DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL
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1 DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL Sco Wsdom, John Hershey 2, Jonahan Le Roux 2, and Shnj Waanabe 2 Deparmen o Elecrcal Engneerng, Unversy o Washngon, Seale, WA, USA 2 Msubsh Elecrc Research Laboraores MERL, Cambrdge, MA, USA Ths repor deals he dervaons or he paper by Wsdom, Hershey, Le Roux, and Waanabe. Secon gves a praccal revew o complex gradens and Wrnger caluclus, and secon 2 deals he graden dervaons or he deep MCGMM.. COMPLEX GRADIENTS AND WIRTINGER CALCULUS Gven a complex-valued uncon o complex-valued daa x u v, wha s he graden x? Wha s he chan rule x s a uncon o an nermedae varable? Ths secon descrbes he complex graden n pracce. Common pracce s o use a compose-real represenaon o complex quanes where every complex number s ransormed no a wo-dmensonal real-valued vecor: u z :. v However, akng he graden o such a represenaon can lead o ncorrec quanes n some suaons 2. Oher mes, algebra usng he real and magnary pars drecly can be arduous, especally or uncons ha have a lo o neracons beween he real and magnary pars. An alernave and equvalen represenaon s augmened-complex ha represens x as a wo-dmensonal complex vecor: x x : x, 2 and consders x and x o be separae, ndependen varables. We wll see ha he compose-real and augmened-complex represenaons are useul n deren conexs... Real-magnary gradens The general case or z R, wh z x y and r s: z x y r r x x r y y r x x R z x r x s I z s y y s y r y s We wll now ls some specal cases ha are useul denes. Case : real, real z, complex x R z r x s Case 2: real, complex z, real x x r y y r Case 3: real, complex z, complex, z a holomorphc uncon o Because o he Cauchy-Remann condons, x r y s x s y 8 r, 6 7
2 hen and z x r x y z s r y s x R z r x x I z s r x s Wrnger calculus Real-magnary gradens are compleely sucen or akng dervaves o complex daa. However, or some uncons, especally nonholomorphc uncons o complex daa, he real and magnary pars can be edous o derve algebracally. To ackle such dervaves, Wrnger calculus also called CR-calculus because o he requen conversons beween complex and real domans becomes useul. The Wrnger dervaves rea z and z as separae, ndependen varables. These dervaves are dened as z : 2 z : 2 x y x y 2 I s a holomorphc uncon.e., s only a uncon o z and no o z, hen complex uncon o z, he conjugae Wrnger dervave sases he deny z 0. I s an arbrary poenally nonholomorphc I s a scalar real-valued uncon o z, we have he deny z z The general case or z R, wh z x y and r s s Usng he deny 3, we have Usng he deny 4, we have. 3 z, 4 z z z z z. 5 z z z. 6 z z z z. 7 z Thus, or hs case, we can see one o he advanages o Wrnger calculus versus real-magnary compose gradens: here, only hree dervaves,, z z, and, are requred, nsead o he our requred by he real-magnary compose graden chan rule 5. z.2.. Graden checkng Usng, we can check he dervaves and by he ollowng procedure, where ɛ s chosen o be a small consan, usually on he x x order o 0 6 : x r x ɛ x r x ɛ x x ɛ x x ɛ x x r x r 2 x x r x r 2 x x x x
