# Patent

## Title
Parameter clustering and sharing for variable-parameter hidden markov models

## Abstract

A speech recognition system uses Gaussian mixture variable-parameter hidden Markov models (VPHMMs) to recognize speech. The VPHMMs include Gaussian parameters that vary as a function of at least one environmental conditioning parameter. The relationship of each Gaussian parameter to the environmental conditioning parameter(s) is modeled using a piecewise fitting approach, such as by using spline functions. In a training phase, the recognition system can use clustering to identify classes of spline functions, each class grouping together spline functions which are similar to each other based on some distance measure. The recognition system can then store sets of spline parameters that represent respective classes of spline functions. An instance of a spline function that belongs to a class can make reference to an associated shared set of spline parameters. The Gaussian parameters can be represented in an efficient form that accommodates the use of sharing in the above-summarized manner.

## Background
BACKGROUND
Speech recognition systems commonly use Hidden Markov Models (HMMs) to automatically recognize speech (e.g., by decoding the linguistic content of the speech). In a training phase, this type of speech recognition system estimates the parameters which define the HMMs; it performs this task by training the HMMs based on a set of speech signals labeled with linguistic content items (e.g., words or other forms). In a recognition phase, the speech recognition system uses the trained HMMs to recognize new speech signals.
One type of HMM is referred to as a Gaussian mixture HMM. This type of HMM uses a combination of Gaussian components (referred to as “mixture components” herein) to model the distribution of speech and noise within the environment for each state. Different environments are characterized by noise having different characteristics. Thus, to provide more robust speech recognition, Gaussian mixture HMMs can be trained using a training set that accounts for speech within different types of environments.
In the classic case, the Gaussian mixture components used by Gaussian mixture HMMs are fixed, meaning that these components do not vary as a function of the characteristics of the environment. In a more recently proposed case, the Gaussian mixture components can vary as a function of a measurable characteristic of the environment, such as the signal-to-noise ratio (SNR). For example, each Gaussian mixture component can include a mean component μ and variance component Σ, each of which varies as a function of SNR. This variable type of HMM model is referred to herein as a variable-parameter hidden Markov model (VPHMM). VPHMMs potentially offer better performance than fixed-component HMMs because the VPHMMs adapt to different types of environments, as opposed to using a single set of parameters to account for every circumstance that may be encountered within an environment.
While VPHMMs have been shown to be effective, there remains room for further improvement in this technology. Particularly, one known version of VPHMM technology uses global polynomial functions to approximate the way in which Gaussian parameters (μ's and Σ's) vary as a function of utterance SNR. There may be various inefficiencies and limitations associated with this approach.
SUMMARY
An illustrative speech recognition system (“recognition system”) is described that decodes the linguistic content of speech using variable-parameter hidden Markov models (VPHMMs). In one case, the VPHMMs include Gaussian mixture components for modeling state observations. Each Gaussian mixture component includes at least one Gaussian parameter that varies as a function of at least one environmental conditioning parameter (referred to in singular in this Summary section for brevity). The recognition system can use spline functions to model the way in which respective Gaussian parameters vary as a function of the environment conditioning parameter. In one implementation, the speech recognition system can be trained, in part, by: clustering the spline functions into a plurality of classes; and, for each class, storing a shared set of spline parameters that represents that class of spline functions. Particular instances of a class of spline functions can make reference to the shared set of spline parameters.
According to another illustrative aspect, the above-mentioned at least one Gaussian parameter includes a mean parameter μ and a variance parameter Σ. The mean parameter μ and the variance parameter Σ can be represented in an efficient form that accommodates the use of parameter sharing in the above-summarized manner.
According to another illustrative aspect, the above-mentioned clustering operation can entail defining distances between respective pairs of spline functions, and using the distances to determine the formation of clusters.
According to another illustrative aspect, for each pair of the spline functions, the distance may be approximated as a difference between knots associated with the two respective spline functions.
According to another illustrative aspect, a spline function can be represented in a store as: K, x(1), x(K), and yi, where: K is a number of knots in the spline function; x(1) and x(K) are first and last x-dimension positions of the knots in the spline function, respectively; and yi are y-dimension positions of the knots in the spline function.
The ensuing Detailed Description describes other illustrative aspects of the speech recognition system. More generally, this Summary is provided to introduce a selection of concepts in a simplified form; these concepts are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows an illustrative recognition system for recognizing speech using variable-parameter hidden Markov models (VPHMM), here showing training aspects of the recognition system.
FIG. 2 shows the recognition system of FIG. 1, here showing recognition aspects of the recognition system.
FIG. 3, which complements FIG. 2, is an illustrative procedure that provides an overview of the recognition aspects of the recognition system.
FIG. 4 shows an example of a piecewise function that can be used to approximate the manner in which a Gaussian parameter varies as a function of an environmental conditioning parameter.
FIG. 5 shows a probability distribution that can be used to determine a range of interpolation for the piecewise function of FIG. 4.
FIG. 6 is an illustrative procedure for training VPHMMs that use piecewise functions.
FIG. 7 is an illustrative procedure for performing clustering of parameters associated with piecewise functions.
FIG. 8 is a graphical depiction of one illustrative technique for approximating a difference between two piecewise functions.
FIG. 9 shows illustrative functionality for mapping an instance of a piecewise function to an associated set of shared parameters.
FIG. 10 shows illustrative processing functionality that can be used to implement any aspect of the features shown in the foregoing drawings.



