W, H given P
	6:

• H denotes conjugate transpose. Algorithm 1

Choosing equal number of NMF/NTF components per source is usually suboptimal, since some sources have more spectral diversity than others, and thus need more components to be well represented. For example, it is evident that "piano" needs more components than "bass guitar", simply because a piano may produce about twice as many possible notes as a bass guitar.

The mixtures are taken from the professionally produced music recordings of SiSEC 2015 evaluation campaign (https://sisec.inria.fr/).

[START_REF] Rafii | A simple user interface system for recovering patterns repeating in time and frequency in mixtures of sounds[END_REF] This number of components is observed to be optimum for music signals in the experiments performed in[START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF].


This text talks about urban and landscape planning in Italy and in its Mediterranean possessions during the period between the two world wars.

The Italian unification dates from 1860 and the capital, Rome, was acquired from the catholic Vatican state after a short war in September 1870. Up to this date, the country -involved in a series of independence warshad no colonial vision abroad.

The first experience was the Assab acquisition by the shipping enterprise Rubattino in 1869. A private venture clearly helped by the government. That practice was, in reality, limited to just a port of call in the Red sea; the 1881 census showed 160 inhabitants on six square kilometres (Labanca, 2002, p. 56).

During the 1880s the Italian government decided to participate to the colonial adventure, of course as the last and the smallest participant. Assab was bought by the government in 1882, after the political defeat within the sovereignty over Tunisia, considered quite an Italian dominion because of the number of Italian residents.

In 1885 the Italian army, shipped to Assab dominion, invaded Eritreathat remained a colony up to 1941and in 1889 two other countries in the extreme east Africa: Somalia and Ethiopia. The first one was a dominion up to 1905 when it became a colony as Somalia Italiana to distinguish it from British Somaliland, and the second one made her free winning the Adua battle in 1896.

The 20 th century started with an important chance for the colonial lobby: due to the Ottoman Empire crisis, some territories were not under a real Turkish control. A pretext allowed the invasion of Cyrenaica and Tripolitania in 1911 (land that was later named Libya, re using the old Roman Empire name) and the Greeks Islands of Dodecanese in 1912. Finally, the ambition of a Mediterranean colony was achieved.

Between the two world wars, the country slipped into the fascist regime that officially detained the power from 1922 to 1943. During this period Italy tried to enlarge its land possession, even through ridiculous proposals as the request to the French colonial government to leave a large corridor from the southern Libyan border to the Atlantic Ocean; just to allow a territorial continuity from Mediterranean to Atlantic shores.

During the '30s the regime betted on colonial expansion, directing the effort to all the Italian colonial fields: Dodecanese islands got the legal status of colony in 1930, the Libyan "rebels" annihilation started with the general Graziani actions, Ethiopia was conquered in 1935 and finally Albania was military occupied in 1939. While the other colonial nations were consolidating their territorial presence, Italy was still in expansion phase.

Even architecture and urban planning reflected the myth of the territorial conquests and the regime used the mythical power of the town foundation and building activities as a really rich area of action and a space of propaganda. The dictator himself played a direct role:

"Mussolini è il costruttore, il regime fascista un cantiere e questa è l'età nella vita del mondo della ricostruzione italiana. La mente del Duce non s'arresta alle fasi intermedie nello sviluppo del pensiero; va immediatamente alla soluzione costruttiva. Egli è l'architetto." (Oragno, 1937, p. 7).

In the '30s, the fascist urban planning and architecture were mature and able to propose a new language to realize new settlements. The structuring myths of conquest and fascism created singular architectural and urban artefacts and similar planning and building paradigms all over the fascist controlled areas, inside and outside the country, can be identified.

The rhetorical process of Italian colonisation was stressed by the regime that amplified some of the expansion myths developed during the first phase of colonisation (from 1880s to the 1920s). To reduce the emigration abroad, for example, it supported colonies dedicated to a massive Italian population, proposed for the first time in 1889 during the liberal Crispi's government, re considered by the Giovanni Giolitti's one in 1911 during the invasion of Libya and realised, in fine, during the 1930s.

