Rapid Computation of I-vector Longting XU 1,2 , Kong Aik LEE 1 , - PowerPoint PPT Presentation

Rapid Computation of I-vector Longting XU 1,2 , Kong Aik LEE 1 , Haizhou Li 1 and Zhen Yang 2 1 Institute for Infocomm Research ( I 2 R ) , Singapore 2 Nanjing University of Posts and Telecomm, China

Introduction • Compression process – an i-vector is a fixed-length low-dimensional representation of a variable-length speech utterance [Dehak et al, 2011]. • i-vector = speaker + recording device + transmission channel + acoustic environment • MAP estimate – posterior mean of the latent variable x in a multi-Gaussian factor analysis model (i.e., total variability model) The alignment of frames to Gaussian could be accomplished using GMM [Kenny et al, 2008] or senone posteriors [Lei et al , 2014]. 2 Odyssey 2016, Bilbao, Spain

Introduction (cont’d) • I-vector extraction is a posterior inference process Prior Observations Posterior          1 x 0 I , , , , o o o x , L 1 2 T    1          T Σ NT T Σ F 1 T 1 1 T 1 L I L • We use a pre-whiten statistics [Matejka et al , 2011] in this work      1      1 T 1 T L I T NT L T F 3 Odyssey 2016, Bilbao, Spain

Objective • Objective : To reduce the computation complexity of i-vector extraction while keeping the memory requirement low with slight degradation in performance. • Why ? – Of particularly interest for i-vector extraction on hand-held devices and large- scale cloud-based applications – The number of senone posteriors is approaching 10k and beyond [Sadjadi et al , 2016]. – The T matrix is trained offline and one-off. Computational load not general seen as a bottleneck. 4 Odyssey 2016, Bilbao, Spain

Problem statement • The main computational load of i-vector extraction is at the computation of the posterior covariance as part of the posterior mean estimation. • Existing solutions: – Simplifying the posterior covariance estimation • Eigen decomposition of posterior covariance [Glembek et al, 2011] • Fixed occupancy count [Aronowitz et al, 2012] • Factorized subspace [Cumani et al, 2014] • Sparse coding [Xu et al, 2015] 5 Odyssey 2016, Bilbao, Spain

Problem statement (cont’d) • Proposed solution: – Estimate directly the posterior mean without the need to evaluate the posterior covariance • Using informative prior • Uniform occupancy assumption 6 Odyssey 2016, Bilbao, Spain

I-vector extraction using INFORMATIVE PRIOR 7 Odyssey 2016, Bilbao, Spain

Posterior inference with informative prior • Conventional i-vector extraction assumes a standard Gaussian prior on the latent variable x   x 0 I , • Consider a more general case where the prior on x has mean 𝛎 p and covariance 𝚻 p   μ Σ x , p p • I-vector extraction with informative prior 8 Odyssey 2016, Bilbao, Spain

Subspace ortho normalizing prior • In this work, we consider the following informative prior      1  Σ Σ T T T x 0 , with p p • I-vector extraction  1          1 T 1 T T L T F L T T T NT    1       T T T T T T NT T F          1    1    T T T T T T I T T T NT T F        1     1 1    T T T T I T T T NT T T T F   9 Odyssey 2016, Bilbao, Spain

Subspace ortho normalizing prior (cont’d) • Using the matrix inversion identity • I-vector extraction with subspace orthonormalizing prior:      1     1 1     T T T T I T T T NT T T T F   Q P P      1     1 1    T T T T T T T I NT T T T F      Projection matrix with 1  T T T T T T T U U 1 1 orthonormal columns 10 Odyssey 2016, Bilbao, Spain

I-vector extraction with RAPID COMPUTATION 11 Odyssey 2016, Bilbao, Spain

Solving the matrix inversion • Singular value decomposition of T     T USV U U SV , 1 2 • It follows that U 1 spans the same subspace as T , and U 1 ⊥ U 2 • Using the above, we solve for the following matric inversion    1     1 1        T T T I N T T T T I N U U       1 1    1       T I N I U U   2 2    1    T I N NU U 2 2 12 Odyssey 2016, Bilbao, Spain