The same numbers are used throughout the disclosure and figures to reference like components and features. Series 100 numbers refer to features originally found in FIG. 1, series 200 numbers refer to features originally found in FIG. 2, series 300 numbers refer to features originally found in FIG. 3, and so on.
DETAILED DESCRIPTION
This disclosure sets forth a speech recognition system (“recognition system”) that uses variable-parameter hidden Markov models (VPHMMs). The recognition system incorporates various features, enumerated below, which potentially enhance the accuracy, flexibility, and/or efficiency of speech training and recognition compared to previous VPHMM solutions.
This disclosure is organized as follows. Section A provides background information regarding VPHMMs. Section B provides an overview of illustrative improvements to VPHMMs. Section C describes the application of piecewise functions to VPHMMs. Section D describes the estimation of environmental conditioning parameters for use in VPHMMs. Section E describes techniques for training VPHMMs. Section F describes the application of clustering and parameter-sharing to VPHMMs. And Section G describes illustrative processing functionality that can be used to implement any aspect of the features described in previous sections.
As a preliminary matter, some of the figures describe the concepts in the context of one or more components, variously referred to as functionality, modules, features, elements, etc. The various components shown in the figures can be implemented in any manner, for example, by software, hardware, firmware, manual processing operations, and so on, or any combination of these implementations. In one case, the illustrated separation of various components in the figures into distinct units may reflect the use of corresponding distinct physical components. Alternatively, or in addition, any single component illustrated in the figures may be implemented by plural physical components. Alternatively, or in addition, the depiction of any two or more separate components in the figures may reflect different functions performed by a single physical component. FIG. 10, to be discussed in turn, provides additional details regarding one illustrative implementation of the functions shown in the figures.
Other figures describe the concepts in flowchart form. In this form, certain operations are described as constituting distinct blocks performed in a certain order. Such implementations are illustrative and non-limiting. Certain blocks described herein can be grouped together and performed in a single operation, certain blocks can be broken apart into plural component blocks, and certain blocks can be performed in an order that differs from that which is illustrated herein (including the performance of certain operations in parallel). The blocks shown in the flowcharts can be implemented by software, firmware, hardware, manual processing, any combination of these implementations, and so on.
As to terminology, the phase “configured to” encompasses any way that any kind of functionality can be constructed to perform an identified operation. The functionality can be configured to perform an operation using, for instance, hardware, software, firmware, etc., and/or any combination thereof.
The term “logic” encompasses any functionality for performing a task. For instance, each operation illustrated in the flowcharts corresponds to logic for performing that operation. In one case, logic may correspond to computer-readable instructions. In another case, logic may correspond to discrete logic components.
A. Background on Fixed-Parameter and Variable-Parameter Hidden Markov Modeling
The improvements to variable-parameter hidden Markov models (VPHMMs) presented here can best be appreciated by providing background information regarding previous VPHMM solutions. For example, Xiadong Cui and Yifan Gong introduce the concept of the VPHMM approach in “Variable Parameter Gaussian Mixture Hidden Markov Modeling for Speech Recognition,” Proceedings of the Conference on Acoustics, Speech, and Signal Processing, Vol. 1, April 2003, pp. 1-12-1-15. These authors also describe their VPHMM solution in “A Study of Variable Parameter Gaussian Mixture Hidden Markov Modeling for Noisy Speech Recognition, in IEEE Transactions on Audio, Speech, and Language Processing, Vol. 15, No. 5, May 2007, pp. 1366-1376, as well as published U.S. Patent Application No. US 2004/0181409. More general background information regarding hidden Markov models can be found in Lawrence R. Rabiner, “A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition,” Proceedings of the IEEE, Vol. 77, No. 2, February 1989, pp. 257-286, as well as B. H. Juang and L. R. Rabiner, “Hidden Markov Models for Speech Recognition,” Technometrics, August 1991, Vol. 33, No. 3, pp. 251-272.
In traditional hidden Markov models (HMMs), a continuous observation density function bi(xr,t) for state i and acoustic observation xr,t at frame t in an utterance r can be expressed as a combination of distribution components. In one particular implementation, the observation density function bi(xr,t) can be expressed as a combination of Gaussian distribution components. These components are referred to herein as Gaussian mixture components. In this implementation, bi(xr,t) is given by:




  
    
      
        
          
            
              
                
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Here, the probability defined by bi(xr,t) is estimated using L Gaussian mixture components, N(xr,t|μi,l, Σi,l), where l refers to the l-th Gaussian mixture component in the combination. Each mixture component is defined by fixed Gaussian parameters, namely a fixed mean component (μi,l) and a fixed variance component (Σi,l). The parameter wi,l defines a positive weight for the l-th Gaussian component, where Σi=l, . . . Lwi,l=1. Further note that xr,t, μi,l, and Σi,l are vectors. A particular dimension of these vectors is represented by d.

HMMs that use Gaussian mixtures components are potentially effective in modeling speech within noisy environments. To facilitate recognition, these types of HMMs can be trained using speech signals (and associated noise) that simulate several different environments. However, these HMMs also have potential drawbacks. For example, as addressed by Cui and Gong, it may not be efficient to use a single set of fixed models to account for different environments in which speech may occur. First, after training, recognition takes place in a single environment at any given point in time; in this case, the great diversity of environments encompassed by the HMMs can complicate recognition (e.g., by providing a large model space that needs to be searched during recognition). This factor, in turn, may potentially degrade the quality of recognition. Further, even though the Gaussian mixture HMMs are robust, they may still be ineffective in modeling all of the environments in which speech may occur. For example, these types of HMMs may have difficulty accounting for new types of noise or known types of noise that occur at newly-encountered intensities.
The VPHMM approach addresses these potential drawbacks by replacing the fixed Gaussian components with variable components. Namely, the Gaussian components (μi,l Σi,l)in the VPHMM approach become functions of an environment-dependent parameter set ζ:




  
    
      
        
          
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That is, the mean parameter μi,l (ζ) now varies as a function of the environmental conditioning parameter ζ. Likewise, the variance parameter Σi,l(ζ) varies as a function of the environmental conditioning parameter ζ. The environmental conditioning parameter proposed by Cui and Gong is utterance signal-to-noise-ratio (SNR).

The parameters μi,l (ζ) and Σi,l(ζ) can be learned from training data during a training phase. More specifically, the Gaussian parameters μi,l (ζ) and Σi,l(ζ) can be constructed using a parametric approach. For example, Cui and Gong use a global polynomial regression function over the utterance SNR to represent the mean parameter μi,l (ζ)and the variance parameter Σi,l(ζ); further, their approach uses the maximum likelihood (ML) algorithm to estimate the parameters in the polynomial functions. A diagonal covariance matrix is assumed. The Gaussian parameters in the d-th dimension are determined by:

μi,l,d(ζr,t,d)=ξ(ζr,t,d|μi,l,d(1), . . . , μi,l,d(K)), and   (3)

Σi,l,d(ζr,t,d)=Σi,l,d(0)eξ(ζr,t,d|Σi,l,d(1), . . . , Σi,l,d(K)).  (4)

Here, μi,l,d(1), . . . , μi,l,d(K) are Σi,l,d(1), . . . , Σi,l,d(K) are polynomial parameters for the mean parameters and variance parameters, respectively, and Σi,l,d(0) is the original variance parameter in a conventional HMM. Note that equation (4) is chosen to guarantee that Σi,l,d (ζr,t,d)>0. In the context of equations (3) and (4), ξ represents the polynomial function that is defined by the polynomial parameters μi,l,d(1), . . . , μi,l,d(K) and Σi,l,d(1), . . . , Σi,l,d(K), respectively.

