Phase Unmixing : Multichannel Source Separation with Magnitude Constraints

PHASE UNMIXING : MUL TICHANNEL SOURCE SEP ARA TION WITH MA GNITUDE CONSTRAINTS Antoine Delefor ge and Y ann T raonmilin Inria Rennes - Bretagne Atlantique, France ABSTRA CT W e consider the problem of estimating the phases of K mixed complex signals from a multichannel observation, when the mixing matrix and signal magnitudes are known. This prob- lem can be cast as a non-con ve x quadratically constrained quadratic program which is known to be NP-hard in gen- eral. W e propose three approaches to tackle it: a heuris- tic method, an alternate minimization method, and a con vex relaxation into a semi-definite program. The last two ap- proaches are showed to outperform the oracle multichannel W iener filter in under -determined informed source separation tasks, using simulated and speech signals. The conv ex relax- ation approach yields best results, including the potential for exact source separation in under -determined settings. Index T erms — Informed Source Separation, Phase Re- triev al, Semidefinite Programming 1. INTR ODUCTION Let M sensors record K complex signals through linear in- stantaneous mixing. The noisy observation y ∈ C M is ex- pressed as: y = A s 0 + n , (1) where s 0 ∈ C K is the source vector , n ∈ C M is the noise vector and A ∈ C M × K is the mixing matrix. This model is very common in signal processing and occurs, for instance, when looking at a single time-frequency bin of the discrete short-time Fourier domain. In that case, each entry a m,k of A may be viewed as a frequency-dependent gain and phase off- set from source k to sensor m . The classical problem of mul- tichannel sour ce separation consists in estimating the source signals s 0 giv en one or several observations y . Ev en with- out noise, this problem is fundamentally ill-posed. This is always true in the blind case where A is unknown, but also when A is perfectly kno wn and full-rank, as long as M < K ( under-determined setting). Due to these ambiguities, source separation under v arious prior kno wledge on A , s 0 or n is a long-standing and still active research topic. Often, spe- cific structures are imposed on A based on physical [1, 2] or learned [3] models of signal propagation. It is also quite com- mon to add statistical assumptions on source and noise sig- nals. For instance, signals may be assumed pairwise statisti- cally independent such as in independent component analysis (ICA [4]), or non-Gaussian and non-white such as in TRINI- CON [5]. If signals are assumed wide-sense stationary , their respectiv e observed (image) cov ariance can be estimated and used to compute the well-known multichannel W iener filter ( e.g . [6]), which is then the optimal linear filter in the least squared error sense. Although very powerful, the stationar- ity assumption is unrealistic for many signals of interest such as speech or music in audio. F or this reason, W iener filter- ing is often used in combination with time-frequency vary- ing source variances and a Gaussian [1, 7] or alpha-stable [8] signal model. T o avoid over -parameterization, v ariances are either assumed to be known such as in informed sourc e sepa- ration [9, 10], or to be provided by a lo w-dimensional model such as nonnegati ve matrix factorization (NMF) [2, 11] or more recently deep neural networks [12]. A common prop- erty of all W iener-filter -based approaches is that they rely on a good model of source v ariances, while the source phases are left unconstrained and estimated from observ ations. The or a- cle W iener-filter corresponds to the case where instantaneous source variances and mixing matrices are kno wn. In this paper , we introduce a slight shift of this paradigm by replacing the prior kno wledge on instantaneous source variances by a prior knowledge on instantaneous source magnitudes. Note that both quantities are related, the for- mer being the maximum-likelihood estimate of the latter for non-stationary Gaussian signals. W e refer to this problem as phase unmixing and focus here on the oracle case where the source magnitudes and the mixing matrix A are exactly known, while only source phases need to be estimated. Ap- plications are hence informed source separation or situations where good magnitude