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Generalized independent lowrank matrix analysis using heavytailed distributions for blind source separation
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 28 (2018)
Abstract
In this paper, statisticalmodel generalizations of independent lowrank matrix analysis (ILRMA) are proposed for achieving highquality blind source separation (BSS). BSS is a crucial problem in realizing many audio applications, where the audio sources must be separated using only the observed mixture signal. Many algorithms for solving BSS have been proposed, especially in the history of independent component analysis and nonnegative matrix factorization. In particular, ILRMA can achieve the highest separation performance for music or speech mixtures, where ILRMA assumes both independence between sources and the lowrankness of timefrequency structure in each source. In this paper, we propose two extensions of the source distribution assumed in ILRMA. We introduce a heavytailed property by replacing the conventional Gaussian source distribution with a generalized Gaussian or Student’s t distribution. Convergenceguaranteed efficient algorithms are derived for the proposed methods, and the relationship between the generalized Gaussian and Student’s t distributions in the source model estimation is revealed. By experimental evaluation, the validity of the heavytailed generalizations of ILRMA is confirmed.
Introduction
Blind source separation (BSS) is a technique for separating individual sources from an observed multichannel mixture without knowing the mixing system, such as the spatial locations of the sensors or sources, in advance. In particular, BSS for multichannel audio signals have been well studied so far. This problem can be divided into two situations: underdetermined (number of microphones < number of sources) and (over)determined (number of microphones ≥ number of sources) cases. In the underdetermined case, the mixing system of the sources has to be estimated using several assumptions. For example, sparsenessassumptionbased methods are popular and reliable approaches [1–3]. In contrast, the determined BSS methods often estimate the inverse system of a mixing process, and highquality separation can be achieved compared with the underdetermined BSS methods. In this paper, we only focus on the determined BSS problem.
The most popular and successful algorithm for solving determined BSS problem is independent component analysis (ICA) [4], which assumes statistical independence between the sources and estimates a demixing matrix (the inverse system of the mixing process). For a mixture of audio signals, because the sources are mixed by convolution owing to the room reverberation, ICA is often applied to the timefrequency signals (spectrograms) of the observed signal, which are obtained by a shorttime Fourier transform (STFT). Frequencydomain ICA (FDICA) [5–8] independently applies ICA to the complexvalued timeseries signals in each frequency bin and estimates a frequencywise demixing matrix. Then, the estimated components in each frequency must be aligned over all frequency bins so that the components of the same source are grouped. This postprocessing of FDICA is the socalled permutation problem [6, 9–11], and several criteria have been used to solve this ambiguity of the signal permutation.
Independent vector analysis (IVA) [12–14] is a sophisticated algorithm that can simultaneously estimate the frequencywise demixing matrix and solve the permutation problem using only one objective function. IVA assumes higherorder dependences (cooccurrence among the frequency bins) of each source by employing a spherical generative model of the source frequency vector, thus avoiding the permutation problem. The original IVA employs the spherical multivariate Laplace distribution as the source model (hereafter referred to as Laplace IVA). To improve the statistical model flexibility and source separation performance, Laplace IVA has been extended by replacing its source model with a spherical generalized Gaussian distribution [15] (GGD, also known as an exponential power distribution) in many papers [16–20] (hereafter referred to as GGDIVA), or with a Gaussian distribution having a timevarying variance [21] (hereafter referred to as timevarying Gaussian IVA). Note that the GGD includes the Laplace and Gaussian distributions as special cases.
As another means of audio source modeling and separation, nonnegative matrix factorization (NMF) [22, 23] has been a very common approach during the last decade. NMF is a nonnegativepartsbased lowrank decomposition of an observed nonnegative data matrix that is typically a power or amplitude spectrogram. The decomposed nonnegative parts (bases and activations) can be used for source separation by clustering the parts into each source [24–28]. Also, NMF can be statistically interpreted as a parameter estimation based on a generative model of data, and the distribution of the model defines the objective function (divergence) in NMF. For example, it was revealed that NMF based on Itakura–Saito divergence (ISNMF) assumes an isotropic complex Gaussian distribution independently defined in each timefrequency slot [29], where the variance of each Gaussian distribution can fluctuate depending on time and frequency. For multichannel audio signals, spatial modeling of the mixing system was introduced into the simple NMF, which is called multichannel NMF (MNMF) [30–32], to solve the BSS problem. MNMF estimates the spatial mixing system, whereas ICAbased BSS techniques optimize the demixing matrix, which yields a more stable and efficient algorithm than MNMF.
Motivated by this issue, a new BSS algorithm called independent lowrank matrix analysis (ILRMA) [33–35] has been proposed^{Footnote 1}. In this method, ISNMFbased lowrank source modeling is introduced into the source model of IVA, namely, a lowrank timefrequency structure (cooccurrence among the timefrequency slots) is estimated for each source by NMF, and the frequencywise demixing matrix is optimized taking the NMF source model into account without causing the permutation problem. Since the vector source model in timevarying Gaussian IVA can be interpreted as NMF with a single spectral basis, ILRMA is a natural extension of IVA, where ILRMA utilizes an arbitrary number of bases in the source model. Also, ILRMA can be considered as a dual problem of MNMF (mixing) because ILRMA estimates the demixing matrix, i.e., the inverse of the mixing system (MNMF model), using the lowrank source modeling with NMF.