The representation of the new-Italian-fascist way of life organized the social structure and the material realizations all over the colonies. But the area of action of the "fascist land control" wasn't limited to the external colonies, a part of the Italian territory and Italian population was considered as a colonisation field too. [START_REF] Fuller | Les chantiers de la colonisation : l'architecture, l'urbanisme et la création de la société moderne dans les colonies italiennes 1869-1943[END_REF] Those political acts needed a material reference field, a tangible manifestation of the ideas. Architecture and urbanism were charged to carry out an important role: they materialized the "fascist order". Architecture was the new order set; shapes and forms paradigms were spread all over the future Empire.

Justified by this vision, we can find a similarity in the building conceiving and realization processes. Some of the similitudes across buildings and town plans are also due to the employment of the same professionals through the "Impero". In the last occupied territory, Albania, the evidence of this question is particularly clear.

We can take as an example, the engineer Luigi Luiggi, a specialist of harbours and harbours neighbourhoods. He used to live and work in Italy, but he was in charge of the first study and urban plan for the port and the old town of Tripoli in 1912 (he sketched the first urban plan, systemized by M. Albino Pasini on a Turkish map, plan approved during September 1912) and, in 1929, he had the same charge for the Albanian port of Durazzo.

The architect Gherardo Bosio was in charge of the Gondar (Ethiopia) town plan in 1936 and the Tirana one in 1939, updating the Brasini first one. Brasini himself, worked in Rome (the Ministry of Colonies for example), but also in Libya. Bosio realized the Gondar theatre in 1940; on a form really similar to the Rhodes' one.

Also enterprises had different bases, like the ETAL (Ente Turistico Alberghiero della Libia) that started an Albanian strategy after the Italian occupation realizing and managing some hotels in Durazzo, Scutari and Tirana.

Due to those links, we can observe and estimate the planning similarity or dissimilarity between Italy and its colonies.

TERRITORIAL DOMINATION, FASCIST RULE AND ARCHITECTURE

The core of the fascist rule integrated territorial domination. The modern discipline of urbanism in Italy developed itself during the years between the two World Wars, a period that was particularly rich in architectural experiences. The national institute for urbanism (INU) was founded in 1930 and his first director was the well-known town planner Alberto Calza Bini, several other architects produced arguments relatives to a modern way of town planning. The most famous were Marcello Piacentini, responsible of several urban master plans and Gustavo Giovannoni, one of the fathers of the nowadays thinking about architectural and urban heritage. The aware of town planning knowledge permitted the regime to trust in the future and on their skills in setting up a new country.

"Per fortuna i segni della rinascita sono quanto mai chiari e promettenti. è infatti in questo momento negli studi urbanistici italiani un magnifico risveglio, che, se nasce dall'opera fervida di pochi, si diffonde rapidamente nella viva massa dei giovani e trascina spesso con sì anche le amministrazioni comunali che fanno a gara a bandire concorsi pei piani regolatori delle città." (Giovannoni, 1931, p.5) New "fascist" territories were "created" mostly enlarging colonial useful surface, recovering reclaimed lands in Italy and beginning a new agrarian property structure. New towns and new settlements were founded and used as media to show and communicate the "new order". Nowadays we are assisting to a quarrel about the number of these new towns founded during the fascist period. About the Italian reality, the number is in between the 12 considered by Diane Ghirardo and Kurt Forster in 1985 and the 130 listed by Antonio Pennacchi in 2003.

Both the estimations look like inexact, the first one because it observe only the bigger and well known ones and the second one because of the too large range used to calculate the number, each small place is considered as a town. However, it doesn't exist a global estimation, considering all the colonies and Italy that we can completely trust on. In the fact, the strategy, even in its minimal configuration, was fundamental and produced an enormous "propaganda sermon" over the importance of towns and settlements foundation.

The use of architecture and town planning as a communication mediabecame so evidentthata artistwriter like Filippo Tommaso Marinetti -the father of the Futurism manifestused terms as "biblical scene" and "fascist fastness" to describe the inauguration of Littoria (nowadays Latina) by the Ducein 1932.