Solving the matrix inversion (cont’d) • Let A = ( I + N )        1      1 1 1        T T T T I N T T T T I N NU U A NU U     2 2 2 2 • Using the matrix inversion lemma       1 1         T 1 1 T 1 T 1 A NU U A A N I U U A N U U A 2 2 2 2 2 2 • Using again the matrix inversion identity      1     1 1          T T 1 1 T 1 T 1 I N T T T T A A NU U I A NU U A     2 2 2 2 13 Odyssey 2016, Bilbao, Spain

Rapid computation of i-vector • Uniform occupancy assumption:            1 1 A N I N N I for 0 1 • Or equivalently N    c c  1 N c • The matrix inversion can be simplified as      1     1 1           T T 1 T 1 T 1 I N T T T T A U U I A NU U A     2 2 2 2 • Since T ⊥ U 2 , the second term diminishes        1         1 1 1      1 T T T T T T T T T I NT T T T F T T T I N F   14 Odyssey 2016, Bilbao, Spain

Computational complexity and memory cost Complexity Memory cost Time ratio Baseline (slow) O ( CFM 2 + M 3 ) O ( CFM ) 106.44 Baseline (fast) O ( CFM + CM 2 + M 3 ) O ( CFM + CM 2 ) 11.99 Proposed (exact) O ( CFM + CM 2 + M 3 ) O ( CFM + CM 2 ) 12.65 Proposed (fast) O ( CFM ) O ( CFM ) 1     1     T T Baseline (slow) I c N T T T F c c c     1     T Baseline (fast) I c N A T F c c       1     Proposed (exact) T T c N 1 T T T F c c c       1    1 Proposed (fast) T T T T T I N F 15 Odyssey 2016, Bilbao, Spain

Posterior covariance • Determined by the zero-order statistics and T matrix. • Might be desired for uncertainty propagation. • Using the same derivation procedure as for the posterior mean. • The computational complexity is O ( CM 2 ), assuming that we pre-computed the matrices . T T T c c 16 Odyssey 2016, Bilbao, Spain

Using informative prior in EM update of T 17 Odyssey 2016, Bilbao, Spain

Rapid computation of i-vector EXPERIMENT 18 Odyssey 2016, Bilbao, Spain

Experimental setup • NIST SRE’10 extended core task, CCs 1 to 9 • UBM – Gender dependent with C = 512 mixtures – 57-dim MFCC – SWB, SRE’04, 05, 06 • T matrix – M = 400 – Trained using the same dataset as UBM • PLDA – LDA to 300-dim and length normalization was performed – 200 speaker factors – Full residual covariance for channel modeling 19 Odyssey 2016, Bilbao, Spain

SRE’10 core -extended (female) • Introducing an informative prior does not seem to degrade the performance. • For the tel-tel CC 5 , the relative degradation is 10.04% in EER and 4.54% in min DCF. • For all the 9 CCs, relative degradation ranges from 10.04% to 16.11% in EER and 0.67% to 20.40% in min DCF. EER MinDCF10 20 Odyssey 2016, Bilbao, Spain

SRE’10 core -extended (female) • We trained the T matrix assuming a subspace-orthonormalizing prior. • Comparing to the results using T trained with standard Gaussian prior, a slightly better results could be observed. EER MinDCF10 21 Odyssey 2016, Bilbao, Spain

Conclusion • We introduced the following for Rapid Computation of I-vector – Subspace ortho nomalizing prior – Uniform occupancy assumption • The computational speed-up is attained by avoiding the need to compute the posterior covariance in order to compute the posterior mean. • The proposed method speed up the i-vector extraction by a factor of 12 compared to fast baseline (and a factor of 106 compared to the slow baseline) with marginal degradation in recognition accuracy. 22 Odyssey 2016, Bilbao, Spain

Thanks for Your Attention! QUESTION? 23 Odyssey 2016, Bilbao, Spain