In this paper, to increase the model flexibility and improve the source separation accuracy, we generalize the source model in ILRMA from the isotropic complex Gaussian distribution of ISNMF to more heavytailed distributions. An important extension is to employ the isotropic complex GGD because it has been reported that GGDIVA can achieve a better separation result in many papers [17, 19, 20]. As another possible generalization, the isotropic complex Student’s t distribution can also be employed in ILRMA. Student’s t distribution includes the Cauchy and Gaussian distributions as special cases and has been used to model audio sources [36, 37]. For use in NMFbased modeling, Cauchy NMF [38], Student’s t NMF (tNMF) [39], and its multichannel extension (tMNMF) [40] have been proposed. The motivation of employing Student’s t distribution is that the Cauchy and Gaussian distributions are a part of the αstable distribution family [41], which has a stable property of a random variable, namely, a linear combination of two independent random variables generated from the same distribution family also has the same distribution up to location and scale parameters. This property is desirable for NMFbased audio source modeling because it justifies the nonnegative linear decomposition of complexvalued signals [42]. For instance, multichannel BSS based on an αstable distribution was recently proposed [43] to benefit from this advantage. However, analytical maximum likelihood (ML) estimation with an αstable distribution is still an open problem because its probability density function (p.d.f.) cannot be represented in a closed form except for several cases. Therefore, instead of employing the αstable distribution family, we adopt Student’s t distribution as the source model in ILRMA, which partly corresponds to the αstable distribution and has the stable property. The relationship among the conventional methods and the proposed ILRMA is depicted in Fig. 1. As shown in this figure, the proposed ILRMA based on the GGD (GGDILRMA) and that based on Student’s t distribution (tILRMA) can be interpreted as a new extension of conventional IVA or ILRMA as well as a computationally efficient solution to the dual problem of MNMF.
Note that this work extends our preliminary work on tILRMA in [44] by developing a new extension, GGDILRMA, and providing additional discussion that explains the theoretical relationship between GGD and tILRMA. Also, the experimental results have been updated with new datasets and conditions for more difficult situations in BSS.
The rest of this paper is organized as follows. Section 2 describes the conventional algorithms including IVA and ILRMA, which are the basis for the proposed GGD and tILRMA described in Section 3. Section 4 reports the validation of the proposed methods by conducting BSS experiments with music and speech sources. Finally, Section 5 concludes this paper.
Conventional method
Formulation
Let \(\tilde {s}_{n}(\tau)\), \(\tilde {x}_{m}(\tau)\), and \(\tilde {y}_{n}(\tau)\) be the source, observed (mixture), and estimated (separated) timedomain signals, respectively, where n=1,⋯,N and m=1,⋯,M are the integral indexes of the sources and channels (microphones), respectively. Also, τ is the integral index of the discrete time. The source signal \(\tilde {s}_{n}(\tau)\) is unknown, and only the observed signal \(\tilde {x}_{m}(\tau)\) can be obtained by using the synchronized multiple microphones. The estimated signal \(\tilde {y}_{n}(\tau)\) is the output data of BSS algorithm. In this paper, these timedomain signals are transformed into the timefrequency domain to treat the convolutive mixture with the room reverberation. The complexvalued timefrequency components of \(\tilde {s}_{n}(\tau)\), \(\tilde {x}_{m}(\tau)\), and \(\tilde {y}_{n}(\tau)\) can be obtained via STFT and are respectively denoted as follows:
where i=1,⋯,I and j=1,⋯,J are the integral indexes of the frequency bins and time frames, respectively, and ^{T} denotes a transpose. We also denote the spectrograms (timefrequency matrices) of the source, observed, and estimated signals as \(\boldsymbol {S}_{n}\ {\in \mathbb {C}^{I{\times }J}}\), \(\boldsymbol {X}_{m}\ {\in \mathbb {C}^{I{\times }J}}\), and \(\boldsymbol {Y}_{n}\ {\in \mathbb {C}^{I{\times }J}}\), whose elements are s_{ij,n}, x_{ij,m}, and y_{ij,n}, respectively. In FDICA, IVA, and ILRMA, the following mixing system is assumed:
where \(\boldsymbol {A}_{i}=(\boldsymbol {a}_{i,1}~\cdots ~\boldsymbol {a}_{i,n}~\cdots ~\boldsymbol {a}_{i,N})\ {\in \mathbb {C}^{M{\times }N}}\) is a frequencywise mixing matrix and a_{i,n} = (a_{i,n1},⋯,a_{i,nm},⋯,a_{i,nM})^{T}is the steering vector for the nth source, which represents the acoustic transfer functions from the nth source to each of the microphones (m=1,⋯,M). The assumed mixing system (4) is called a linear timeinvariant mixture or rank1 spatial model [45] because the spatial covariance of each source image (multichannel observation of each source signal) is restricted to a rank1 matrix in this system [34]. If the mixing system is determined, namely, M=N, and A_{ i } is a nonsingular matrix for all i, we can define the frequencywise demixing matrix \(\boldsymbol {W}_{i}=(\boldsymbol {w}_{i,1} \cdots \boldsymbol {w}_{i,n} \cdots \boldsymbol {w}_{i,N})^{\mathrm {H}}=\boldsymbol {A}_{i}^{1}\) that recovers the source signal, and the estimated signal y_{ ij } is obtained as
where w_{i,n} is the demixing filter for the nth source and ^{H} denotes a Hermitian transpose. The goal of BSS based on FDICA, IVA, or ILRMA is to estimate W_{ i } and obtain y_{ ij } from only the observations x_{ ij } by assuming statistical independence between s_{ij,n} and \(s_{ij,n'}\phantom {\dot {i}\!}\), where n^{′}≠n. In this paper, we only focus on BSS with the determined situation M=N. For the overdetermined situation M>N, principal component analysis is often applied to x_{ ij } for dimensionality reduction so that M=N [46].
IVA
IVA [12–14] is an elegant solution of the permutation problem [6, 9–11], which considers not the frequencywise component x_{ij,m} but the vector of all frequency components, \(\bar {\boldsymbol {x}}_{j,m}=(x_{1j,m},~\cdots ~x_{Ij,m})^{\mathrm {T}}\ {\in \mathbb {C}^{I{\times }1}}\), as an independent variable as shown in Fig. 2. Thus, in IVA, ICA is applied to the timeseries vectors \(\bar {\boldsymbol {x}}_{1,m}~\cdots ~\bar {\boldsymbol {x}}_{J,m}\) while assuming the spherical Idimensional nonGaussian distribution \(p(\bar {\boldsymbol {s}})\approx p(\bar {\boldsymbol {y}})\). For example, the generative model in GGDIVA [16–18, 20] is represented as
where ∥·∥_{2} denotes the L_{2} norm and β>0 is the shape parameter of GGD. Laplace IVA [12–14] corresponds to β=1. Since the probability of (6) only depends on the norm of \(\bar {\boldsymbol {y}}_{j,n}\) (spherical property), the components in the vector \(\bar {\boldsymbol {y}}_{j,n}\) have higherorder dependence. Therefore, frequency components that have similar activations, such as a fundamental frequency and its harmonic components, will be merged as one source avoiding the permutation problem.