Outside the mainland and in Mediterranean area, the subjugation of new lands was a part of the "new roman empire" myth. Also in this case, the concept was introduced by formers liberal governments and fascism found it and used it, amplifying the activities and exploiting it strongly.

In the aim to establish legitimacyfor the Libyan colonisation, for example, the Italian government had immediately instituted an antiquity service: the Ministry of colonies opened officially the Libyan archaeological service in 1913. M. Salvatore Aurigemma was the first director of the soprintendenza and soon started the archaeological digs on Leptis Magna and Cirene sites. A Royal decree (n. 1271 of September 29, 1914) passed quite immediately the Libya' archaeological rules. The most part of the considered patrimony was, of course, ancient Roman Empire ruins, as Sabratha that started to be digged in 1921; while fascist period that took place in 1922.

The Cesareo in Cyrene was seriously retouched and rebuilt as it was done with the Rhodes ruins on the hill that dominate the ancient town that were assembled with a large concrete use.

A clear vision of the role of architecture inside the rules appears in the letter wrote by M. Italo Balbo, governor of Tripolitania and Cyrenaica, dated December 31 st , 1934, to M. Giacomo Guidi soprintendenteresponsible for antiquitiessince 1928.

"non v'è dubbio che la ricostruzione di opere monumentali colpisce la fantasia della massa dei visitatori più del frammento archeologico che pur forma la delizia degli studiosi, ed Ella comprende agevolmente quale importanza sia dal punto di visto politico che turistico riveste il fatto di poter suscitare così vasta impressione tra tutti coloro, italiani e stranieri, per i quali la visita agli scavi è maggiormente fonte di diletto che non di erudizione" (Sangiovanni, 1993, p.93) The political will to use architectural imaginary as a communication support was clear and clearly explicated and new settlements played an important role. Both in communication and propaganda channels: as a product of propaganda and communication but also as a communication and propaganda imagery producers. New settlements were advertising pads; they showed the way to the new era.

Fascism used, as a lot of other authoritarian governments, communication control as a power lever. At the same time, finally during the same years, Italy produced a strong relationship with monuments. Historical monuments and old heritage were legitimating a part of the new era, so the "new era" restored and rebuilt a part of the historical monuments in the aim to transform at its needs.

At the same time, a new monument construction strategy was established, the new monument that has to mark the fascism. The language of signs and symbols was really powered. In a poor country with a high percentage of population with reading difficulties, the old visual show was boosted, messages passed through iconographical instrument as a new cathedral era. The interest of introducing art product in buildings was really high; it occurred, under Giuseppe Bottai ministerial period to a specific law that considered, for public building, 2% of the building budget to be allocated to the art programme.

AGRICULTURE EMPOWERMENT AND DEMOGRAPHY

Italy is an historical urban oriented country; in spite of that, fascist government considered the agrarian development as a priority.

During the '20s, some areas of the country had been already identified as largely underdeveloped and the poorness of some rural zones pushed to the improvement of productive land surface. Conjugate agricultural needs and territorial expansion interests had, as logical answer, the drainage of extended areas and the foundation of several settlements. The Agro Pontino -a large marshy district immediately south of Rome -was the most important drainage project but others localities were taken into consideration in Italy and in the colonies. Government had to find new agricultural areas; new agricultural villages and new towns all over the future empire was the answer.

Some of the positions of the fascist party on these topics were officially presented during the Mussolini speech to the parliament on the Ascension Day, 1927 named "the Ascension speech". Far to be a real bible of fascist positions, this speech contains some clear references to the priority granted to rural areas, contrasting with the urban and industrial development. vi spiegherete quindi come io non ammetta in Italia che le industrie sane, le quali industrie sane sono quelle che trovano da lavorare nell'agricoltura e nel mare" [START_REF] Mussolini | Discorso dell'ascensione, Rome: Camera dei Deputati[END_REF] In the same text, the theme of demographic increment to become a powerful country was also presented. To Mussolini, and his entourage, the number of inhabitants was a direct power indicator.

The demographical theorem was translated in an unambiguous politic, an instrument to attain the sought result.