B. Overview of Illustrative VPHMM Improvements
An improved speech recognition system is described herein that uses Gaussian mixture variable-parameter hidden Markov models (VPHMMs) to recognize speech (e.g., by decoding the linguistic content of speech). According to one feature, the recognition system uses spline functions to model the way in which its Gaussian parameters change as a function of one more environmental conditioning parameter (rather than global single-piece polynomial functions). According to another feature, the recognition system 100 uses an instantaneous measure the environmental conditioning parameter (or parameters) (rather than utterance SNR). According to another feature, recognition system expresses the Gaussian components in a way in which a Gaussian-specific part is separated from a sharable part. According to another feature, the recognition system uses a discriminative training operation to generate the parameters of the VPHMMs. The discriminative training operation is more effective and efficient than other solutions (for instance, dispensing with the use of quantization on the conditioning parameter); this efficiency is made possible, in part, due to the effective way in which the Gaussian components are expressed.
The speech recognition system also facilitates sharing of spline parameters. For instance, according to one feature, the speech recognition system applies a clustering operation to identify classes of spline functions. The speech recognition system can provide a shared store that provides spline parameters associated with the identified classes. Particular instances of spline functions in the VPHMMs can make reference to the shared store to obtain corresponding spline parameters; this reduces the need for spline functions to individually provide “local copies” of their corresponding spline parameters. The clustering is made feasible, in part, by the efficient way in which the Gaussian components are expressed.
The above list of features is representative, not exhaustive. Additional details regarding the features enumerated above will be provided in the following sections.
Before discussing the individual features in detail, this section will provide an overview of the recognition system with reference to FIGS. 1-3. These figures show a recognition system 100 for recognizing speech. The recognition system 100 operates in a training phase and a recognition phase. The training phase involves determining the parameters of the VPHMMs; these parameters define the VPHMMs. The recognition phase entails using the VPHMMs to recognize speech. FIG. 1 shows the training aspects of the recognition system 100, while FIGS. 2 and 3 describe the recognition aspects of the recognition system 100.
As shown in FIG. 1, the recognition system 100 is trained by presenting a training set to the recognition system 100. The training set represents a set of speech signals that occur within a testing environment 102. The training set labels the speech signals to indicate phonemes and/or words that the speech signals represent. The testing environment 102 may encompass plural environments in which speech occurs. The environments may be characterized by one or more environmental conditions which affect the recognition of speech. For example, the environmental conditions may take the form of noise of different types, and/or noise of the same type at different intensities. Alternatively, or in addition, the environmental conditions may take the form of different speaking rates, and/or different speech accents (e.g., a French accent, a German accent, etc.), and/or different speech styles, and so on. No limitation is placed herein on what environmental conditions may affect the recognition of speech.
A feature determination module 104 extracts features from the training set. The features pertain to information which characterizes the speech. Any feature extraction functionality can be used to perform this task, such as Mel-frequency cepstral coefficient (MFCC) functionality.
A condition determination module 106 measures one or more of the above-mentioned environmental conditions that affect the ability of the recognition system 100 to recognize speech. The condition determination module 106 formulates these measurements in terms of one or more environmental conditioning parameters. For example, one or more environmental conditioning parameters may be generated that are based on noise within the testing environment 102. For example, one environmental conditioning parameter can measure signal-to-noise-ratio (SNR) within the testing environment 102. Another environmental conditioning parameter may reflect a level of noise within the testing environment 102. Another environmental conditioning parameter may reflect a power of noise within the testing environment 102. One or more other types of environmental conditioning parameters may reflect aspects of the speech which are attributed to the speaker himself or herself. For example, another environmental conditioning parameter may reflect a rate of speech. Another environmental conditioning parameter may reflect an accent of speech. Another environmental conditioning parameter may reflect a style of speech by a speaker, and so on. No limitation is placed herein on what may constitute an environmental conditioning parameter; broadly, an environmental conditioning parameter reflects any aspect of the testing environmental 102 that has an impact on the ability of the recognition system 100 to recognize speech.
In one case, the recognition system 100 can operate on the basis of a single environmental conditioning parameter. In other cases, the recognition system 100 can operate on the basis of plural environmental conditioning parameters. In one scenario of this latter case, the recognition system 100 can operate on the basis of plural variables which reflect the plural environmental conditioning parameters. In another scenario, the recognition system 100 can operate on the basis of a single variable, and that single variable, in turn, can be based on plural environmental conditioning parameters. To facilitate explanation, the recognition system 100 will be described below as acting on a single environmental conditioning parameter, such as signal-to-noise-ratio (SNR). However, it should be kept in mind that the recognition system 100 can operate on the basis of plural environmental conditioning parameters in the manner described above.
In one case, the condition determination module 106 can provide an instantaneous measurement of the environmental conditioning parameter to be operated on by the recognition system 100. For example, the recognition system 100 can operate based on instantaneous sign-to-noise-ration (SNR). In contrast, previous solutions describe a recognition system which operates based on utterance SNR. Utterance SNR appropriates SNR on the basis of the noise that is manifested over an entire unit of speech, such as an entire phoneme or an entire word or an entire sentence, etc. The use of utterance SNR introduces delay when decoding the speech.
Section D (below) will provide additional details regarding the condition determination module 106.
A training module 108 receives features extracted by the feature determination module 104 and at least one environmental conditioning parameter provided by the condition determination module 106. The training module 108 proceeds by generating the parameters which define the VPHMMs. In a broad sense, the training tries to find those models that best account for the observations in the labeled training set. The training module 108 stores the parameters which define the VPHMMs in a data store 110. These parameters are used in the recognition phase of the recognition system 100 (to be described below).
The training module 108 applies a discriminative training approach which differs from previous VPHMM solutions. For instance, the training module 108 represents the Gaussian parameters of the VPHMMs using piecewise functions, such as cubic spline functions. Further, the training module 108 represents the Gaussian parameters in a way in which a Gaussian-specific part is separated from a sharable part (associated with the knots of the spline functions). The way in which the Gaussian parameters are expressed allows the training module 108 to perform its training using a different approach than previous solutions. For example, the training module 108 trains the VPHMM parameters directly from a single conventional HMM instead of from a set of them trained under quantized SNR conditions. Section E provides additional details regarding the training operation.
Further, the training module 108 can optionally apply a clustering operation which identifies classes of spline functions. Parameters which describe these common classes can be stored in a shared data store. An instance of a spline function that belongs to one of these classes can make reference to the stored parameters which define the class. This avoids the need for this instance of a spline function to store an individual copy of the spline function's parameters. Section F provides additional details regarding the clustering operation.
It should be noted that the clustering operation is qualified as being optional. This means that the recognition system 100 can express each spline function using individual copies of the parameters which define the spline function, rather than making reference to a common pool of spline parameters.
Advancing to FIG. 2, this figure shows the recognition aspects of the recognition system 100. In this case, the recognition system 100 accepts a new speech signal in a recognition environment 202. It is not known a priori what information is being imparted by this new speech signal. The goal of the recognition system 100 in the recognition phase is to recognize speech based on the speech signal.
The recognition system 100 recognizes speech using a speech recognition module 204. The speech recognition module 204 operates based on input received from the feature determination module 104 and the condition determination module 106. These modules (104, 106) operate in the same manner as described above with respect to FIG. 1. That is, the feature determination module 104 extracts features from the speech signal that characterize the informational content of the speech signal. The condition determination module 106 forms at least one environmental conditioning parameter which characterizes one or more environmental conditions which impact the recognition of speech. In the following discussion, to facilitate explanation, the condition determination module 106 is described as providing a single environmental conditioning parameter. In one example, the condition determination module 106 can provide a measure of instantaneous SNR.
The speech recognition module 204 also interacts with the parameters provided in the data store 110. These parameters describe the VPHMMs that have been generated in the training phase. The VPHMMs include Gaussian parameters that are a function of the environmental conditioning parameter; thus, the VPHMMs, in effect, are customized based on the measured value of the environmental conditioning parameter.
The speech recognition module 204 recognizes speech by attempting to find the VPHMMs that best account for the features identified by the feature determination module 104. This procedure can make use of known search algorithms for finding the most appropriate VPHMMs within a total search space defined by the VPHMMs.
FIG. 3 shows a procedure 300 which provides a general overview of the recognition aspects of the recognition system 100. In block 302, the recognition system 100 receives a speech signal. The speech signal may represent informational content associated with a human utterance, as well as one or more environmental conditions (such as noise) which may impact speech recognition. In block 304, the recognition system 100 determines features associated with the informational content of the speech signal, as well as an environmental conditioning parameter which expresses the environmental condition in the recognition environment 202. In block 306, the recognition system uses the stored VPHMMs to recognize the speech—e.g., to determine the informational content of the speech signal.
C. Approximation of Gaussian Components Using Piecewise Functions
As summarized above, the recognition system 100 can use piecewise functions (each of which is denoted by the symbol ξ) to represent the way in which Gaussian parameters (the mean parameter μ and the variance parameter Σ) vary as a function of the environmental conditioning parameter (which is denoted by the symbol ζ). More specifically, in one illustrative case, the recognition system 100 uses spline functions as the piecewise functions, such as cubic spline functions. The following discussion will be framed in the context of the use of cubic spline functions, but it should be emphasized that other implementations can use other types of piecewise functions.
FIG. 4 shows one spline function that expresses the way in which a Gaussian parameter (the mean parameter μ or the variance parameter Σ), plotted along a y dimension varies as a function of the environmental conditioning parameter (ζ), plotted along an x dimension. In this merely illustrative case, the spline function includes four spline segments (S1, S2, S3, and S4). The spline segments include respective endpoints, referred to as knots. Each spline segment represents a polynomial curve that can be parametrically defined. In general, the goal of the spline-based curve fitting is to select the parameters which define the spline function such that the spline function best approximates the change of the Gaussian parameter over the measured environmental conditioning parameter (ζ). Although not shown in FIG. 4, the Gaussian parameters and environmental conditioning parameter are vectors with multiple dimensions. Thus, spline-based curve fitting can be performed in multiple dimensions d.
More formally, in one illustrative implementation, it can be assumed that the covariance matrices are diagonal and that each dimension d of the mean and variance vector can be approximated with a cubic spline ξ as:




  
    
      
        
          
            
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In these expressions, μi,l,d(0) and Σi,l,d(0) are the Gaussian-component-specific mean and variance, and μ ω(i,l,d)(1), . . . , μ ω(i,l,d)(K) and Σ ω(i,l,d)(1), . . . , Σ ω(i,l,d)(K) are the spline knots that can be shared across different Gaussian components.

The notation ω(i,l,d) refers to a regression class. That is, in one implementation, the training module 108 can identify parameters that are shared by multiple Gaussian components. A data store can store these shared parameters. Different regression classes represent different sets of shared parameters. In this approach, if a particular Gaussian component has parameters that are described by one of the regression classes, it can make reference to the appropriate regression class (by providing information, ω(i,l,d), which identifies the regression class). In this manner, multiple Gaussian components that make reference to the same shared set of parameters are not required to store duplicate copies of these parameters. However, the use of sharing and regression classes is optional. This means that each Gaussian component can provide its own instantiations of its parameters. This further means that the equations developed herein can dispense with the ω(i,l,d) notation. Section F provides additional information regarding the use of clustering and sharing in the VPHMM approach disclosed herein.
Note that (6) is different from (4) and can lead to a significantly simplified re-estimation formula in the training phase (as described in Section E).
Given K knots {(x(i),yi)|i=1, . . . , K; x(i)<x(i+1(} in the cubic spline, the value of a data point x can be estimated by:




  
    
      
        y
        =
        
          
            
              ay
              
                (
                j
                )
              
            
            +
            
              by
              
                (
                
                  j
                  +
                  1
                
                )
              
            
            +
            
              c
              ⁢
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  y
                
                
                  ∂
                  
                    x
                    2
                  
                
              
            
          
          ⁢
          
            |
            
              x
              =
              
                x
                
                  (
                  j
                  )
                
              
            
          
          ⁢
          
            
              +
              d
            
            ⁢
            
              
                
                  ∂
                  2
                
                ⁢
                y
              
              
                ∂
                
                  x
                  2
                
              
            
          
          ⁢
          
            |
            
              x
              =
              
                x
                
                  (
                  
                    j
                    +
                    1
                  
                  )
                
              
            
          
          .
        
      
    
    
      
        (
        7
        )
      
    
  




Here, a, b, c, and d are interpolation parameters, and [x(j),x(j+1)] refers to a section where the point x falls. The interpolation parameters (a, b, c, d) are given by:





  
    
      
        
          a
          =
          
            
              
                x
                
                  (
                  
                    j
                    +
                    1
                  
                  )
                
              
              -
              x
            
            
              
                x
                
                  (
                  
                    j
                    +
                    1
                  
                  )
                
              
              -
              
                x
                
                  (
                  j
                  )
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          c
          =
          
            
              1
              6
            
            ⁢
            
              (
              
                
                  a
                  3
                
                -
                a
              
              )
            
            ⁢
            
              
                (
                
                  
                    x
                    
                      (
                      
                        j
                        +
                        1
                      
                      )
                    
                  
                  -
                  
                    x
                    
                      (
                      j
                      )
                    
                  
                
                )
              
              2
            
          
        
        ,
        and
      
    
    
      
        (
        8
        )
      
    
  




b=1−a, and d=1⁄6(b3−b)(x(j+1)−x(j))2.   (9)

Assume that that x(j) are evenly distributed, and that the following expression applies:

h=x(j+1)−x(j)=x(k+1)−x(K)>0, ∀j, k ε{1, . . . K−1}  (10)

Further assume that a natural spline is used. In this case, equation (7) can be written as:

y=(ExT+FxTC−1D){tilde over (y)},   (11)

where:

{tilde over (y)}=[y(1). . . y(K)]T.   (12)





  
    
      
        
          
            E
            x
          
          =
          
            
              [
              
                0
                ⁢
                
                  
                
                ⁢
                ...
                ⁢
                
                  
                
                ⁢
                
                  a
                  
                    }
                    j
                  
                
                ⁢
                
                  
                
                ⁢
                
                  b
                  
                    }
                    
                      j
                      +
                      1
                    
                  
                
                ⁢
                
                  
                
                ⁢
                ...
                ⁢
                
                  
                
                ⁢
                0
              
              ]
            
            T
          
        
        ,
      
    
    
      
        (
        13
        )
      
    
  
  
    
      
        
          
            F
            x
          
          =
          
            
              [
              
                0
                ⁢
                
                  
                
                ⁢
                ...
                ⁢
                
                  
                
                ⁢
                
                  c
                  
                    }
                    j
                  
                
                ⁢
                
                  
                
                ⁢
                
                  d
                  
                    }
                    
                      j
                      +
                      1
                    
                  
                
                ⁢
                
                  
                
                ⁢
                ...
                ⁢
                
                  
                
                ⁢
                0
              
              ]
            
            T
          
        
        ,
      
    
    
      
        (
        14
        )
      
    
  








  
    
      
        
          C
          =
          
            
              h
              6
            
            ⁡
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    0
                  
                
                
                  
                    1
                  
                  
                    4
                  
                  
                    1
                  
                  
                    0
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    0
                  
                
                
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                
                
                  
                    ...
                  
