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

Rapid Computation of I-vector

1Institute for Infocomm Research

(I2R), Singapore

2Nanjing University of Posts and

Telecomm, China

Longting XU 1,2, Kong Aik LEE 1, Haizhou Li 1 and Zhen Yang 2

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].

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Introduction (cont’d)

• I-vector extraction is a posterior inference process
• We use a pre-whiten statistics [Matejka et al, 2011] in this work

 

, x 0 I

Prior Observations Posterior

 

1 2

, , ,

T

• o
• 

 

1

, 



x L

1 T 1



 

  L T Σ F

 

1 1 T 1   

  L I T Σ NT

1 T





  L T F

 

1 1 T   

 L I T NT

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Objective

• Objective:
• 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.

To reduce the computation complexity of i-vector extraction while keeping the memory requirement low with slight degradation in performance.

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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]

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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

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INFORMATIVE PRIOR

I-vector extraction using

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Posterior inference with informative prior

• Conventional i-vector extraction assumes a standard Gaussian prior on the

latent variable x

• Consider a more general case where the prior on x has mean 𝛎p and

covariance 𝚻p

• I-vector extraction with informative prior

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 

, x 0 I

 

p p

, x μ Σ

Subspace orthonormalizing prior

• In this work, we consider the following informative prior
• I-vector extraction

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   

1 T p p

with ,



 x Σ Σ T T

1 T





  L T F

1 1 T T  

      L T T T NT

   

 

   

1 T T T 1 1 T T T T 1 1 1 T T T T



     

                     T T T NT T F T T I T T T NT T F I T T T NT T T T F

Subspace orthonormalizing prior (cont’d)

• Using the matrix inversion identity
• I-vector extraction with subspace orthonormalizing prior:

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   

1 1 1 T T T T



  

       I T T T NT T T T F P Q P Projection matrix with

• rthonormal columns

 

1 T T T 1 1 

 T T T T U U

   

1 1 1 T T T T



  

      T T T I NT T T T F

RAPID COMPUTATION

I-vector extraction with

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Solving the matrix inversion

• Singular value decomposition of T
• It follows that U1 spans the same subspace as T, and U1 ⊥ U2
• Using the above, we solve for the following matric inversion

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 

1 2

,   T USV U U SV

     

1 1 1 T T T 1 1 1 T 2 2 1 T 2 2     

                          I N T T T T I N U U I N I U U I N NU U

Solving the matrix inversion (cont’d)

• Let A = (I + N)
• Using the matrix inversion lemma
• Using again the matrix inversion identity

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   

1 1 T 1 1 T 1 T 1 2 2 2 2 2 2      

    A NU U A A N I U U A N U U A

     

1 1 1 1 T T T T 2 2 2 2    

             I N T T T T I N NU U A NU U

   

1 1 1 T T 1 1 T 1 T 1 2 2 2 2       

           I N T T T T A A NU U I A NU U A

Rapid computation of i-vector

• Uniform occupancy assumption:
• Or equivalently
• The matrix inversion can be simplified as
• Since T ⊥ U2 , the second term diminishes

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 

1 1

for 0 1  

 

     A N I N N I 1

c c

N c N    

   

1 1 1 T T 1 T 1 T 1 2 2 2 2



     

            I N T T T T A U U I A NU U A

     

 

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

Computational complexity and memory cost

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Complexity Memory cost Time ratio Baseline (slow) O(CFM2 + M3) O(CFM) 106.44 Baseline (fast) O(CFM + CM2 + M3) O(CFM + CM2) 11.99 Proposed (exact) O(CFM + CM2 + M3) O(CFM + CM2) 12.65 Proposed (fast) O(CFM) O(CFM) 1

Baseline (slow) Baseline (fast) Proposed (exact)

 

1 T T c c c c N





  



I T T T F

 

1 T c c c N





  



I A T F

 

 

1 T T

1

c c c c N





  



T T T F

 

 

1 1 T T



 

  T T T I N F

Proposed (fast)

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(CM2), assuming that we pre-computed

the matrices .

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T c c

T T

Using informative prior in EM update of T

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EXPERIMENT

Rapid computation of i-vector

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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

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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.

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EER MinDCF10

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.

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EER MinDCF10

Conclusion

• We introduced the following for Rapid Computation of I-vector

– Subspace orthonomalizing 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.

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QUESTION?

Thanks for Your Attention!

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