"l'Italia, per contare qualche cosa, deve affacciarsi sulla soglia della seconda metà di questo secolo con una popolazione non inferiore ai sessanta milioni di abitanti … Se si diminuisce, signori, non si fa l'Impero, si diventa una colonia!" [START_REF] Mussolini | Discorso dell'ascensione, Rome: Camera dei Deputati[END_REF] The other Mussolini assumption pronounced in 1928: "In un'Italia tutta bonificata, coltivata, irrigata, disciplinata, cioè fascista, c'è posto e pane ancora per dieci milioni di uomini", resume clearly the political context and justify the new settlements strategy. A large legislation i and considerable funds boosted the recover of reclaimed land. This politic represented also a safe base to legitimize the fascism himself; the agrarian politic was a link with a large part of the population, already in crisis because of poorness and difficulty in integrate the new economical structure.

To combine prioritiesrurality anddemographya powerful propaganda machine pushed to the development agriculture. Mussolini appeared in agricultural postures as wheat threshing and harvesting or awarding prizes to exceptionally large rural families -a duty to tax singles was also introduced. In fact, during the 20s and the 30s, demography by direct born rate increased faster in rural area than in urban ones. Some historians consider that the priority to rural area was directly connected with the purpose to limit social riots. Obviously also urban population increased during the fascist period due to the benefits ofindustrial growth.

THE '30S: RIPE PROPOSALS IN ITALY AND ABROAD

The fascist regime realized an enormous number of constructions, new buildings and master plans for the major Italian towns like Rome, Naples and Milan. At the same time, new master plans and large improvements were planned and realized all over the possessions like Addis Ababa, Rhodes and Tirana.

Concerning the new settlements, mainly dedicated to agricultural purpose, the regime realized towns, villages and small service centres. Skipping the numeric discussion due to the polemic on the number and status of realized settlements, I clarify the differences in ranking. The conceptual base of these realizations was new land organization and exploiting system. The political aim was to dominate new territories with new structures; applying new fascist mankind.

Pragmatism suggests considering three different levels in subdivision: town, villages and small secondary settlements. The attribution of a one tier standard produce a confused definition and, at the same time, refuse to consider small settlements as a realized part of the land domination, giving an image that's too restricted and out of sense.

Beside this, the most of the realizations were done during the '30s; in 1940 Italy joinedthe Second World War and the construction activities slowed down. The regime fell down in 1943; that left the development agenda not fulfilled: town and territorial complexity, in fact, can't be fully realized in a decade.

To analyse architecture and urbanism it is necessary to take into consideration two different external elements: development of rationalism and modern style and, starting from 1936, the autarchy established in reason of Society of Nations embargo, reacting to Italian invasion of Ethiopia.

Since most of the best Italian architects and town planners were integrated in the fascist system, the new settlements and architecture enjoyed a high quality of project.

"Non è mai esistita, di fatto, un'opposizione al regime fascista operata dagli architetti italiani attivi tra il Venti e il Quaranta, anzi, se mai, fu vero proprio il contrario: si attiva una specie di rincorsa per raggiungere … Futuristi, razionalisti, classicisti, novecentisti … una posizione di predominio e di relativo monopolio culturale" (Muratore, 2002, p. 20).

In the agricultural foundation structures, we can find "Gli anni '20 del nostro secolo testimoniano di una profonda cesura rispetto all'architettura tradizionale, ma in Italia la tradizione subì una ben calcolata riformulazione, con una singolare mescolanza di moderno e di tradizionale in architettura e in urbanistica.

Di tale fenomeno Littoria e Guidonia costituiscono esempi particolarmente efficaci, con tipi storici identificabili, abilmente riadattati per seguire criteri urbanistici moderni, inseriti in nuove versioni di vecchi prototipi.