                  
                    0
                  
                  
                    1
                  
                  
                    4
                  
                  
                    1
                  
                  
                    0
                  
                  
                    ...
                  
                
                
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                
                
                  
                    0
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    0
                  
                  
                    1
                  
                  
                    4
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        15
        )
      
    
  
  
    
      and
    
    
      
        
      
    
  
  
    
      
        D
        =
        
          
            
              1
              h
            
            ⁡
            
              [
              
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    0
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      2
                    
                  
                  
                    1
                  
                  
                    0
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    0
                  
                
                
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                
                
                  
                    ...
                  
                  
                    0
                  
                  
                    1
                  
                  
                    
                      -
                      2
                    
                  
                  
                    1
                  
                  
                    0
                  
                  
                    ...
                  
                
                
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                
                
                  
                    0
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    0
                  
                  
                    1
                  
                  
                    
                      -
                      2
                    
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    ...
                  
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        16
        )
      
    
  




It follows that:




  
    
      
        
          
            d
            y
          
          
            d
            
              y
              ~
            
          
        
        =
        
          
            
              (
              
                
                  E
                  x
                  T
                
                +
                
                  
                    F
                    x
                    T
                  
                  ⁢
                  
                    C
                    
                      -
                      1
                    
                  
                  ⁢
                  D
                
              
              )
            
            T
          
          .
        
      
    
    
      
        (
        17
        )
      
    
  




Since a, b, c, d are functions of x, Ex and Fx are also functions of x. However, C−1D is independent of x. This allows C−1D to be pre-calculated, stored, and shared across different splines (making this solution computationally efficient). Cubic splines provide several other favorable features. First, for these functions, interpolation is smooth up to the second-order derivative. Second, an interpolation value depends only on the nearby data points (knots). Third, the interpolation accuracy can be improved by increasing the number of knots.
The following expressions are now defined:




  
    
      
        
          
            
              μ
              ~
            
            
              π
              ⁡
              
                (
                
                  i
                  ,
                  l
                  ,
                  d
                
                )
              
            
          
          =
          
            
              [
              
                
                  μ
                  
                    π
                    ⁡
                    
                      (
                      
                        i
                        ,
                        l
                        ,
                        d
                      
                      )
                    
                  
                  
                    (
                    1
                    )
                  
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  μ
                  
                    π
                    ⁡
                    
                      (
                      
                        i
                        ,
                        l
                        ,
                        d
                      
                      )
                    
                  
                  
                    (
                    K
                    )
                  
                
              
              ]
            
            T
          
        
        ,
      
    
    
      
        (
        18
        )
      
    
  
  
    
      
        
          
            
              v
              
                π
                ⁡
                
                  (
                  
                    i
                    ,
                    l
                    ,
                    d
                  
                  )
                
              
              T
            
            ⁡
            
              (
              
                ζ
                
                  r
                  ,
                  t
                  ,
                  d
                
              
              )
            
          
          =
          
            
              E
              
                ζ
                
                  r
                  ,
                  t
                  ,
                  d
                
              
              T
            
            +
            
              
                F
                
                  ζ
                  
                    r
                    ,
                    t
                    ,
                    d
                  
                
                T
              
              ⁢
              
                C
                
                  -
                  1
                
              
              ⁢
              D
            
          
        
        ,
      
    
    
      
        (
        19
        )
      
    
  
  
    
      
        
          
            
              Σ
              ~
            
            
              π
              ⁡
              
                (
                
                  i
                  ,
                  l
                  ,
                  d
                
                )
              
            
          
          =
          
            
              [
              
                
                  Σ
                  
                    π
                    ⁡
                    
                      (
                      
                        i
                        ,
                        l
                        ,
                        d
                      
                      )
                    
                  
                  
                    (
                    1
                    )
                  
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  Σ
                  
                    π
                    ⁡
                    
                      (
                      
                        i
                        ,
                        l
                        ,
                        d
                      
                      )
                    
                  
                  
                    (
                    K
                    )
                  
                
              
              ]
            
            T
          
        
        ,
        
          
        
        ⁢
        and
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          
            ς
            
              π
              ⁡
              
                (
                
                  i
                  ,
                  l
                  ,
                  d
                
                )
              
            
            T
          
          ⁡
          
            (
            
              ζ
              
                r
                ,
                t
                ,
                d
              
            
            )
          
        
        =
        
          
            E
            
              ζ
              
                r
                ,
                t
                ,
                d
              
            
            T
          
          +
          
            
              F
              
                ζ
                
                  r
                  ,
                  t
                  ,
                  d
                
              
              T
            
            ⁢
            
              C
              
                -
                1
              
            
            ⁢
            
              D
              .
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




Using the above expressions, the parametric form of equations (5) and (6) can be rewritten succinctly as:




  
    
      
        
          
            
              μ
              
                i
                ,
                l
                ,
                d
              
            
            ⁡
            
              (
              
                ζ
                
                  r
                  ,
                  t
                  ,
                  d
                
              
              )
            
          
          =
          
            
              μ
              
                i
                ,
                l
                ,
                d
              
              
                (
                0
                )
              
            
            +
            
              
                
                  v
                  
                    π
                    ⁡
                    
                      (
                      
                        i
                        ,
                        l
                        ,
                        d
                      
                      )
                    
                  
                  T
                
                ⁡
                
                  (
                  
                    ζ
                    
                      r
                      ,
                      t
                      ,
                      d
                    
                  
                  )
                
              
              ⁢
              
                
                  μ
                  ~
                
                
                  π
                  ⁡
                  
                    (
                    
                      i
                      ,
                      l
                      ,
                      d
                    
                    )
                  
                
              
            
          
        
        ,
        and
      
    
    
      
        (
        22
        )
      
    
  
  
    
      
        
          
            Σ
            
              i
              ,
              l
              ,
              d
            
          
          ⁡
          
            (
            
              ζ
              
                r
                ,
                t
                ⁢
                
                  
                
                ,
                d
              
            
            )
          
        
        =
        
          
            
              
                Σ
                
                  i
                  ,
                  l
                  ,
                  d
                
                
                  (
                  0
                  )
                
              
              ⁡
              
                (
                
                  
                    
                      ς
                      
                        π
                        ⁡
                        
                          (
                          
                            i
                            ,
                            l
                            ,
                            d
                          
                          )
                        
                      
                      T
                    
                    ⁡
                    
                      (
                      
                        ζ
                        
                          r
                          ,
                          t
                          ,
                          d
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      Σ
                      ~
                    
                    
                      π
                      ⁡
                      
                        (
                        
                          i
                          ,
                          l
                          ,
                          d
                        
                        )
                      
                    
                  
                
                )
              
            
            
              -
              2
            
          
          .
        