By assuming the independence between the source vectors, the objective function (negative loglikelihood function of the observed signal) in IVA can be obtained as
where \(G(\bar {\boldsymbol {y}}_{j,n}) = \log p(\bar {\boldsymbol {y}}_{j,n})\) is called a contrast function and detW_{ i } denotes the determinant of a matrix W_{ i }. Note that the separated signal y_{ij,n} in \(\bar {\boldsymbol {y}}_{j,n}\) includes the variable W_{ i } as \(y_{ij,n}=\boldsymbol {w}_{i,n}^{\mathrm {H}}\boldsymbol {x}_{ij}\).
As another generative model of source signals, an isotropic complex Gaussian distribution with timevarying variance can be utilized in IVA [21], which is represented as
where r_{j,n} is the timevarying variance shared over all frequency bins. Similar to (6), (8) also has a spherical property. Note that even though (8) consists of Gaussian distributions, its marginal distribution over j becomes a superGaussian distribution because the variance can fluctuate depending on j [17].
Regarding the optimization of W_{ i }, a fast and stable optimization algorithm called iterative projection (IP), which is based on a majorizationminimization (MM) algorithm [47], has been derived for ICA [48], Laplace IVA [49], GGDIVA [17], and timevarying Gaussian IVA [21]. IP can achieve better convergence than classical gradientbased algorithms.
ILRMA based on Gaussian distribution
Generative model
ILRMA [33–35] is a method unifying IVA and ISNMF, namely, we assume both statistical independence between sources and the lowrankness of the timefrequency structure in each source. Similar to ICA or IVA, we must assume a nonGaussian distribution as the generative model of source signals to solve the BSS problem. In ILRMA, the following distribution is assumed for the spectrogram of each source:
where t_{ik,n}≥0 and v_{kj,n}≥0 are the nonnegative basis and activation elements (NMF variables) of \(\boldsymbol {T}_{n}\in \mathbb {R}_{\geq 0}^{I{\times }K}\) (basis matrix) and \(\boldsymbol {V}_{n}\in \mathbb {R}_{\geq 0}^{K{\times }J}\) (activation matrix), respectively, k=1,⋯,K is the integral index of the basis, and K is the number of NMF bases (spectral patterns). Also, r_{ij,n}≥0 is a sourcewise timefrequencyvarying variance that corresponds to the lowrank source model. Therefore, the nonnegative matrix T_{ n }V_{ n } represents the rankK model spectrogram of the nth source as Y_{ n }^{.2}≈T_{ n }V_{ n }, where ·^{.q} for matrices denotes the elementwise absolute and qthpower operations. Because of the fluctuation of the variance r_{ij,n} of the time and frequency, the marginal distribution of the generative model (9) over j becomes a superGaussian distribution, which can be used for independencebased BSS.
The local distribution p(y_{ij,n}) is circularly symmetric in the complex plane, and the probability only depends on the power y_{ij,n}^{2}. For this reason, the variance r_{ij,n} corresponds to the expectation value of the power spectrum y_{ij,n}^{2}, namely, r_{ij,n}=E[y_{ij,n}^{2}]. In addition, if we assume that the source spectrogram y_{ij,n} consists of K components c_{ij,nk}, namely, \(y_{ij,n} = {\sum \nolimits }_{k} c_{ij,nk}\), the generative model of c_{ij,nk} also becomes the complex Gaussian distribution because of the stable property as follows:
Note that the variances in p(y_{ij,n}) and p(c_{ij,nk}) are \(r_{ij,n}={\sum \nolimits }_{k} t_{ik,n}v_{kj,n}\) and t_{ik,n}v_{kj,n}, respectively, and they correspond to the expectation values of y_{ij,n}^{2} and c_{ij,nk}^{2} as r_{ij,n}=E[y_{ij,k}^{2}] and t_{ik,n}v_{kj,n}=E[c_{ij,nk}^{2}], respectively. Even if \(y_{ij,n}={\sum \nolimits }_{k} c_{ij,nk}\), the additivity of the power spectra does not hold (\(y_{ij,n}^{2}\neq {\sum \nolimits }_{k} c_{ij,nk}^{2}\)) because of the phase cancelation. However, (9) and (11) mean that the additivity of expectations t_{ik,n}v_{kj,n}=E[c_{ij,nk}^{2}] is satisfied as \(r_{ij,n}={\sum \nolimits }_{k} t_{ik,n}v_{kj,n}\) because of the stable property in Gaussian distribution. Therefore, the generative model (9) theoretically justifies to linearly decompose the power spectrogram y_{ij,n}^{2} into K nonnegative parts t_{ik,n}v_{kj,n}. This advantage was extended to a more general domain in [42] using an αstable distribution, which is a distribution family ensuring the stable property. When α=2, αstable distribution is equal to Gaussian distribution (9) and the additivity of power spectra holds in the expectation sense. When α=1, αstable distribution converges to Cauchy distribution, which ensures the additivity of amplitude spectra in the expectation sense [38].
Objective function and update rules
The objective function of ILRMA is the negative loglikelihood function of the observed signal x_{ ij } and can be obtained from (9) by assuming independence between all sources as
where X={X_{1},⋯,X_{ M }} and Y={Y_{1},⋯,Y_{ N }} are the set of the observed and estimated signals and the independence between sources, \(p(\mathsf {Y}) = \prod _{n} p(\boldsymbol {Y}_{n})\), is assumed. The first and third terms in (13) correspond to the objective function in timevarying Gaussian IVA [21], and the second and third terms correspond to the objective function in ISNMF [29]. The task of the ILRMA algorithm is to minimize the objective function \({\mathcal {L}}\) w.r.t. T_{ n }, V_{ n }, and W_{ i }.
For the optimization of the demixing matrix W_{ i }, similar to IVA, IP can be used for minimizing \(\mathcal {L}\). The update rules based on IP are expressed as follows:
where e_{ n } denotes the N×1 unit vector with the nth element equal to unity. By iterating these algorithms, the demixing matrix W_{ i } is updated so that the objective function (14) decreases. Note that IP does not include any stepsize parameter in its update rules. Regarding the NMF variables T_{ n } and V_{ n }, the following convergenceguaranteed update rules based on the MM algorithm have been derived [50]:
From the above, the objective function can be efficiently optimized by iterating the update rules (16)–(22). However, a scale ambiguity exists between W_{ i } and r_{ij,n} because both of them can determine the scale of the separated signal y_{ij,n}. Therefore, W_{ i } or r_{ij,n} has a risk of diverging during the optimization. To avoid this problem, the following normalization should be applied at each iteration:
where λ_{ n } is an arbitrary sourcewise normalization coefficient such as the sourcewise average power \(\lambda _{n} = \left [ (IJ)^{1} {\sum \nolimits }_{i,j} y_{ij,n}^{2} \right ]^{\frac {1}{2}}\). These normalizations do not change the value of the objective function (13). The scale of the separated signal y_{ij,n} can be restored by applying the following backprojection technique [51] after the optimization:
where \(\hat {\boldsymbol {y}}_{ij,n}=(\hat {y}_{ij,n1}~\cdots ~\hat {y}_{ij,nM})^{\mathrm {T}}\) is a separated source image whose scale is fitted to the observed signals at each microphone and ∘ denotes the Hadamard product (entrywise multiplication). The detailed implementation can be found in [52].
Figure 3 shows the separation principle of ILRMA. When the original sources have a lowrank spectrogram S_{ n }^{.2}, the spectrogram of their mixture, X_{ m }^{.2}, should be more complicated, where the rank of X_{ m }^{.2} should be greater than that of S_{ n }^{.2}. On the basis of this assumption, in ILRMA, the lowrank constraint for each estimated spectrogram Y_{ n }^{.2} is introduced by employing NMF. The demixing matrix W_{ i } is estimated so that the spectrogram of the estimated signal Y_{ n }^{.2} becomes a lowrank matrix modeled by T_{ n }V_{ n }, whose rank is at most K. The estimation of W_{ i }, T_{ n }, and V_{ n } can consistently be carried out by minimizing (13) in a fully blind manner. ILRMA is theoretically equivalent to conventional MNMF only when the rank1 spatial model (4) is assumed, which yields a stable and computationally efficient algorithm for ILRMA. This issue has been well discussed in [34, 35].
Proposed generalization of ILRMA
Motivation and strategy
The conventional ILRMA described in Section 2.3 is based on the isotropic complex Gaussian distribution (9) with a timefrequencyvarying variance r_{ij,n}. For independencebased BSS, nonGaussianity of the source signals is required for the separation, and the model (9) relies on only the fluctuation of the variance r_{ij,n}. If the variance r_{ij,n} is a constant value for all i and j, the model (9) becomes completely Gaussian and the independencebased BSS collapses because the ICA algorithm cannot distinguish multiple Gaussian sources. Therefore, it is worth generalizing the distribution in ILRMA to a more flexible nonGaussian source model. In fact, several approaches based on a nonGaussian distribution with a timefrequencyvarying parameter, such as tNMF, have been proposed, and it has been reported that NMF audio source modeling based on a nonGaussian distribution provides better separation performance [39]. From the IVA side, the source distribution has also been generalized by employing the GGD in many studies [16–20], which gave more accurate BSS results.
For the reasons mentioned above, in this section, we propose two generalizations of the source distribution (generative model) in ILRMA using heavytailed distributions: the isotropic complex GGD and the isotropic complex Student’s t distribution. The former is a natural extension of the conventional generative model (9) and has often been used for the generalization of Laplace IVA or timevarying Gaussian IVA as GGDIVA. The GGD has a shape parameter that controls the super or subGaussianity. In particular, the GGD includes Laplace and Gaussian distributions as special cases. Since most audio sources follow superGaussian distributions, in this paper, we only focus on GGDILRMA with a superGaussian region.
The latter generalization was inspired by a recently developed framework [42] that ensures the stable property of complexvalued random variables, i.e., audio modeling based on an αstable distribution. In this model, similar to ISNMF (11), the decomposition of a complexvalued spectrogram into several nonnegative parts is theoretically justified by the stable property of this distribution family. Student’s t distribution has a degreeoffreedom parameter that determines the shape of the distribution and its superGaussianity. Similar to the GGD, Student’s t distribution includes Cauchy and Gaussian distributions as special cases, which are also special cases of the αstable distribution. Therefore, NMF source modeling (decomposition of complexvalued spectrogram Y_{ n }) in tILRMA is partially justified when the Gaussian or Cauchy distribution is assumed, which is theoretically preferable for audio signal processing.
In addition, we introduce a new domain parameter for NMF modeling in GGD and tILRMA because the generative model of a spectrogram strongly depends on the data domain, such as the selection of the amplitude or powerdomain spectrogram to be used. By controlling both the generative model and the modeling domain of data, we can find a suitable statistical assumption for the audio BSS problem.
ILRMA based on GGD
Generative model and objective function in GGDILRMA
In GGDILRMA, we assume the isotropic complex GGD as the source generative model, which is independently defined in each timefrequency slot as follows:
where σ_{ij,n} is the timefrequencyvarying scale parameter, Γ(·) is a gamma function, and p is the domain parameter in the NMF modeling. The distribution (28) is depicted in Fig. 4a. The p.d.f. becomes identical to (9) when β=2. For β=1, (28) corresponds to the complex Laplace distribution. Similar to (9), the probability of (28) only depends on y_{ij,n}, and the phase of y_{ij,n} is uniformly distributed. From (28), the objective function in GGDILRMA can be obtained as follows by assuming independence between sources:
It is obvious that GGDILRMA (30) coincides with the conventional ILRMA (13) when β=p=2.
Derivation of update rules for GGDILRMA
First, we derive the iterative update rules for obtaining W_{ i } that optimizes (30). Since it is difficult to directly calculate the partial derivative of (30) w.r.t. w_{i,n}, we use an MM algorithm, i.e., we minimize the majorization function (upperbound function) instead of the original objective function. This approach can indirectly minimize the original function (30). Unlike the conventional ILRMA (13), GGDILRMA (30) includes the term \(y_{ij,n}^{\beta } = \left \boldsymbol {w}_{i,n}^{\mathrm {H}}\boldsymbol {x}_{ij}\right ^{\beta }\). If we bound this term by y_{ij,n}^{2}, the MMalgorithmbased efficient optimization, IP, can be used for GGDILRMA because the objective function becomes identical to the conventional ILRMA (13) w.r.t. w_{i,n}. To achieve this, we use the following inequality:
to design a majorization function of (30), where γ_{ij,n}>0 is an auxiliary variable and the equality of (31) holds if and only if
Note that the inequality (31) holds only for 0<β<2, and the other values of β are beyond the scope of this paper. By applying (31) to (30), the majorization function of (30) can be designed as
where \({\mathcal {C}}_{1}\) includes the constant terms that do not depend on w_{i,n}. Since (33) has the same form as the conventional ILRMA (14) w.r.t. w_{i,n}, we can apply IP to the majorization function (33). The update rules for w_{i,n} are derived as (34) with (32) and (17)–(19), where (34) coincides with (16) when β=p=2.
