User Guide
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1. 2 2 GMMs initialization The VQ implements a hierarchical clustering algorithm to provide a first estimate of the GMMs models It does not consider uncertainty Therefore the only input parameters are the actual data function means covars index VQ x niveau_max diag_covs_flag means covars index VQ x niveau_max diag_covs flag An hierarchical clustering vector quantization VQ or K means algorithm input m x observations T vectors of length n n x T matrix niveau max desired number of splits in VQ diag_covs_flag opt estimate diagonal covariances flag def 1 output i i means n x Q matrix of cluster means covars cluster full or diagonal covariances that can be n x n x Q tensor in case of full covariances dias coves ilas i n x Q matrix in case of diagonal covariances diag_covs_flag 0 index 2 niveau_max 1 1 length vector of splits where Q lt 2 niveau_max is the resulting number of clusters Figure 2 VQ function and the number of binary splits and the type of covariance matrix to be estimated full or diagonal covariance After a hierarchical split of the data operated along the axes of maximal variance the function outputs a first estimate of the GMM parameters see Figure 2 for further details 3 Experiment on Synthetic Data The code discussed here has been2. SNR and clean co SNR e subdir_name_train SNR condition for testing On a standard 2 GHz machine with one core the example shown on Figure 5 with 3 speakers needs about an hour The evaluation of one training testing condition for example 9dB SNR at training and 0 dB SNR at testing is expected to take one day for the 34 speakers available So replicating the results of the full table is expected to take about 48 days References 1 A Ozerov M Lagrange and E Vincent GMM based classification from noisy features in Proc 1st Int Workshop on Machine Listening in Multisource Environments CHiME Flo rence Italy September 2011 pp 30 35 2 Uncertainty based learning of gaussian mixture models from noisy data Computer g org Speech and Language 2011 submitted 3 A P Dempster N M Laird and D B Rubin Maximum likelihood from incomplete data via the EM algorithm Journal of the Royal Statistical Society Series B Methodological vol 39 pp 1 38 1977 function EXAMPLE_2__real_19D_MFCC_data comments data_dir_name SP_ REC_Uncrt _MFCC speaker_ids 2 4 6 can be any from 1 to 34 subdirenamemtielsit mods ccanm be 0d bln ods sods es OdBe as mod silewmodn subdir sane train m6dB u IV can be OdB 3dB edb 9dB msds m de Figure 5 Main parameters of the speaker recognition task
3. wle H oig w O mo os end TEST log_likelihoods zeros nbClasses nbSequences_test nbClasses for cl 1 nbClasses fprintf Test for class d of d classes n cl nbClasses for seq 1 nbSequences_test for cl_model 1 nbClasses dummy1 dummy2 dummy3 dummy4 log_likelihoods cl seq cl_model gt GMM_EM_uncertainty_learning squeeze x_noisy_test cl seq squeeze cE_test cl seq est_gmms train_mode cl_model u_gmm est_gmms train_mode cl_model c_gmm est_gmms train_mode cl_model w_gmm 1 0 0 no log i DS end end end compute score end visualization Figure 3 Code of the experiment on synthetic data 4 Speaker recognition experiment on Speech Data In order to experiment with a more realistic task we considered in 2 a speaker recognition task on noisy speech data In this case the uncertainty is not know as a prior and estimated using a method based on Wiener filtering and Vector Taylor Series VTS expansion see Section 4 1 5 of 2 for more details Data GMMs Noisy data Clean data lt GMM of class 1 ae lt Class 1 GMM of class 2 j lt Class 2 4 GMM of class 3 4b lt Class 3 4 4 z4 tae 4 Conventional training clean data Uncertainty training noisy data 2 1 5 0 5 10 10 5 0 5 10 Figure 4 GMMs estimated using di
4. considered to generate the results discussed in 1 with the additional case of conventional noisy training and uncertainty decoding 3 1 Data The data consists in synthetically generated features using GMM sampling The uncertainty is here sampled from a signal Gaussian and supposed to be known both for the training and testing phases see 1 for further details 3 2 Code The function EXAMPLE_1__synthetic_2D_data demonstrate the use of uncertainty learning for the purpose of classifiying synthetic data As can be seen on Figure 3 it mainly consists in training and testing phases for each classes using the proposed approach Uncertainty training on noisy data and conventional ones Conventional training on clean or noisy data The Figure 4 gives a sense of the results for a 2 dimensional case function EXAMPLE_1__synthetic_2D_data comments and generation of synthetic data for train_mode train_modes display info TRAIN for cl 1 nbClasses fprintf Train model d of d models n cl nbClasses x_noisy_train_cl squeeze x_train_CUR cl cE_train_cl squeeze cE_train_CUR cl uXe cXe VQ x_noisy_train_cl log2 nbGaussians 0 0 initialize full covariances wXe ones 1 nbGaussians wXe wXe sum wXe est_gmms train_mode cl u_gmm est_gmms train_mode cl c_gmm est_gmms train_mode cl w_gmm GMM_EM_uncertainty_learning x_noisy_train_cl cE_train_cl uXe cXe
5. Gaussian Mixture Model Uncertainty Learning GMMUL Version 1 0 User Guide Alexey Ozerov Mathieu Lagrange and Emmanuel Vincent INRIA Centre de Rennes Bretagne Atlantique Campus de Beaulieu 35042 Rennes cedex France 2 IRCAM STMS UPMC CNRS UMR 9912 1 place Igor Stravinsky 75004 Paris alexey ozerov emmanuel vincent inria fr mathieu lagrange ircam fr January 10 2012 1 Introduction This user guide describes a method that can be considered for learning Gaussian Mixture Models GMMs while acknowledging the fact that the data can be uncertain The level of uncertainty can be known or estimated In order to fully grasp the theoretical aspects of the method as well as the practical facts that motivated the proposal of this method the reader is strongly encouraged to read the following papers 1 2 that will continuously be referred to in the remaining of this document This guide is organized as follows Section 2 describes the two core functions that allows the user to learn GMMs from uncertain data Those functions are then presented within two evaluation frameworks respectively considered in 1 and 2 Section 3 describes the first one that focus on synthetic data i e observed data and uncertainty is generated from sampling of GMMs Section 4 describes the second one that considered noisy speech data where the uncertainty is known as a prior or estimated 2 Processing Functions 2 1 GMM EM uncertainty learn
