Auxiliary function based IVA using a source prior exploiting fourth order relationships Yanfeng Liang, Jack Harris, Gaojie Chen, Syed Mohsen Naqvi, Christian Jutten, Jonathon Chambers To cite this version: Yanfeng Liang, Jack Harris, Gaojie Chen, Syed Mohsen Naqvi, Christian Jutten, et al.. Auxiliary function based IVA using a source prior exploiting fourth order relationships. EUSIPCO 2013 - 21th European Signal Processing Conference, Sep 2013, Marrakech, Morocco. pp.EUSIPCO 2013 1569746061. hal-00867131 HAL Id: hal-00867131 https://hal.archives-ouvertes.fr/hal-00867131 Submitted on 27 Sep 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EUSIPCO 2013 1569746061 1 2 3 4 AUXILIARY FUNCTION BASED IVA USING A SOURCE PRIOR 5 EXPLOITING FOURTH ORDER RELATIONSHIPS 6 7 Yanfeng Liang∗, Jack Harris∗,†, Gaojie Chen∗, Syed Mohsen Naqvi∗,Christian Jutten†, Jonathon Chambers∗ 8 9 ∗ 10 School of Electronic, Electrical and System Engineering 11 Loughborough University, UK 12 {y.liang2, g.chen, s.m.r.naqvi, j.a.chambers}@lboro.ac.uk 13 †GIPSA-Lab, CNRS UMR5216, 14 Universite´ de Grenoble, France 15 {jack.harris, christian.jutten}@gipsa-lab.grenoble-inp.fr 16 17 18 19 ABSTRACT algorithmically. The permutation ambiguity can theoretically be avoided by using a multivariate source prior to retain the 20 Independent vector analysis (IVA) can theoretically avoid the dependency between different frequency bins of each source 21 permutation ambiguity present in frequency domain indepen- [7][8][9]. IVA can thereby solve the permutation problem 22 dent component analysis by using a multivariate source prior during the convergence process without post processing. In 23 to retain the dependency between different frequency bins of recent years, different modified IVA methods have been pro- 24 each source. The auxiliary function based independent vector posed. An adaptive step size IVA method is proposed to in- 25 analysis (AuxIVA) is a stable and fast update IVA algorithm crease the convergence speed [10]. The fast fix-point IVA 26 which includes no tuning parameters. In this paper, a particu- method adopts Newton’s update method to obtain a fast con- 27 lar multivariate generalized Gaussian distribution source prior vergence form of IVA [11]. Video information is introduced 28 is therefore adopted to derive the AuxIVA algorithm which to form the audio-video based IVA method [12]. In 2011, the 29 can exploit fourth order relationships to better preserve the auxiliary function based form of independent vector analy- 30 dependency between different frequency bins of speech sig- sis (AuxIVA) was proposed, which is a stable and fast form 31 nals. Experimental results confirm the improved separation of IVA algorithm. By using the auxiliary function technique 32 performance achieved by using the proposed algorithm. AuxIVA can guarantee a monotonic decrease of the cost func- 33 Index Terms— AuxIVA, multivariate generalized Gaus- tion without the need for tuning parameters such as step size 34 sian distribution, fourth order relationships [13]. 35 36 The source prior is important to all IVA methods, be- 37 1. INTRODUCTION cause it is used to derive the nonlinear score function which 38 is used to keep the dependency between different frequency 39 Independent component analysis (ICA) is the central tool for bins. For the original IVA algorithm, the multivariate Laplace 40 the blind source separation (BSS) problem [1][2]. The most distribution is adopted as the source prior [8]. However, it is 41 famous BSS problem is the cocktail party problem [3][4], in not the only form for source prior; an improved source prior 42 which one speaker must be selected from a mixture of sounds. which can better preserve the relationships between differ- 43 In the real room environment, due to reverberation, it be- ent frequency bins is still needed. In this paper, we adopt 44 comes a convolutive blind source separation (CBSS) problem. a generalized multivariate Gaussian distribution as the source 45 Therefore time domain methods are not appropriate because prior, which can preserve the fourth order terms within the 46 of the computational complexity [1]. Thus frequency domain score function. Then the AuxIVA algorithm based on this 47 methods are proposed to solve the CBSS problem [5]. How- particular source prior is derived. The proposed method will 48 ever, the permutation problem inherent to BSS needs to be be shown to better preserve the relationships between differ- 49 solved to achieve the separation result. Many methods ex- ent frequency bins, thereby improving the separation perfor- 50 ploiting the positions or spectral structures of the sources have mance. 