Next, we derive the update rules for T_{ n } and V_{ n }. They can also be derived by designing a majorization function and applying the MM algorithm. Since the term \(\left ({\sum \nolimits }_{k} t_{ik,n}v_{kj,n}\right)^{\frac {\beta }{p}}\) in (30) is always convex for all values of β>0 and p>0, we can bound this term using Jensen’s inequality as
where δ_{ij,nk}>0 is an auxiliary variable that satisfies \({\sum \nolimits }_{k} \delta _{ij,nk}=1\). Also, the term \(\log {\sum \nolimits }_{k} t_{ik,n}v_{kj,n}\) in (30) can be bounded by the tangentline inequality as
where ε_{ij,n}>0 is an auxiliary variable. The equalities of (35) and (36) hold if and only if
respectively. By applying (35) and (36) to (30), the majorization function of (30) can be designed as
where \({\mathcal {C}}_{2}\) includes the constant terms that do not depend on t_{ik,n} or v_{kj,n}. By setting the partial derivative of (39) w.r.t. t_{ik,n} to zero, we have
The solution of this equation is
Then, we can obtain the following update rule for t_{ik,n} by substituting (37) and (38) into (40):
Similar to (41), we can obtain the update rules for v_{kj,n} as
These algorithms can be interpreted as NMF based on the GGD (hereafter called GGDNMF). Since the derivations of the update rules are based on the MM algorithm, they ensure the monotonic decrease in the objective function in each iteration.
ILRMA based on Student’s t distribution
Generative model and objective function in tILRMA
In tILRMA, the isotropic complex Student’s t distribution is independently assumed in each timefrequency slot as the following source generative model:
where ν>0 is the degreeoffreedom parameter that controls the superGaussianity of Student’s t distribution and σ_{ij,n} is defined as (29). The distribution (43) is depicted in Fig. 4b. Similar to (28), this p.d.f. also becomes identical to (9) when ν→∞, and the probability of (28) does not depend on the phase of y_{ij,n}. For ν=1, (43) corresponds to the complex Cauchy distribution. The objective function of tILRMA can be obtained from (43) as
When ν → ∞ and p = 2, (44) coincides with (13).
Derivation of update rules for tILRMA
Similarly in Section 3.2.2, we first derive the iterative update rules for W_{ i } that optimizes (44) using the MM algorithm. In the case of tILRMA, the objective function (44) includes the term \(y_{ij,n}^{2}= \left \boldsymbol {w}_{i,n}^{\mathrm {H}}\boldsymbol {x}_{ij}\right ^{2}\) inside of the logarithm function. Therefore, we bound this term by a linear function of y_{ij,n}^{2} using the following tangentline inequality:
where ζ_{ij,n}>0 is an auxiliary variable and the equality of (45) holds if and only if
By applying (45) to (44), the majorization function of (44) can be designed as
where \({\mathcal {C}}_{3}\) includes the constant terms that do not depend on w_{i,n}. Since (48) has the same form as the conventional ILRMA (14) w.r.t. w_{i,n}, we can apply IP to the majorization function (48). The update rules for w_{i,n} are derived as (49) with (46) and (17)–(19), where (49) coincides with (16) when ν→∞ and p=2.
Next, we derive the update rules for T_{ n } and V_{ n } in the same manner as for GGDNMF. Since the term \(\left ({\sum \nolimits }_{k} t_{ik,n}v_{kj,n}\right)^{\frac {2}{p}}\) in (47) is always convex for any value of p, we can bound this term in the same manner as (35), i.e.,
where η_{ij,nk}>0 is an auxiliary variable that satisfies \({\sum \nolimits }_{k} \eta _{ij,nk}=1\). The equality of (50) holds if and only if
Also, the term \(\log {\sum \nolimits }_{k} t_{ik,n}v_{kj,n}\) in (47) can be bounded by (36). By applying (50) and (36) to (47), we can design a further majorization function of (47) as follows:
where \({\mathcal {C}}_{4}\) includes the constant terms that do not depend on t_{ik,n} or v_{kj,n}. By setting the partial derivative of (52) w.r.t. t_{ik,n} to zero, we have
The solution of this equation is obtained as
Then, we can obtain the following update rule for t_{ik,n} by substituting (51) and (38) into (54):
where
Similar to (55), we can obtain the update rules for v_{kj,n} as
These update rules are similar to those in tNMF [39], but they include the new domain parameter p. Similar to GGDILRMA, all the derivations of the update rules are based on the MM algorithm, thus ensuring their theoretical convergence.
Relationship between GGD and tILRMA
The update rules for t_{ik,n} and v_{kj,n} (the GGD and tNMF parts in GGD and tILRMA, respectively) have an interesting relationship. To clarify this issue, we here interpret these two NMF models in relation to the ISNMF used in the original ILRMA. In GGD and tNMF, we introduced a new parameter p that determines the signal domain of the lowrank modeling, whereas ISNMF is typically applied to the observed power spectrogram (p=2) [29]. To fill the gap in the formulation between ISNMF and GGD or tNMF, we use the following generalized version of the update rules for ISNMF:
where b is a new exponent parameter. Note that (58) and (59) with b=0.5 were originally derived on the basis of the MM algorithm [50], then the update rules with b=1 were derived using the majorizationequalization (ME) algorithm [53]. Recently, we have proven that (58) and (59) with any value of b in the range (0,1] can be interpreted as valid update rules of ISNMF, which are obtained by applying the parametric ME algorithm to the objective function in ISNMF, and can be used for ISNMF or ILRMA without losing the theoretical convergence [54]. This parameter b controls the optimization speed of the NMF variables t_{ik,n} and v_{kj,n}, and b=1 provides the fastest convergence in ISNMF.
For GGDNMF, (41) can be reformulated as
where
The update rule of GGDNMF (60) corresponds to that of ISNMF (58) by assuming the observed signal as (61), which is the “geometric mean” of the data y_{ij,n} and the lowrank model σ_{ij,n} with a ratio of β/p to 1−(β/p). In contrast, for tNMF, (55) can also be rewritten as
As mentioned in [39], the update rule of tNMF (62) corresponds to that of ISNMF (58) by assuming the observed signal to be (63), which is the “harmonic mean” of y_{ij,n}^{2} and \(\sigma _{ij,n}^{2}\) with a ratio of ν to two. The same reformulation can be found for the variable v_{kj,n}.