and mixing models are available. W e show that multichannel phase unmixing can be expressed as a non-con ve x quadratically constrained quadratic minimization problem, which is kno wn to be NP-hard in general and hard to solve in practice. W e propose three different approaches to tackle it: a heuristic approach, an alternated minimiza- tion approach, and a con vex relaxation of the problem into a semi-definite program (SDP). Some preliminary theoretical insights and a detailed e xperimental study on simulated data are presented to compare the proposed methods to the oracle multichannel Wiener filter . A task of informed multichannel speech source separation is also performed. The proposed con ve x scheme yields particularly encouraging results, in- cluding stability to noise and the potential for exact source separation in under-determined settings. Related work. The considered problem of phase unmixing is related but not to be confused with the problem of phase r etrieval , which has triggered considerable research interest ov er the past 30 years [13, 14] and has recently regained mo- mentum thanks to novel methodologies [15, 16]. In phase re- triev al, only the magnitudes of y are observed while s 0 is completely unknown. This problem occurs in applications such as adapti ve optics [17] or X-ray crystallography [18], where the phases of the F ourier transform are intrinsically lost during measurement. Phase retriev al is also sometimes used to find phase estimates which are consistent [19] with mag- nitudes estimates of a single-channel signal of interest. This consistency may be imposed by the properties of the short- time Fourier transform [19, 20] or sinusoidal source models [21]. In contrast, the proposed framework solely relies on complex multichannel observations to perform recovery and does not require any structure on phases. 2. PHASE UNMIXING Under model (1), we consider the problem of estimating the phases of s 0 ∈ C K giv en strictly positi ve magnitudes b = | s 0 | , the multichannel observation y ∈ C M and the mixing matrix A ∈ C M × K , which we assume full rank. A natural approach is to minimize the Euclidean norm of the residual: ˆ s = argmin s k A s − y k 2 2 s.t. | s k | 2 = b 2 k , k = 1 . . . K. ( Φ LS) W e refer to this problem as phase least-squares ( Φ LS), be- cause without constraints, it becomes a standard least-squares problem. Least squares has infinitely many solutions in the under-determined case ( K > M ) and a unique closed- form solution ˆ s LS = A † y otherwise, where {·} † denotes the Moore-Penrose pseudo-in verse. On the other hand, ( Φ LS) is an instance of quadratically constrained quadratic program (QCQP). These problems are non-con ve x, and solving them or ev en finding whether they hav e a solution is NP-hard in general [22]. While branch-and-bound methods exist to solve non-conv ex QCQPs [22, 23], the y are extremely slo w in practice 1 . A generally e xact and ef ficient solution to ( Φ LS) is thus most likely out-of-reach, but we propose in the following three practical approaches to tackle it. Normalized multichannel Wiener filter . The multichan- nel Wiener filter (MWF) is one of the most widely used methods in signal processing [6]. One interpretation of MWF is that it is the maximum a posteriori estimator of s 0 giv en y , assuming that source and noise signals are zero-mean complex circular-symmetric Gaussian ( e .g. [1]). For an i.i.d. noise with v ariance σ 2 n and independent sources with v ari- ances b 2 1 , . . . , b 2 K , the MWF estimate ˆ s MWF is 2 ˆ s MWF = argmin s 1 σ 2 n k A s − y k 2 2 + P K k =1 | s k | 2 /b 2 k (2) = ( σ 2 n Diag { b } − 2 + A H A ) − 1 A H y (3) where {·} H denotes Hermitian transposition. While ( Φ LS) as- sumes that source magnitudes are known, MWF assumes that source v ariances are known, which is the same quantity of 1 Solving one instance of ( Φ LS) using the Matlab version of BAR ON [23] on a regular laptop takes o ver a minute with M = 2 , K = 3 . 