… architettura e urbanistica sono vincolate a forme e schemi urbani tratti da tipi antichi o medievali … non si può certo parlare di un'autentica modernità … Anche le città non erano vere città -nel senso di centri culturali, sociali, politici e commerciali -e non erano nella stessa misura in cui costituivano invece una parte integrante di una struttura amministrativa destinata a collegare i singoli aricoltori od operai alle organizzazioni fasciste e, attravaerso queste ultime, agli indirizzi razionali del fascismo. (Ghirardo and Forster, 1985, pages 630, 631 and 670) In this text, Ghirardo and Forster presented the well-known "traditional" view of fascist social interpretation of the rural society. Far from regime rehabilitation, we should consider a new analysis of the theme.

Structures of villages answer to the archetypical idea of small rural municipality in northern Italy, with a central social and service squarewith the fascist building as social collector in this caseand a large diffusion of workers independent houses in the agricultural fields.

The formal idea of rural spreading properties produce a territorial conception funded over central nucleus surrounded by arrayed family houses on a large territorial net. The '70s literature, considered that land occupation directly deriving from the regime fear of social and political agglomeration out of order.

During the '90s, due to new references and some personal interview with people who lived in Libyan villages and in the Latina province, I was convinced to re-examine from a different point of view. The diffuse habitat directly on field is something traditional in mixed agricultural and breeding production, large families groups used to live in that way, far to consider themselves as isolated. Small neighbourhood activities acted during the working week and there were weekly general meetings in the "burgs". This created a kind of proximity, of course, less dense than the town and factory one.

There are some concentration areas in the fascist new settlements: three in Italy -Agro Pontino, Sicily and northern Puglieand one on the northern part of Libya. The same strategy was used in AOI (Africa Orientale Italiana, Italian Eastern Africa), Albania, Greece and nowadays Croatia but these experiences had less inhabitants. The Sardinia' new towns were dedicated mainly to mining activities, so their social and architectural structures are completely different (Mussolinia di Sardegna, now Arborea, is an exception as it was founded before the regime existence).

The Agro Pontino model is the first one and a clearly successful one, due to the interesting and rich actors' structure and strategy and because of his subsequent development,realized after the war.

The most important of the new towns was Littoria; nowadays called Latina is second town in Lazio after Rome. Projected by Oriolo Frezzottiit was inaugurated by Mussolini on June the 30 th , 1932. The town is the largest urban result of the ONC (Opera Nazionale Combattenti -Organization for I WW Veterans) activities, with a traditional three squares system and high representative buildings. Littoria/Latina was conceived as the centre of the new reclaimed area and it is still playing that role.

In the same large zone, several other towns and villages were founded; the second most well-known is Sabaudia, which was founded directly by Mussolini, on August 1933, who understood the using foundation mythology importance.

The urban plan of this second town was realized by the establishment town planner Luigi Piccinato, in cooperation with G. Cancellotti, E. Montuori and A. Scalpelli.

Around those two towns, 19 small new towns and villages were born within the same landscape programme. The reclaimed area is inhabited nowadays by nearly 230 thousands inhabitants; in 1921 they were 7.667, so the growth is more than 3.000% (Pennacchi, 2003).

This experience leadedin town planning and landscape -the others in Italy, as the northern Puglie and the Sicilian one. Really important experiences were realized in colonies.

Libya had a special position in the new settlements reality. Agricultural use of the land was considered immediately as the colony core business and a concession system was realised. Due to the difficulties in conquest and to the installing process, up to the '20s the human presence was not really high and composed mainly by large extensive capitalistic exploitations. Starting from the '20s, a new process was introduced, with really heavy infrastructure investments realised by the colonial government in the aim to make Libya suitable to receive a large population.

The colonization strategy instituted a system where the land property passed from the State to labourer families after several years of work on site. The principal actors of this strategy were the Azienda Tabacchi Italiani (Italians Tobacco Enterprise), the Ente per la Colonizzazione della Cirenaica (Company for Cyrenaica Colonization) and, later, the Ente per la Colonizzazione della Libia (Company for Libyan Colonization).

The 1937 agricultural census shows a total of 12288 people composing 2771 families, with more than 80% of them working directly in agriculture and about 700 families located in the new districts developed by the colonisation organisms.

The situation radically changed with the "Ventimila" massive operation organised under Italo Balbo governorate mandate.