These facts mean that both NMF algorithms approximate the virtual observation z_{ij,n} by the lowrank model σ_{ij,n} in the ISNMF sense. Since z_{ij,n} consists of the geometric or harmonic mean of the real observation y_{ij,n} and the current lowrank model σ_{ij,n}, lowrankness of the estimated (updated) model σ_{ij,n} tends to be more emphasized compared with the ISNMF decomposition using only the observation y_{ij,n}. In other words, the geometric or harmonic mean in z_{ij,n} prevents σ_{ij,n} from an overfitting to y_{ij,n} by ignoring sparse outliers in y_{ij,n}, which enhances the lowrank decomposition. In (61) or (63), the shape parameter β or ν controls the intensity of such lowrank enhancement in NMF decomposition. However, intriguingly, the domain parameter p also affects the estimation of the lowrank model σ_{ij,n}. In GGDNMF (61), by setting p<β, the geometric mean corresponds to the point externally dividing y_{ij,n} and σ_{ij,n}, which mitigates the intensity of the lowrank enhancement mentioned above. Also, in tNMF, p<2 causes the same behavior because the term \(\sigma _{ij,n}^{p2}\) exists in (63), where the inverse of \(\sigma _{ij,n}^{2p}\) (2−p>0) mitigates the lowrankness.
In summary, as shown in Table 1, smaller β and ν, which correspond to the sparse signal model, can inject the lowrank nature in GGD and tILRMA, whereas a smaller p mitigates the property; the optimal balance among them will be discussed later on the basis of experimental evaluations. For ILRMAbased BSS, we can expect that such lowrank enhancement in NMF leads to the more accurate estimation of W_{ i }. This is because the estimation of the lowrank model σ_{ij,n} becomes robust against outliers in the separated signal y_{ij,n}, and we can correctly capture the inherent spectral parts in the timefrequency structure of each source.
In addition, it is worth mentioning that the exponent value of the NMF update rules, b, is also important for ILRMA. It has been experimentally revealed that a smaller value of b is preferable for achieving better separation performance, although the optimized speed of r_{ij,n} becomes slow. This may be to avoid trapping at a poor local minimum in the early and middle stages of the iteration in ILRMA because the optimization balance between W_{ i } and r_{ij,n} is significant for converging toward a better solution. In GGD or tILRMA, the exponent value in (60) or (62) is defined as p/(β+p) or p/(p+2), respectively. These values become small when p is small and β is large, which may result in a better separation result.
Results and discussion
To evaluate our proposed algorithms, we conducted some BSS experiments using music and speech mixtures. We first compared various conventional methods using observed signals in the case of two sources and two microphones. Then, we compared the conventional and proposed ILRMA in a more difficult situation with three sources and three microphones.
Dataset
We artificially produced monaural dry music sources of the four melody parts depicted in Fig. 5 using a YAMAHA MU1000 PCMbased MIDI tone synthesizer, where several musical instruments were chosen to play these melody parts as shown in Table 2 [55]. The sources were selected to construct typical combinations of instruments with different melody parts (because the sources that simultaneously play the same melody are rare), where only the six combinations, Music 1–Music 6, were adopted for the sake of avoiding combinatorial explosion. For the speech signals, we used the monaural dry speech sources from the source separation task in SiSEC2011 [56] whose data names are dev1_female4 and dev1_male4 [57]. The detailed conditions of these speech signals are described in [56, 57].
BSS experiment with two sources
Conditions
In this experiment, we compared the seven methods shown in Fig. 1, namely, Laplace IVA (optimized by IP) [49], GGDIVA (optimized by IP) [17], MNMF (based on a multivariate complex Gaussian distribution) [32], tMNMF [40], ILRMA (based on a complex Gaussian distribution with a timefrequencyvarying variance) [34], GGDILRMA, and tILRMA. The dry sources used in this experiment are shown in Table 3. To simulate a reverberant mixture, the mixture signals were produced by convoluting the impulse response E2A, which was obtained from the RWCP database [58], with two spatial arrangements, E2A_1 and E2A_2. The recording conditions of the impulse responses in E2A_1 and E2A_2 are depicted in Fig. 6. The other conditions are shown in Table 4. As the evaluation score, we used the improvement of the signaltodistortion ratio (SDR) [59], which indicates the overall separation quality.
Results using fixed parameters
Figures 7, 8, 9, and 10 show examples of the average SDR improvements for Music 1, Music 4, Speech 2, and Speech 4, respectively, with the E2A_1 spatial arrangement. Ten trials with different random seeds were performed for all the methods. Note that conventional ILRMA and GGDILRMA with β=p=2 are the same method. Also, for GGDIVA and tMNMF, the results are shown for the best parameter settings β and ν, as described in the caption of each figure. Similar to the E2A_1 results, we show examples of results for Music 2, Music 3, Speech 1, and Speech 3 with the E2A_2 spatial arrangement in Figs. 11, 12, 13, and 14, respectively. Table 5 indicates the overall average results for all music and speech signals with E2A_1 and E2A_2, respectively, with the best parameter settings. From these results, we can confirm that the conventional and proposed ILRMA mostly outperform the other methods and that there are several settings of p and β or ν that outperform the conventional ILRMA based on the Gaussian distribution. In particular, the proposed methods with p=1.0 or p=0.5 often outperform the same methods with other values of p. However, regarding the parameters β and ν, smaller values produce poor separation results except for tILRMA in Fig. 8 (Music 4). This is because the NMF source model with the heavytailed distribution excessively enhances the lowrankness in the early stage of the iterative optimization, which can cause the serious problem of the sourcewise NMF model incorrectly capturing the spectrogram of the mixture signal by ignoring the important components for discriminating the sources, and the estimated signals become a distorted mixture signal and an artificial residue.
Results using parameter tempering
To solve the problem described in Section 4.2.2, we applied a tempering approach to the parameters in GGD or tILRMA. The detailed tempering process is shown in Fig. 15. In the first half of the optimization, we perform GGDILRMA with β=2 and p=1. Then, the NMF source model T_{ n }V_{ n } is retrained using a temporary estimated signal. After that, ILRMA with the desired distribution (desired parameters p_{T} and β_{T} or ν_{T}) is performed using the pretrained W_{ i }, T_{ n }, and V_{ n }. The intermediate NMF process is based on the same parameters (p_{T} and β_{T} or ν_{T}) as the subsequent ILRMA in the second half of the optimization. This can be considered as a binary tempering approach that avoids overfitting of the source model to the mixture signal. Note that we also attempted a more precise tempering approach involving continuously changing the parameters in every iteration, but the binary tempering approach shown in Fig. 15 achieved the most accurate and stablest results. The reason why we started from not p=2 but p=1 is that a small exponent value, p/(β+p) in (41) and (42) or p/(p+2) in (55) and (57), in the NMF update rules provides better separation as revealed in [54], where the exponent value monotonically decreases as a value of p decreases. Indeed, the results in Section 4.2.2 showed outstanding performance for p=1 rather than p=2.