      
    
    
      
        (
        23
        )
      
    
  




D. Estimation of Conditioning Parameters
As explained above, different types of environmental conditioning parameters can be used to model the environmental conditions present in the environment. In one illustrative case, the environment-conditioning parameter can represent instantaneous posterior SNR in the cepstral domain, which can be expressed as:




  
    
      
        
          
            
              
                ζ
                d
              
              =
              
              ⁢
              
                
                  ∑
                  i
                
                ⁢
                
                  
                    a
                    
                      d
                      ,
                      i
                    
                  
                  ⁢
                  log
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      σ
                      
                        i
                        ,
                        y
                      
                      2
                    
                    
                      σ
                      
                        i
                        ,
                        n
                      
                      2
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∑
                  i
                
                ⁢
                
                  
                    
                      a
                      
                        d
                        ,
                        i
                      
                    
                    ⁡
                    
                      (
                      
                        
                          log
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            
                              i
                              ,
                              y
                            
                            2
                          
                        
                        -
                        
                          log
                          ⁢
                          
                            
                          
                          ⁢
                          
                            σ
                            
                              i
                              ,
                              n
                            
                            2
                          
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




Here, ad,i is the inverse discrete cosine transformation (IDCT) coefficient, σi,y2 is the power of noisy speech signal y associated with the i-th Mel-frequency filter, and σi,n2 is the power of a noise component n of the noisy speech signal y associated with the i-th Mel-frequency filter. The noise power σi,n2 can be estimated using a minimum-controlled recursive moving-average noise tracker. One such noise-tracker that can be used is described in Israel Cohen, “Noise Estimation by Minima Controlled Recursive Averaging for Robust Speech Enhancement,” IEEE Signal Processing Letters, Vol. 9, No. 1, January 2002.

FIG. 5 graphically illustrates how measurements of the environmental conditioning parameter (such as, but not limited to, instantaneous SNR) can be used to define a range of interpolation associated with a spline function. As indicated there, the distribution associated with the environmental conditioning parameter can be assumed to be Gaussian. The Gaussian distribution is characterized by a mean (μζd) and a standard deviation (σζdd), which can be estimated from the training data. The range of interpolation can be defined by identifying a portion of this Gaussian distribution. For example, the range of interpolation can be selected to extend from a μζd−ασζd position in the distribution to a μζd+ασζd in the distribution, where, in one merely illustrative case, α is 2. In other words, μζd−ασζd defines a first knot in the spline function, while μζd+ασζd defines a last knot in the spline function. Although not shown, it is possible to extrapolate beyond the range of measured data using the spline-based fitting approach.
E. Discriminative Training for a VPHMM-Based Recognition System
FIG. 6 shows a procedure 600 that sets forth operations in an illustrative training operation performed by the training module 108 of FIG. 1. To review, the general goal of the training operation is to derive parameters which define the VPHMMs. The training operation shown in FIG. 6 can be generally characterized as a growth-transformation (GT) based discriminative training algorithm. The training operation operates in an efficient and powerful way that is made possible through the manner in which the Gaussian parameters are expressed (as set forth above). For instance, the training operation can dispense with the use of quantization performed by previous solutions.
As stated above, the training operation takes place in the context of an implementation that makes use of shared spline parameters, and thus, which makes reference to regression classes ω. However, other implementations can omit the use of shared spline parameters.
Block 602 involves training a conventional HMM using any training technique.
Block 604 entails initializing the training operation by copying μi,l,d(0) and Σi,l,d(0) from the conventional HMM, μ ω(i,l,d)(1), . . . , μ ω(i,l,d)(K) to zero, and setting Σ ω(i,l,d)(1), . . . , Σ ω(i,l,d)(K) to one.
Block 606 entails performing the discriminative training operation. In each discriminative training iteration, block 606 entails first re-estimating μ ω(i,l,d)(1), . . . , μ ω(i,l,d)(K), then Σ ω(i,l,d)(1), . . . , Σ ω(i,l,d)(K), and finally μi,l,d(0) and Σi,l,d(0) with the rest of the parameters fixed. Illustrative equations to compute these parameters are described below. (In the following equations, the superscript ‘refers to a current parameter set. The algorithm operates by iteratively estimating a new parameter set based on the current estimated parameter set.)
Block 608 entails storing the final parameters which define the VPHMMs.
Before presenting the training algorithm perform in block 606, the following expression is defined:




  
    
      
        
          d
          ⁡
          
            (
            
              r
              ,
              t
              ,
              i
              ,
              l
            
            )
          
        
        =
        
          
            ∑
            s
          
          ⁢
          
            
              
                d
                ′
              
              ⁡
              
                (
                s
                )
              
            
            ⁢
            
              
                p
                ⁡
                
                  (
                  
                    
                      
                        q
                        
                          r
                          ,
                          t
                        
                      
                      =
                      i
                    
                    ,
                    
                      l
                      |
                      s
                    
                    ,
                    
                      Λ
                      ′
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




Here, Λ′ is the current parameter set, qr,t is the state at time t in the r-th utterance, s is the label sequence, and d′(s) is a parameter that controls the convergence speed. The occupation probability of Gaussian mixture component l of state i, at time t in the r-th utterance can be denoted as:

γi,l,r,sr(t)=p(qr,t=i,l|Xr,sr, Λ′),   (26)

which can be obtained through an efficient forward-backward algorithm. Further, the following expression is defined:





  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            γ
            ⁡
            
              (
              
                i
                ,
                l
                ,
                r
                ,
                t
              
              )
            
          
        
        =
        
          
            ∑
            s
          
          ⁢
          
            
              p
              ⁡
              
                (
                
                  
                    s
                    |
                    X
                  
                  ,
                  
                    Λ
                    ′
                  
                
                )
              
            
            ⁢
            
              (
              
                
                  C
                  ⁡
                  
                    (
                    s
                    )
                  
                
                -
                
                  O
                  ⁡
                  
                    (
                    
                      Λ
                      ′
                    
                    )
                  
                
              
              )
            
            ⁢
            
              
                
                  γ
                  
                    i
                    ,
                    l
                    ,
                    r
                    ,
                    
                      s
                      r
                    
                  
                
                ⁡
                
                  (
                  t
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




Here, O(Λ′) is the discriminative training criterion and C(s) is the weighting factor. For conciseness in presenting the re-estimation formulas, where appropriate, the remainder of the this section simplifies the notation of ζr,t,d as ζ, ω(i,l,d) as ω, xr,t,d as x, μi,l,d as μ, Σi,l,d as Σ, Δγ(i,l,r,t) as Δ, and d(r,t,i,l) as d.