6. a is divided in two main directories test and train that respectively contains the data used for training and testing For each one clean no noise addition and several Signal to Noise ratios are considered from 6dB to 9dB of SNRs For each SNR the 3 above discussed conditions mix ssep ssep_uncrt are available For the latter the full covariance uncertainty estimated using the Wiener VTS approach is provided It shall be noted that the MFCCs of this condition are the same as in the ssep condition as the VTS estimator does not change the actual MFCC values see Equation C 2 in 2 4 2 Code The function EXAMPLE_2__real_19D_MFCC_data can be considered to replicate the full results of the experiments reported in Table D 7 of 2 One run computes the results for one training testing condition for the following 4 cases 1 Conventional training Conventional decoding without signal enhancement 2 Conventional training Conventional decoding with signal enhancement 3 Conventional training Uncertainty decoding with signal enhancement 4 Uncertainty training Uncertainty decoding with signal enhancement Figure 5 shows the first line of the function that set the main processing parameters e data_dir_name path to the data repository e speaker_ids selection vector for the speaker to consider 1 34 will consider all the speakers available e subdir_name_test SNR condition for training from m6dB 6 dB SNR to 9dB 9 dB
7. fferent learning conditions By considering the uncertainty in the learning algorithm the resulting GMMs are much closer to the original ones when considering noisy data 4 1 Data The data is provided separately from this toolbox and can be freely obtained at http www irisa fr metiss ozerov Software SP_REC_Uncrt_MFCC zip It consists of Mel Frequency Cep stral Coefficients MFCCs computed over 3 different inputs 1 mix raw addition of clean speech and noise 2 ssep output of a state of the art source separation algorithm fed with the raw addition of clean speech and noise 3 ssep_uncrt same but with an estimate of the uncertainty of the output done with the VTS method Details about how and on what kind of audio data the MFCCs have been generated is provided in Section 4 1 2 of 2 Data is provided as MAT file that can be opened with Matlab or any parser that can read HDF5 files The naming convention is as follows s lt speakerId gt _ lt utteranceId gt _mfcc mat The speakerlId corresponds to the numeric id of the speaker from 1 to 34 For the 3 different inputs described above the MAT file contains the mfcc variable which is a 2 dimensional floating point matrix of size 20 x nf where np is the number of frames that have been considered for computing the 20 dimensional MFCCs For the last input a second variable mfcc_covar is available It encodes the uncertainty as a 3 dimensional tensor of size 20 x 20 x ny The dat
8. ing The core function is GMM_EM_uncertainty_learning that implements a new Expectation Maximization EM algorithm for learning GMMs from uncertain data It can be considered for the purpose of training GMMs and decoding GMMs both with handling of uncertainty function uXe cXe wXe log_like_N 1 GMM_EM_uncertainty_learning y cE uX cX wX nbIterations print_log_flag Yo ues een we los Hke NSn GMM_EM_uncertainty_learning y cE uX cX wX nbIterations print_log flag Expectation maximization EM algorithm for training Gaussian mixture models GMMs from noisy observations with Gaussian uncertainty input ence y observations N vectors of length M Mx N matrix cH opt Gaussian uncertainty full or diagonal covariance matrices that can be MxMxN tensor in case of full covariances Mx N matrix in case of diagonal covariances empty equivalent to conventional training def uX initial GMM Gaussian means M x K matrix cX initial GMM Gaussian full or diagonal covariances that can be Mx Mx K tensor in case of full covariances M x K matrix in case of diagonal covariances wX initial GMM Gaussian weights K length vector nbIterations opt number of EM algorithm iterations def 30 ouput log flas ni opi printinge loge tiag den 1 where K number of gaussians in GMM M dimensionalit
9. y of feature vector N number of observations output T uXe estimated GMM Gaussian means M x K matrix cXe estimated GMM Gaussian full or diagonal covariances that can be Mx Mx K tensor in case of full covariances M x K matrix in case of diagonal covariances NOTE cXe has the same dimentionality as its initialization cX wXe estimated GMM Gaussian weights K length vector log like N N length array of log likelihoods for each observation 1 nbIterations length array of global log likelihoods over EM algorithm iterations Figure 1 Header of the main processing function For decoding GMMs one simply needs to provide the data optionally with a Gaussian uncer tainty and the parameters of the model In this case the number of iteration can be set to 1 and the log likelihood computed at the Expectation step is considered For training GMMs one needs to provide the data with a Gaussian uncertainty a first estimate of the parameters of the model and a number of iterations that is sufficient to reach convergence In all the experiments reported in 1 and 2 those estimates are obtained with the function VQ discussed below Details about inputs and outputs are described In both cases if the uncertainty is not provided the algorithm reduces to a standard EM algorithm for training GMMs 3 Detailed description of the input and output parameters is given in Figure 2 1
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