Figures 16, 17, 18, 19, 20, 21, 22, and 23 show examples of results with the proposed tempering approach, where the signals correspond to Figs. 7, 8, 9, 10, 11, 12, 13, and 14 with parameter tempering, and Table 6 shows the overall average results for all the signals. The results show that the parameter tempering improves the separation, particularly in ILRMA with heavytailed source models. Also, it further improves the results obtained using fixed values of the parameters. In total, the proposed generalization of ILRMA can achieve approximately 1.2 dB improvement in the SDR compared with the conventional ILRMA with the Gaussian model, which is a significant gain in BSS tasks with two sources.
Performance for various signal lengths
In BSS framework, the length of observed signal is important to achieve the better separation performance. This is because the accuracy of statistical estimation decreases when the number of time frames J is insufficient [60, 61]. In the extreme case, the demixing matrix W_{ i } cannot be updated by IP when J=1 because the rank of U_{i,n} in (16), G_{i,n} in (34), or H_{i,n} in (49) becomes unity. However, it is not clarified whether the heavytailed source distribution provides more robust statistical estimation for fewer time frames or not. Thus, in this subsection, we experimentally compare the separation performance of ILRMA, GGDILRMA, and tILRMA for the observed signals with fewer and more time frames.
To simulate the short and long source signals, we utilized the dry sources described in Table 3. As the short dry sources, the music and speech signals were trimmed only to the former half, and their signal lengths were 2.5 s (music) and 5.0 s (speech), respectively. In contrast, the long music and speech signals were produced by repeating the entire length of the dry sources twice, namely, the signal lengths of the long music and speech dry sources become 10.0 s (music) and 20.0 s (speech), respectively. These dry sources were convoluted with E2A_1 to produce the observed mixture signal with two sources, where the combinations of dry sources were the same as those described in Table 3. The other experimental conditions were the same as those in Section 4.2.2.
Figures 24 and 25 show the results of Music 1 for shorter and longer signals, respectively. Also, Table 7 shows the overall average results for all the signals. By comparing these figures and Fig. 7 (the results of Music 1 with the original length), we can confirm that the separation performance of all the methods improves in proportion to the number of time frames J. Similarly to ILRMA, GGD and tILRMA also suffer from the degradation of separation performance depending on the decrease of J regardless of the heavy tail property.
BSS experiment with three sources
To emphasize the advantage of the proposed methods, we investigated a more difficult situation with three sources. In this experiment, for the sake of simplicity, we only compared the conventional ILRMA and the proposed GGD and tILRMA. The used dry sources are shown in Table 8, which were convoluted with the impulse response depicted in Fig. 26. The other conditions were the same as those in Section 4.2.
Table 9 shows the overall average results of each method. Similar to the previous results, the proposed methods outperform the conventional ILRMA, and the tempering approach slightly improves the quality of separation compared with GGD or tILRMA with fixed parameters.
Comparison of computational times
To demonstrate the optimization efficiency of ILRMA, we compared the computational times of Laplace IVA, GGDIVA, MNMF, tMNMF, ILRMA, GGDILRMA, and tILRMA. The update calculation for the NMF parameters in each algorithm was almost the same, but the estimation of the spatial parameter (W_{ i } for ILRMAbased methods and the spatial covariance for MNMFbased methods) was different. Although ILRMAbased methods require one inverse of W_{ i }U_{i,n} for each i and n, MNMFbased methods require J inverses and two eigenvalue decompositions of the M×M matrix. Table 10 shows relative computational times normalized by that of Laplace IVA based on IP [49], where the conditions are the same as in Table 4 and we used MATLAB 9.2 (64bit) with an AMD Ryzen 7 1800X (8 cores and 3.6 GHz) CPU. From this table, we can confirm that the computational time of ILRMAbased methods is not significantly larger than that of IVA, whereas that of MNMFbased methods is significantly larger.
Conclusions
In this paper, we proposed two generalizations of the source distribution assumed in ILRMA that introduce a heavytailed property by using the GGD and Student’s t distribution. The GGD can be considered as a natural extension of the conventional Gaussian source model, and Student’s t distribution partially satisfies the stable property of complexvalued random variables, which is desirable for NMFbased lowrank decomposition. We derived efficient optimization algorithms for GGD and tILRMA, which ensure a monotonic decrease in the objective function and provide faster computation than existing MNMFbased BSS methods. Also, we revealed an interesting relationship between GGD and tNMF: GGDNMF is equivalent to ISNMF upon assuming the geometric mean of the data and the lowrank model as an observation, whereas tNMF corresponds to the same algorithm with the harmonic mean of the data and the lowrank model as previously mentioned. These properties lead to more accurate parameter estimation in an ILRMAbased BSS framework, resulting in higher separation accuracy than the conventional ILRMA with the Gaussian source distribution. From the experiments, it is confirmed that the proposed generalized ILRMA improves the separation accuracy, especially for the music mixture signals. However, the improvement for speech mixture signals is still limited. This is because typical speech sources do not have an apparent lowrank timefrequency structure, and NMFbased source model in ILRMA cannot capture the precise spectral structures in speech sources even if the source model is generalized by the heavytailed distributions. The better modeling for speech sources remains as a future work.
Abbreviations
 BSS:

