Optimal state dependent spectral representation for HMM modeling : a new theoretical framework

In this paper we propose a theoretical framework to extend classical continuous density HMM in order to consider different spectral representations depending on the state. We stress the need for a reference space and for spectral transformations between the model spectral representation spaces and the reference space. We show that this framework permits to obtain more precise pdfs in the reference space. Preliminary speech recognition experiments for two spectral representations MFCC and linear frequency scale cepstral coefficients show no improvements ; however they identify that the choice of the spectral representations is crucial and the determination of the spaces transformations is a complex problem 2. THEORETICAL FRAMEWORK 2.1 Model Definition Classically an HMM λ, with Q states and Gaussian sub-processes pdfs, is defined using the parameters : • π = {πi} i=1,...,Q : the probabilities of occupying the state i at the first instant. • A = {aij}i,j=1,...,Q : the transition probabilities from state i to state j. • B = {bi(Xτ)} i=1,...,Q : the Gaussian subprocesses associated with the different states of the model. Each Gaussian distribution has two sets of parameters : its mean μi and its covariance matrix Γi. Here, we propose to associate a given spectral representation identified by its index α with each state in the model. We assume that the different feature vectors belongs to R. This assumption is necessary in order to perform ‘‘Maximum Likelihood’’ training and classification. In such case the sub-processes pdfs are defined : • B = {bi(Xτ,αi)} i = 1,...,Q and αi ∈ {1, ..., M} : where each distribution is identified by the spectral representation index in addition to the classical mean vector and covariance matrix. However, as defined, the new HMM cannot be trained nor used for classification since the observed outputs do not belong to a predefined and unique space, even if the different representation spaces have the same dimension p. Actually, the observation spectral representation depends on the state of the HMM which is hidden and not observed. This means that the spectral representation of frame may vary for several word hypotheses and within a single hypothesis during the training, making difficult to tie the measured likelihood to the probability. A solution to this problem consists in defining a reference space with a spectral representation α̂ (MFCC for example). Considering that a function T α/α̂ permits to match the space X α on to the space X α̂ : X τ,α̂ = T α/α̂(X τ,α) (1) then the density for a state i can be written : bi(X τ,α̂) = bi(X τ,αi) / ||J(X τ,αi)|| (2) where J(X τ,αi) is the Jacobian matrix whose (k,l) element is : Jk,l(X τ,α) = ∂T α/α̂(X τ,α)k ∂X τ,α l (3) Using the Eq. (2) and (3) the observations of the new defined HMM belong to a unique space X α̂. A very important question appears at this stage : • In what the new model differs from an HMM completely defined in the space X α̂? Actually, the new model differs from the X α̂ model in the form of its sub-processes distributions. It may be more reliable to approximate with a Gaussian the distribution of the data in a space X α for a given state, than to approximate with a Gaussian the distribution of the same data in the reference space X α̂. Thus, the data in X α̂ may be distributed following a more complex and precise probability density function depending on the space transformation. Hence, through the Jacobian matrices, the corresponding distribution in the reference space would be more precise. Moreover, more precise distributions lead generally to a more precise and robust model. Nevertheless, the estimation of these Jacobian matrices remains a major problem as we will see in the following. 2.2 Computation of the Space Transformation function The determination of the transformation T α/α̂ that allows to match a given space on to the reference space is not always obvious. Thus, one can approximate this transformation by a simpler function chosen for some mathematical attractiveness such as a regression matrix, a LMR, a linear quadratic function, a Volterra function, etc. Once defined the parameters of these functions must be estimated. This can be done using classical criteria such as ‘‘Minimum Mean Square Error’’ (MMSE). We can notice that the transformation parameters estimation may be easier here since there is no alignment problem since the different analysis techniques are derived for the same frames of signal. 2.3 Training Algorithm The parameters of the new model may be trained using the classical EM algorithm. In the ‘‘Estimate’’ step the auxiliary function relative to a given state is computed for the M possible spectral representations. During the ‘‘Maximize’’ step, in order to maximize the auxiliary function, the mean and covariance matrices would be determined for all the spectral representations, and for a given distribution the optimal spectral representation is chosen following : αopt=argmax α ∑ τ [-1/2log||Γi,α|| log||J(X τ,α)||] (4) Looking to this equation it appears clearly that the Jacobian impact is crucial to compensate for decreasing the variations in the state. 3. PARTICULAR APPLICATION : VARIABLE RESOLUTION CEPSTRAL COEFFICIENTS Different spectral representations may be obtained by varying the spectral resolution of the filter-bank used to compute the MFCC coefficients. In [6], a computational method using FFT is proposed for spectra with nonuniform resolution on the frequency scale. A procedure for implementing a frequency warping all-pass transformation, known as bilinear transformation, estimates the AR parameters (of order p) using the outputs of p successive filters. These dephasing filters are preceded by a corrective filter in order to obtain an all-pass filter. If first order dephasing filters are considered then, the frequency transformation is expressed by the following relation in the normalized frequency domain :