2 In the under-determined case, the expression (3) is equiv alently replaced by Diag { b } 2 A H ( A Diag { b } 2 A H + σ 2 n I M ) − 1 y for numerical stability . prior information. Note that adding the magnitude constraint | s k | 2 = b 2 k to (2) recov ers ( Φ LS). A simple heuristic approach to ( Φ LS) is thus to normalize ˆ s MWF by changing its magni- tudes to b while keeping its phases unchanged. W e refer to this as normalized multichannel W iener filtering (NMWF). Alternated minimization. A second approach to solve ( Φ LS) is by alternated minimization w .r .t. each coordinate s i until conv ergence, i.e. , by coordinate descent. The La- grangian of ( Φ LS) writes: L ( s , λ ) = k A s − y k 2 2 + P K k =1 λ k ( | s k | 2 − b 2 k ) . (4) Finding the zeros of the deri vati ves of (4) w .r .t. to the real and imaginary parts of s i and λ i yields s i = b i h y − A : ,i c s i c , a i i / |h y − A : ,i c s i c , a i i| , (5) where a i is the i -th column of A , s i c is s depriv ed of its i -th element and A : ,i c is A depriv ed of its i -th column. Note that (5) is almost surely well-defined (see 3 for detailed deri va- tions). Gi ven an initial guess s (0) ∈ C K , repeatedly applying (5) for i = 1 . . . K until conv ergence yields the phase unmix- ing by alternated minimization (PhUnAlt) method, described in Alg. 1. PhUnAlt con ver ges because the nonnegativ e resid- ual error r ( p ) decreases at each iteration. Since ( Φ LS) is not con ve x, it does not generally con verge to a global minimum but to a local minimum which depends on the initial guess. Lifting scheme. W e first note the follo wing identity: k A s − y k 2 2 =    A − y   s 1    2 2 = k e A x k 2 2 = x H e A H e A x = trace  e A H e A xx H  = trace ( CX ) , (6) where e A = [ A , − y ] , x = [ s , 1] > ∈ C K +1 , X = xx H ∈ C ( K +1) 2 and C = [ A , − y ] H [ A , − y ] ∈ C ( K +1) 2 . W e can no w consider the following con ve x relaxation of ( Φ LS): b X = argmin X trace ( CX ) s.t. diag { X } = e b , X  0 (PhUnLift) where e b = [ b . 2 , 1] > and PhUnLift stands for phase un- mixing by lifting . Note that the rank-1 constraint on X has been remov ed. Using (6) and observing that the diagonal of X contains the squared magnitudes of s , it is easy to see that if b X = ˆ x ˆ x H is a rank-1 solution of (PhUnLift), then ˆ s = ˆ x 1: K / ˆ x K +1 = b X 1: K,K +1 is a global solution of ( Φ LS). Hence, the NP-hard quadratic problem ( Φ LS) has been relaxed to a simple con vex, linear semi-definite pr ogram (SDP). SDPs hav e been extensiv ely studied over the past decades, and many methods are available to solve them ef ficiently [24, 25]. Following [16], we propose to use the particularly inexpensi ve block-coordinate descent (BCD) method of [25]. Algorithm 2 shows the method adapted to (PhUnLift). Each iteration only requires two matrix- vector multiplications, which makes iterations of PhUnAlt and PhUnLift of comparable computational complexity . 3 http://people.irisa.fr/Antoine.Deleforge/ supplementary_material_ICASSP17.pdf . Algorithm 1 PhUnAlt Input: y ∈ C M , A ∈ C M × K , b ∈ R K + , s (0) ∈ C K . Output: Source estimate ˆ s with | ˆ s | = b . 1: p := 0 ; r (0) := + ∞ ; 2: repeat 3: for i = 1 → K do 4: s ( p ) i := b i h y − A : ,i c s ( p ) i c , a i i / |h y − A : ,i c s ( p ) i c , a i i| ; 5: s ( p +1) i := s ( p ) i ; 6: end for 7: p := p + 1 ; r ( p ) := k y − A s ( p ) k 2 2 ; // Residual err or 8: until ( r ( p − 1) − r ( p ) ) / r ( p ) < 10 − 3 9: return s ( p ) The problem is that solutions of (PhUnLift) are not nec- essarily rank-1. T o understand why this relaxation may still be a viable approach to phase unmixing, let us first draw a connection to related works. W ithout loss of generality , e b can be set to 1 by changing C to Diag { e b } C Diag { e b } . Then, (PhUnLift) has the same form as the SDP PhaseCut , recently introduced in [16] for phase retriev al. The name PhaseCut was chosen because the real rank-1 counterpart of the problem is known to be equiv alent to the classical NP- hard MaxCut problem in graph theory . An SDP relaxation of MaxCut was proposed 20 years ago [26], and since then many extensions ha ve been de veloped. For the phase retrie val ap- plication, it was showed through a connection to the method PhaseLift [15] that PhaseCut does yield a rank-1 solution with high-probability when some stringent conditions on C are verified [16]. In the presence of noise, solutions are no longer rank-1 but some stability results are available. Unfor - tunately , none of these results can be directly transposed to the phase unmixing