Blind source separation
 FDICA:

Frequencydomain independent component analysis
 GGD:

Generalized Gaussian distribution
 GGDILRMA:

Independent lowrank matrix analysis based on generalized Gaussian distribution
 GGDIVA:

Independent vector analysis based on spherical multivariate generalized Gaussian distribution
 ICA:

Independent component analysis
 ILRMA:

Independent lowrank matrix analysis
 IP:

Iterative projection
 ISNMF:

Nonnegative matrix factorization based on Itakura–Saito divergence
 IVA:

Independent vector analysis
 Laplace IVA:

Independent vector analysis based on spherical multivariate Laplace distribution
 MM:

Majorizationminimization
 MNMF:

Multichannel nonnegative matrix factorization
 NMF:

Nonnegative matrix factorization
 p.d.f.:

Probability density function
 SDR:

Signaltodistortion ratio STFT: Shorttime Fourier transform
 tILRMA:

Independent lowrank matrix analysis based on Student’s t distribution
 tMNMF:

Multichannel nonnegative matrix factorization based on Student’s t distribution
 tNMF:

Nonnegative matrix factorization based on Student’s t distribution
 Timevarying Gaussian IVA:

Independent vector analysis based on Gaussian distribution having a timevarying variance
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Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped improve the quality of this manuscript.
Funding
This work was partly supported by the ImPACT Program of Council for Science, SECOM Science and Technology Foundation, and JSPS KAKENHI Grant Numbers JP16H01735, JP17H06101, and JP17H06572.
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Kitamura, D., Mogami, S., Mitsui, Y. et al. Generalized independent lowrank matrix analysis using heavytailed distributions for blind source separation. EURASIP J. Adv. Signal Process. 2018, 28 (2018). https://doi.org/10.1186/s1363401805495
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Keywords
 Blind audio source separation
 Independent lowrank matrix analysis
 Nonnegative matrix factorization
 Student’s t distribution
 Generalized Gaussian distribution