The follow illustrative formulas can be used in the re-estimation of μi,l,d(0) and Σi,l,d(0):




  
    
      
        
          μ
          
            i
            ,
            l
            ,
            d
          
          
            (
            0
            )
          
        
        =
        
          
            
              Z
              ⁢
              
                
              
              ⁢
              1
            
            
              Z
              ⁢
              
                
              
              ⁢
              2
            
          
          +
          
            
              μ
              
                i
                ,
                l
                ,
                d
              
              
                ′
                ⁡
                
                  (
                  0
                  )
                
              
            
            .
          
        
      
    
    
      
        (
        28
        )
      
    
  
  
    
      
        Here
        ,
      
    
    
      
        
      
    
  
  
    
      
        
          
            Z
            ⁢
            
              
            
            ⁢
            1
          
          =
          
            
              ∑
              r
            
            ⁢
            
              
                ∑
                t
              
              ⁢
              
                
                  Δγ
                  ⁡
                  
                    (
                    
                      x
                      -
                      
                        
                          μ
                          π
                          ′
                        
                        ⁡
                        
                          (
                          ζ
                          )
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    Σ
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    ζ
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        29
        )
      
    
  
  
    
      and
    
    
      
        
      
    
  
  
    
      
        
          Z
          ⁢
          
            
          
          ⁢
          2
        
        =
        
          
            ∑
            r
          
          ⁢
          
            
              ∑
              t
            
            ⁢
            
              
                (
                
                  Δγ
                  +
                  d
                
                )
              
              ⁢
              
                
                  
                    Σ
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    ζ
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  








  
    
      
        Further
        ,
      
    
    
      
        
      
    
  
  
    
      
        
          Σ
          
            i
            ,
            l
            ,
            d
          
          
            (
            0
            )
          
        
        =
        
          
            Σ
            
              i
              ,
              l
              ,
              d
            
            
              ′
              ⁡
              
                (
                0
                )
              
            
          
          ⁢
          
            
              
                Z
                ⁢
                
                  
                
                ⁢
                3
              
              
                Z
                ⁢
                
                  
                
                ⁢
                4
              
            
            .
          
        
      
    
    
      
        (
        31
        )
      
    
  
  
    
      
        Here
        ,
      
    
    
      
        
      
    
  
  
    
      
        
          
            Z
            ⁢
            
              
            
            ⁢
            3
          
          =
          
            
              ∑
              r
            
            ⁢
            
              
                ∑
                t
              
              ⁢
              
                (
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        γ
                        ⁡
                        
                          (
                          
                            x
                            -
                            
                              μ
                              ⁡
                              
                                (
                                ζ
                                )
                              
                            
                          
                          )
                        
                      
                      2
                    
                    ⁢
                    
                      
                        Σ
                        
                          ′
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        ζ
                        )
                      
                    
                  
                  +
                  d
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        32
        )
      
    
  
  
    
      and
    
    
      
        
      
    
  
  
    
      
        
          Z
          ⁢
          
            
          
          ⁢
          4
        
        =
        
          
            
              ∑
              r
            
            ⁢
            
              
                ∑
                t
              
              ⁢
              Δγ
            
          
          +
          
            d
            .
          
        
      
    
    
      
        (
        33
        )
      
    
  




The following illustrative formulas can be used in the re-estimation of μ ω(i,l,d):

μ ω(i,l,d)A ω−1B ω(i,l,d).   (34)

In expression (34), A ω(i,l,d) is matrix whose element at the k-th row and the j-th column is:




  
    
      
        
          A
          
            π
            ⁡
            
              (
              
                i
                ,
                l
                ,
                d
              
              )
            
          
          
            (
            
              k
              ,
              j
            
            )
          
        
        =
        
          
            ∑
            r
          
          ⁢
          
            
              ∑
              t
            
            ⁢
            
              
                ∑
                
                  
                    
                      i
                      ′
                    
                    ,
                    
                      l
                      ′
                    
                    ,
                    
                      
                    
                    ⁢
                    
                      st
                      .
                    
                  
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            i
                            ′
                          
                          ,
                          
                            l
                            ′
                          
                          ,
                          d
                        
                        )
                      
                    
                    =
                    
                      π
                      ⁡
                      
                        (
                        
                          i
                          ,
                          l
                          ,
                          d
                        
                        )
                      
                    
                  
                
              
              ⁢
              
                
                  (
                  
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      γ
                    
                    +
                    d
                  
                  )
                
                ⁢
                
                  
                    Σ
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    ζ
                    )
                  
                
                ⁢
                
                  
                    v
                    π
                    
                      (
                      k
                      )
                    
                  
                  ⁡
                  
                    (
                    ζ
                    )
                  
                
                ⁢
                
                  
                    
                      v
                      π
                      
                        (
                        j
                        )
                      
                    
                    ⁡
                    
                      (
                      ζ
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




B ω(i,l,d) is a vector whose k-th value is:




  
    
      
        
          B
          
            π
            ⁡
            
              (
              
                i
                ,
                l
                ,
                d
              
              )
            
          
          
            (
            k
            )
          
        
        =
        
          
            
              ∑
              r
            
            ⁢
            
              
                ∑
                t
              
              ⁢
              
                
                  ∑
                  
                    
                      
                        i
                        ′
                      
                      ,
                      
                        l
                        ′
                      
                      ,
                      
                        
                      
                      ⁢
                      
                        st
                        .
                      
                    
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              i
                              ′
                            
                            ,
                            
                              l
                              ′
                            
                            ,
                            d
                          
                          )
                        
                      
                      =
                      
                        π
                        ⁡
                        
                          (
                          
                            i
                            ,
                            l
                            ,
                            d
                          
                          )
                        
                      
                    
                  
                
                ⁢
                
                  
                    
                      ΔγΣ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      ζ
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      x
                      -
                      
                        μ
                        
                          
                            i
                            ′
                          
                          ,
                          
                            l
                            ′
                          
                          ,
                          d
                        
                        
                          (
                          0
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      v
                      π
                      
                        (
                        k
                        )
                      
                    
                    ⁡
                    
                      (
                      ζ
                      )
                    
                  
                
              
            
          
          +
          
            
              ∑
              r
            
            ⁢
            
              
                ∑
                t
              
              ⁢
              
                
                  ∑
                  
                    
                      
                        i
                        ′
                      
                      ,
                      
                        l
                        ′
                      
                      ,
                      
                        
                      
                      ⁢
                      
                        st
                        .
                      