3 2. GRADIENTS FOR THE DEEP MCGMM The gradens or backpropagaon are as ollows. All gradens are Wrnger gradens, as descrbed n 2, and hus use he Wrnger chan rule. In he ollowng, he operaors and ndcae broadcased addon, and margnalzng mulplcaon, respecvely. For example, broadcased addon o wo ensors o sze N P and M P resuls n a ensor o sze N M P, where any dmensons o are broadcased over he dmenson o he oher ensor. Margnalzng mulplcaon or Wrnger gradens s dened as ollows. I y/x s a graden o a N y M y P y ensor y wh respec o a N x M x P x ensor x, and x/ s a graden o x wh respec o a N M P ensor, hen we wll denoe y/x as havng sze Ny My Py N x M x P x and x/ as havng sze Nx Mx Px N M P. Then margnalzng mulplcaon beween he wo Wrnger gradens s y y x x N x M x P x n m p y x :,:,: m,n,p x m,n,p :,:,: N x M x P x n m p y x :,:,: m,n,p x In erms o mplemenaon or he mulchannel GMM, varables and gradens are sored as ensors ha all have he same maxmum sze, I J Z F T. I a parcular graden does no have a parcular dmenson, hen ha dmenson s smply se o. For example, he I J F channel model B s sored as a ensor o sze I J F. In rare cases when more dmensons are needed or example, he ˆΣ xx xx sample spaal covarance whch s J J F, he unused dmensons can be used. For example, ˆΣ s sored as a J J F ensor. We wll now descrbe he gradens nvolved n backpropagaon. For reerence,, gure 2 s helpul. To nd all compuaonal pahs, sar a he cos uncon D and ollow arrows n he oppose drecon. m,n,p :,:,: Las layer K In he las layer, he ollowng pahs exs o λ K log γ K : D K µ K γ K λ K π K L K π K L K µ K γ K π K L K γ K, 9 and he pahs o A K and b K are D K π K L K A K, b K 20 These pahs requre he ollowng gradens: 2, X j, L j,z,k µ j,z,k, D ESR j,k, D ESR j,k, j,k, µ j,z,k, µ j,z,k, γ j,z,k j,k, π j,z,k π j,z,k L j,z,k L j,z,k µ j,z,k, L j,z,k γ j,z,k, X j, 2 j,k, 2 2 X j, 2 22, X j, π j,z,k 23 µj,z,k, γ j,z,k 24 µ j,z,k, 25 somax 2 γ j,z,k γ j,z,k j,z,k π 26 µ j,z,k, 27 µ j,z,k, 2 28
4 where somax s he dervave o he somax uncon. γ j,z,k exp λ j,z,k λ j,z,k 29 L j,z,k λ j,z,k L j,z,k A j,k 30 π j,z,k 3 L j,z,k b j,k Lower layers k < K To proceed downward hrough he nework rom layer k o layer k, here are wo man choke-pons : µ k and B k. The pah o µ k s D... k µ k 33 L k. On he way o B k, pahs hrough γ k are requred: Fnally, he pahs o B k are wh Wrnger gradens γ j,z,k B k µ j,z,k, B k D... µ k γ k L k D... µ k B k µ j,k, B k k γj B k γ j,z,k γ j,z,k { B k γ k B k y T ψ ψ ψ, j j 0, j j. { 0, j j T \j,k, j,k,, j j, B k T :,\j ψ, j j 0, j j, Connung downward, we go hrough k : D... µ k B k k 39 The op pah uses graden µ j,z,k, j,k, B k γ j,z,k The boom pah s a good example o a Wrnger graden chan rule, and s gven by ψ B k :,\j. 40 D j,k, D B k D B k D 0 B k Bk j,k, Bk j,k, D B k D B k y, dag Bk j,k, k B ˆΣ,k j,k, 4
5 Nex we proceed o π k : wh B k π j,z,k D... k π k B k ˆΣ Y k ˆΣ,k 2 γ j,z,k µ j,z,k, The graden rom π k o L k s, as beore, he dervave o he sgmod uncon. Now we proceed o µ k : D... k µ k B k L k wh B k µ j,z,k, B k µ j,z,k, ˆΣ Y,k ˆΣ k 2 j,z,k π µ j,z,k, 45 As a las sep, we proceed o B k, whch requres nermedae pahs hrough γ k : D... µ k γ k L k B k 46 wh B k γ j,z,k ˆΣ Y k ˆΣ k 2 πj,z,k 2 γ j,z,k. 47 Fnally, we reach B k, whch concludes he requred gradens down hrough layer k. D... µ k B k γ k. 48 The gradens wh respec o he ranable parameers λ k, A k, and b k are he same as n layer K, and use gradens 29, 30, 3, and REFERENCES S. Wsdom, J. Hershey, J. Le Roux, and S. Waanabe, Deep Unoldng or Mulchannel Source Separaon, n submsson o ICASSP, K. Kreuz-Delgado, The Complex Graden Operaor and he CR-Calculus, arxv: mah, June 2009, arxv:
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