problem. Indeed, the PhaseCut/PhaseLift equiv alence only occurs when C has a specific form in volving an orthogonal projection matrix [16], which is not the case for phase unmixing. W e provide here a first stability theorem for PhUnLift in the determined case only ( K ≤ M ). A proof of this theorem is av ailable in the supplementary material 3 : Theorem 1 Let y = A s 0 + n , b = | s 0 | , A be full-rank and K ≤ M . Let ˆ s be the output of Algorithm 2. W e have: k ˆ s − s 0 k 2 ≤ 2 √ 2 σ min ( A ) k n k 2 (7) wher e σ min ( A ) is the smallest singular value of A . In other words, PhUnLift recov ers the true source vector up to an error proportional to the noise level. In particular , a rank-1 solution and exact recov ery is obtained in the noiseless case. Note that for K ≤ M , bounds similar to (7) can also be obtained for the least-squares, MWF and NMWF estimates. For the more interesting under-determined case, a different approach is needed because then σ min ( A ) = 0 . A theoretical extension of theorem 1 to K > M and additional properties on A is likely intricate to obtain, although numerical results of section 3 do suggest that this is possible. Algorithm 2 PhUnLift (Block-coordinate descent) Input: y ∈ C M , A ∈ C M × K , b ∈ R K + , ν > 0 , typically small [25] (in fact, ν = 0 w orked well in practice). Output: Source estimate ˆ s with | ˆ s | = b . 1: C = [ A , − y ] H [ A , − y ] ; 2: p := 0 ; r (0) := + ∞ ; X (0) := I K +1 ; 3: repeat 4: for i = 1 → K do 5: z := X ( p ) i c ,i c C i c ,i ; γ := z H C i c ,i ; X ( p +1) i c ,i := 6: X ( p +1)H i,i c := − q b i − ν γ z for γ > 0 , 0 otherwise; 7: end for 8: p := p + 1 ; r ( p ) := trace ( CX ( p ) ) ; // Residual err or 9: until ( r ( p − 1) − r ( p ) ) / r ( p ) < 10 − 3 10: return ˆ s with the same phases as X ( p ) 1: K,K +1 and | ˆ s | = b 3. EXPERIMENTS AND RESUL TS W e no w compare the efficienc y of MWF (3), NMWF , PhUnAlt (Alg.1) and PhUnLift (Alg. 2) on the task of estimating the phases of s 0 giv en an M -channel mixture y = A s 0 + n . All these methods are compared in the oracle setting: the true magnitudes b = | s 0 | and the true mixing matrix A are pro- vided. Moreover , W iener-filter -based methods are giv en the true v ariance σ 2 n used to generate the noise in all experiments (interestingly , this is not needed by PhUnAlt or PhUnLift). Three initializations are considered for PhUnAlt: random phases (PhUnAlt), the output of NMWF (NMWF+) or the output of PhUnLift (PhUnLift+). Moreover , a brute-for ce ap- proach (PhUnAlt*5) is considered, which picks the PhUnAlt estimate with smallest residual out of 5 randomly initialized runs. Phase unmixing problems are generated by randomly picking all the elements of A , s 0 and n from i.i.d. zero-mean complex Gaussian distributions of respeci ve standard devi- ations σ A , σ s and σ n . In each experiment, σ A and σ s are randomly uniformly picked in [0 , 2] while σ n is adjusted to the desired signal-to-noise-ratio SNR = k A s 0 k 2 2 / ( M σ 2 n ) . For each considered combination of ( M , K, SNR ) , all meth- ods are ran on 1000 random tests. Fig. 1(a) shows the mean relativ e error k ˆ s − s 0 k 2 2 / k s 0 k 2 2 as a function of M when K = M (determined case), under low-noise conditions (SNR = 60dB ). As predicted by The- orem 1, near -exact recovery is possible with PhUnLift, and MWF and NMWF yield similarly low errors. PhUnLift+ does not improve ov er PhUnLift suggesting that the global mini- mum of ( Φ LS) is already reached, while NMWF+ leads to the same solution as PhUnLift. PhUnAlt and PhUnAlt*5 per- form relati vely less well due to local-minimum con vergence. Fig. 1(b)-(c) sho ws the same e xperiment in under -determined settings. PhUnLift seems to yield solutions sufficiently close to the global optimum to achie ve again near -exact reconstruc- tion with PhUnLift+ for M sufficiently large. This does not seem to be the case with other methods. This is further illus- trated in Fig. 1(d) and Fig. 1(e), which sho w the probability of exact reconstruction of PhUnLift and PhUnAlt for different Number of sensors M 0 2 4 6 8 10 Mean Relative Error 10 -6 10 -4 10 -2 10 0 (a) K = M sources (d) Phase Transition (PhUnLift+) Number of sources K 1 3 5 7 9 11 13 15 Number of sensors M 1 3 5 7 9 11 13 15 (e) Phase Transition (PhUnAlt) Number of sources K 1 3 5 7 9 11 13 15 Number of sensors M 1 3 5 7 9 11 13 15 % Exact Source Recovery 0 20 40 60 80 100 Number of sensors M 0 2 4 6 8 10 Mean Relative Error 10 -6 10 -4 10 -2 10 0 (b) K = M+1 sources Number of sensors M 0 2 4 6 8 10 Mean Relative Error 10 -6 10 -4 10 -2 10 0 (c) K = M+2 sources MWF NMWF NMWF+ PhunAlt PhunAlt*5 PhunLift PhunLift+ SNR (dB) 0 20 40 60 Mean Relative Error 10 -2 10 -1 10 0 (g) M < K < M+3 SNR (dB) 0 20 40 60 Mean Relative Error 10 -6 10 -4 10 -2 10 0 (f) M=K Fig. 1 . (a)-(c) : Mean relativ e error for a fixed SNR of 60dB. (d)-(e) Probability of e xact reconstruction with PhUnLift+ and PhUnAlt (noiseless). (f)-(g): Robustness to noise in determined cases and under -determined cases ( M = 2 . . . 10 ). M , K → 2 , 2 2 , 3 2 , 4 4 , 4 4 , 5 4 , 6 Input 0.49 -2.70 -4.66 -4.51 -5.76 -7.27 Rand -6.44 -6.28 -4.99 -4.41 -4.46 -5.28 MWF 59.6 21.7 17.0 58.6 27.9 25.0 NMWF 59.9 21.6 16.9 59.4 28.6 25.9 NMWF+ 59.9 22.8 19.2 59.7 34.1 31.6 PhUnAlt 22.7 17.6 15.3 25.8 21.5 22.0 PhUnAlt*5 43.5 35.4 21.5 47.7 37.3 33.6 PhUnLift 59.9 37.6 22.3 59.0 57.3 40.9 PhUnLift+ 59.9 39.6 21.2 58.0 59.3 44.8 T able 1 . Mean SDR (dB) for 1-second M -channel mixtures of K speech sources. Means are ov er the K sources for each mixture. values of ( M , K ) . Here, exact means a relativ e error lower than 10 − 8 . 100% e xact recovery seems possible with PhUn- Lift+ in a number of under-determined cases where PhUnAlt only achieves around 80% . Fig. 1(f) illustrates that the error of PhUnLift is proportional the noise error when K ≤ M , as predicted by Theorem 1. This is also true for MWF , NMWF and NMWF+, but not for PhUnAlt due to local-minima. In the under-determined setting showed in Fig. 1(g), stability to noise is less obvious. PhUnLift and PhUnLift+ perform best, closely follo wed by PhUnAlt*5. In general, the fair results obtained with PhUnAlt*5 suggests that the number of local- minima is often not too high, making multiple initialization of PhUnAlt a feasible approach. From a computational point-of- view , the non-iterativ e methods MWF and NMWF are much faster but also perform less well. The computational times of other methods depend on the number of iterations needed for conv ergence. PhUnAlt generally con ver ges in a few hun- dred iterations. The same is observed for PhUnLift, except in under-determined cases with high SNRs ( ≥ 30 dB), where tens of thousands of iterations are often needed. This calls for using an SDP solver with faster con ver gence rate. W e finally conduct an informed speech separation task us- ing random 1 second utterances from the TIMIT dataset [27]. The clean speech signals are sub-sampled at 16 kHz and trans- formed to the short-time Fourier (STF) domain using a 64 ms sliding windo w with 50 % overlap, yielding F = 512 positiv e- frequency bins and T = 33 time bins. They are then mixed using a global gain g ( m, k ) ∈ [ − 5dB , +5dB] and a discrete time-domain delay τ ( m, k ) ∈ [0 , 50] in samples from each source k to each microphone m . The corresponding mixing matrices used in the STF domain are defined by A f ( m, k ) = 10 g ( m,k ) / 20 exp( j τ ( m, k ) f /F ) were f = 0 . . . F − 1 is the frequency index. For each experiment, both gains and delays are uniformly pick ed at random such that mixing matrices re- main full rank. T o save computational time, when a source has its magnitude lo wer than -40dB in a gi ven time-frequenc y bin, the source is ignored and assigned a random phase by all methods. Mean signal-to-distortion-ratios (SDRs) calcu- lated with [28] for each considered method are showed in ta- ble 1 ( Rand means random phases with correct magnitudes). PhUnLift and PhUnLift+ outperform the other methods in under-determined settings, while MWF , NMWF , NMWF+, PhUnLift and PhUnLift+ performs similarly for K = M . 4. CONCLUSION The problem of oracle phase unmixing, i.e. , multichannel source separation with known magnitudes and mixing matrix, was introduced and cast as a non-con ve x quadratic problem. Three approaches were proposed to tackle it, including a lifting scheme which showed best performance in practice. The proposed methods outperformed the oracle multichannel W iener filter in under-determined settings. 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