51 been proposed to solve the permutation problem, but all of The paper is organized as follows, in Section 2, the par- 52 these methods need post processing to address the problem ticular source prior is analyzed. In Section 3, the AuxIVA al- 53 [6]. gorithm with our source prior is derived and the advantage of 54 Independent vector analysis (IVA)is proposed to solve the this source prior is discussed. Then, the experimental results 55 frequency domain blind source separation (FD-BSS) problem are shown to confirm the advantages of the proposed method 56 57 60 61 1 1 2 in Section 4. Finally, conclusions are drawn in Section 5. 3. AUXIVA USING THE PARTICULAR SOURCE 3 PRIOR 4 2. A MULTIVARIATE GENERALIZED GAUSSIAN The auxiliary function technique is developed from the 5 DISTRIBUTION SOURCE PRIOR expectation-maximization algorithm, and avoids the step size 6 tuning [14]. In the auxiliary function technique, an auxiliary 7 For the convolutive blind source separation problem, the basic function is designed for optimization. Instead of minimiz- 8 noise free model in the frequency domain is described as: ing the cost function, the auxiliary function is minimized in 9 terms of auxiliary variables. The auxiliary function technique 10 x(k)=H(k)s(k) (1) can guarantee monotonic decrease of the cost function, and 11 (k)=W (k) (k) y x (2) therefore provides effective iterative update rules [13]. 12 T The contrast function for AuxIVA is derived from the 13 where x(k)=[x1(k), ··· ,xm(k)] is the observed sig- T source prior [14]. For the original IVA algorithm, 14 nal vector, while s(k)=[s1(k), ··· ,sn(k)] and y(k)= T [y1(k), ··· ,yn(k)] 15 are the source signal vector and the es- G(yi)=GR(ri)=ri (6) 16 timated source vector respectively in the frequency domain; T 17 and (·) denotes vector transpose. H(k) is the mixing matrix where ri = yi2. 18 with m×n dimension, and W (k) is the unmixing matrix with By using the proposed source prior, we can obtain the fol- 19 n × m dimension. The index k denotes the k − th frequency lowing contrast function k =1, ··· ,K 20 bin, and . The unmixing matrix can be defined 2 3 21 as G(yi)=GR(ri)=ri . (7) h 22 W (k)=[w1(k) ···wn(k)] (3) 23 The update rules contain two parts, i.e. the auxiliary vari- (·)h 24 where denotes Hermitian transpose. able updates and unmixing matrix updates. In summary, the 25 For traditional CBSS approaches, the scalar Laplace dis- update rules are as follow: 26 tribution is widely used for the source prior. However, the K 27 resultant nonlinear score function is a univariate function, r = |wh(k)x(k)|2 28 which can not retain the dependency between different fre- i i (8) k=1 29 quency bins for each source. In [8], IVA exploits a dependent 30 multivariate Laplace distribution as the source prior. In this GR(ri) h 31 paper we adopt a particular multivariate distribution as the Vi(k)=E[ x(k)x(k) ] (9) ri 32 source prior which can be written as: 33 −1 3 h −1 wi(k)=(W (k)Vi(k)) ei (10) 34 p(si) ∝ exp − (si − μi) Σi (si − μi) (4) 35 T wi(k) 36 =[s (1), ··· ,s (K)] i wi(k)= . where si i i is the -th vector source, and h (11) −1 wi (k)Vi(k)wi(k) 37 μi and Σi are respectively the mean vector and covariance 38 matrix. The covariance matrix is then assumed to be a di- In equation (10), ei is a unity vector, the i-th element of 39 agonal matrix due to the orthogonality of the Fourier bases, which is unity. 40 which implies that each frequency bin sample is mutually un- During the update process of the auxiliary variable Vi(k), G (r ) 41 correlated. Moreover, if the source signal is zero mean and R i we notice that ri is used to keep the dependency between 42 unity variance, then the source prior becomes: different frequency bins for source i. In this paper, as we 2 43 3 K 1 2 defined previously, GR(ri)=ri . Therefore 44 2 3 3 p(si) ∝ exp − |si(k)| = exp − si2 (5) 45 G (r ) 2 2 k=1 R i = = r 4 (12) 46 i 3r 3 3 K 2 2 i 3 ( k=1 |yi(k)| ) 47 where |·|denotes the absolute value, and ·2 denotes the 48 Euclidean norm. This new source prior can be taken as a mul- If we expand the equation under the cubic root in equation 49 tivariate generalized Gaussian distribution with the shape pa- 2 (12), then 50 rameter 3 , and the original multivariate Laplace distribution K K 51 also belongs to this family with the shape parameter 1.