On the 1938 and 1939 March on Rome anniversary (October, 28 th ), 30 000 colons arrived in Libya and reached their villages. It was the largest new settlement operation covering a total of more than 133 000 ha.

The number of Italian farmer families passed from 331 in 1930/31 to 6 166 in 1940, while farm cottages passed from 347 to 5 752. In that context natives' families in native' villages conditions were 131 in 1939.

38new villages were founded between 1933 and 1940 (Gresleri, 1993, p. 310): the most of them constituted by a symbolic square, representing the core of the villageusually hosting school, church, basic shops, police station and fascist party building -, and several colonial houses spread all over the agricultural land.

On a quite similar urban structure, we can find similar styles of architecture. The most of the buildings are a combination of modernism, of the myth of fascist order, and of the idea of the Mediterranean house. A reference to the metaphysic aesthetic is quite unforgettable.Only some case differs to those references. D'Annunzio village,for example, is roofed on tiles as a traditional central and northern Italy village.

AFTER THE WORLD WAR

The intersection within the myth of the conquest, the modern style and the genius historici recovering, created singular architectural and urban artefacts.

That combination produced some interesting territorial models that, in spite of their similarity, were soon included by different states. The Italian defeat in the IIWW spread those settlements into different countries (Italy, Greece, Libya, Albania, Yugoslavia/Croatia and the former AOI), producing a plurality of nowadays cultural heritage status.

The interest to that heritage was directly connected to political events. The cultural relationship to colonial period and particularly to fascist one, became a large reference of its identity. It was, and it still is, considered as a political container.

Immediately after the war, Italian political situation was not clear. A large part of the political parties were interested in going on with the Italian colonial adventure, so they tried to conserve the former colonies. Only the Communist party decided to condemn colonisation.

During the instable period between April 25 th , 1945the official end of the IIWW in Italyand June the 2 nd , 1946date of the referendum that decided the passage from kingdom to republiccolonial topics were integrated in the national political restructure.

Even if largely cut off from real contacts with the old-colonies, topics related to Africa took part in national political reconstruction. The evidence of the defeat was the Italian territorial resize, the lost of the colonies and the transfer of a part of Italy to Yugoslavia. The territorial domination remained a special issue of fascism.

As a matter of fact, the double defeatmilitary and diplomaticwas completely assigned to the Mussolini regime and forget it as fast as possible became the easiest solution. The cut off with the former kingdoma not complete cut off that left a large space to permeabilitywas done negating a part of the memory.

Relationship with colonial settlements, favourite site of fascist propaganda, became vague, even in scientific fields. The R. Mariani statement, mainly based on the fascist new towns in 1978, could be enlarged to the global panel of new fascist settlements.

"Tuttavia, mentre lo stato fascista sfruttava al massimo il potenziale propagandistico rappresentato dalle città nuove, gli storici sono stati per lungo tempo riluttanti a strudiarle correndo il rischio persino, di avallare un regime totalmente sureditato. Motivi politici ne decretano la fondazione, e motivi politici ne hanno sancito, per quasi quarant'anni, una pressoché totale misconoscenza" [START_REF] Mariani | Le città nuove del periodo fascista. Com'erano, perché furono costruite, come sono adesso[END_REF].

Memory is on going to (re) discover the experience. During the last 20 years of the woth century, interest on colonisations grows up. Archives where finally opened to historians, scientific results started to be published and political events, in Italy and abroad, allowed the rediscovery of some historical "forbidden" and lost passages.

"

  Questo ancora non basta. C'è un tipo di urbanesimo che è distruttivo, che isterilisce il popolo, ed è l'urbanesimo industriale … l'urbanesimo industriale porta alla sterilità le popolazioni; secondo che altrettanto fa la piccola proprietà rurale. Aggiungete a queste due cause d'ordine economico la infinita vigliaccheria morale delle classi cosiddette superiori della società.

The royal decree 3256/23 (D.R.L.December 30, 1923, n. 3256) and the Serpieri law 753/24(May 18, 1924, n. 753), the law 3134/28 (December

24, 1928, n. 3134 "bonifica integrale") and the decree 215/33 (R.D.L.February 13, 1933 n. 215).