                    
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            
                              i
                              ′
                            
                            ,
                            
                              
                                l
                                ′
                              
                              ⁢
                              d
                            
                          
                          )
                        
                      
                      =
                      
                        π
                        ⁡
                        
                          (
                          
                            i
                            ,
                            l
                            ,
                            d
                          
                          )
                        
                      
                    
                  
                
                ⁢
                
                  d
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      Σ
                      
                        -
                        1
                      
                    
                    ⁡
                    
                      (
                      ζ
                      )
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          μ
                          ′
                        
                        ⁡
                        
                          (
                          ζ
                          )
                        
                      
                      -
                      
                        μ
                        
                          
                            i
                            ′
                          
                          ,
                          
                            l
                            ′
                          
                          ,
                          d
                        
                        
                          (
                          0
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      v
                      π
                      
                        (
                        k
                        )
                      
                    
                    ⁡
                    
                      (
                      ζ
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




The following illustrative formulas can be used in the re-estimation of {tilde over (Σ)} ω(i,l,d):

{tilde over (Σ)} ω(i,l,d)={tilde over (Σ)} ω(i,l,d)−(F ω(i,l,d))−1E ω(i,l,d)   (37)

Here, F ω(i,l,d) is a vector whose k-th row and j-th column is:




  
    
      
        
          F
          
            π
            ⁡
            
              (
              
                i
                ,
                l
                ,
                d
              
              )
            
          
          
            (
            
              k
              ,
              j
            
            )
          
        
        =
        
          
            ∑
            r
          
          ⁢
          
            
              ∑
              t
            
            ⁢
            
              
                ∑
                
                  
                    
                      i
                      ′
                    
                    ,
                    
                      l
                      ′
                    
                    ,
                    
                      
                    
                    ⁢
                    
                      st
                      .
                    
                  
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            i
                            ′
                          
                          ,
                          
                            l
                            ′
                          
                          ,
                          d
                        
                        )
                      
                    
                    =
                    
                      π
                      ⁡
                      
                        (
                        
                          i
                          ,
                          l
                          ,
                          d
                        
                        )
                      
                    
                  
                
              
              ⁢
              
                
                  [
                  
                    
                  
                  ⁢
                  
                    
                      
                        
                          
                            Δ
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                γ
                                ⁡
                                
                                  (
                                  
                                    Σ
                                    
                                      i
                                      ,
                                      l
                                      ,
                                      d
                                    
                                    
                                      (
                                      0
                                      )
                                    
                                  
                                  )
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  
                                    (
                                    
                                      x
                                      -
                                      
                                        μ
                                        ⁡
                                        
                                          (
                                          ζ
                                          )
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                                +
                                
                                  
                                    Σ
                                    π
                                    ′
                                  
                                  ⁡
                                  
                                    (
                                    ζ
                                    )
                                  
                                
                              
                              )
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          2
                          ⁢
                          
                            
                              d
                              ⁡
                              
                                (
                                
                                  Σ
                                  
                                    i
                                    ,
                                    l
                                    ,
                                    d
                                  
                                  
                                    (
                                    0
                                    )
                                  
                                
                                )
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            Σ
                            ′
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                    ς
                    π
                    
                      (
                      k
                      )
                    
                  
                  ⁡
                  
                    (
                    ζ
                    )
                  
                
                ⁢
                
                  
                    
                      ς
                      π
                      
                        (
                        j
                        )
                      
                    
                    ⁡
                    
                      (
                      ζ
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




Further, E ω(i,l,d) is a vector whose k-th value is:




  
    
      
        
          E
          
            π
            ⁡
            
              (
              
                i
                ,
                l
                ,
                d
              
              )
            
          
          
            (
            k
            )
          
        
        =
        
          
            ∑
            r
          
          ⁢
          
            
              ∑
              t
            
            ⁢
            
              
                ∑
                
                  
                    
                      i
                      ′
                    
                    ,
                    
                      l
                      ′
                    
                    ,
                    
                      
                    
                    ⁢
                    
                      st
                      .
                    
                  
                  
                    
                      π
                      ⁡
                      
                        (
                        
                          
                            i
                            ′
                          
                          ,
                          
                            l
                            ′
                          
                          ,
                          d
                        
                        )
                      
                    
                    =
                    
                      π
                      ⁡
                      
                        (
                        
                          i
                          ,
                          l
                          ,
                          d
                        
                        )
                      
                    
                  
                
              
              ⁢
              
                
                  
                    Δγ
                    ⁡
                    
                      (
                      
                        Σ
                        
                          i
                          ,
                          l
                          ,
                          d
                        
                        
                          (
                          0
                          )
                        
                      
                      )
                    
                  
                  
                    
                      -
                      1
                    
                    /
                    2
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    Σ
                    
                      ′
                      -
                      
                        1
                        2
                      
                    
                  
                  ⁡
                  
                    (
                    ζ
                    )
                  
                
                ⁢
                
                  (
                  
                    
                      
                        (
                        
                          x
                          -
                          
                            μ
                            ⁡
                            
                              (
                              ζ
                              )
                            
                          
                        
                        )
                      
                      2
                    
                    -
                    
                      
                        Σ
                        ′
                      
                      ⁡
                      
                        (
                        ζ
                        )
                      
                    
                  
                  )
                
                ⁢
                
                  
                    
                      ς
                      π
                      
                        (
                        k
                        )
                      
                    
                    ⁡
                    
                      (
                      ζ
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        39
        )
      
    
  




F. Application of Clustering and Sharing
Each of the spline functions described above is characterized by spline parameters. That is, a set of spline parameters describes the shape of each spline function. This section describes one approach that can be used to cluster the spline functions into different classes, referred to as regression classes ω. In effect, the classes pertain to groups of spline functions that are similar to each other. Once the classes are formed, individual instances of the spline functions can make reference to their associated classes. This approach makes it possible for each individual spline functions to omit a “local copy” of the spline function's parameters; instead, the individual spline function can obtain the relevant spline parameters from a shared store of spline function parameters.
There are at least two advantages to the clustering approach. First, the use of a common store of shared spline parameters can reduce the complexity and storage requirements of the recognition system 100. More specifically, parameter sharing can be carried out so that the otherwise very large number of parameters in the VPHMMs can be effectively reduced with practically feasible amounts of training data. Second, each class of spline functions can provide a set of parameters that is potentially robust; this is because the class reflects a consensus among different spline functions which were used to define the class. Clustering cannot readily be performed on previous approaches that use global (non-piecewise) polynomial functions to approximate the Gaussian parameters because it is more difficult to find commonality among global polynomial functions. In other words, global polynomial functions are more unique and parameter-sensitive than individual spline segments; this is because a relatively small change in the parameters of a global polynomial function can affect the shape of the polynomial function in a relatively significant way.
As a point of clarification, the clustering described below pertains to the grouping of spline parameters. It is also possible to cluster the Gaussian-specific parameters. Known algorithms can be used to perform the clustering of Gaussian-specific parameters.
FIG. 7 is a procedure 700 which presents an overview of the clustering operation. This procedure 700 can be implemented as part of the training operation performed by the recognition system 100.
In block 702, the recognition system 100 can perform training without the application of shared spline functions. This operation generates a plurality of individual spline functions which are used to approximate the way in which the Gaussian parameters change as a function of the environmental conditioning parameter.
In block 704, the recognition system 100 can perform clustering to organize the spline functions into classes. This approach involves defining distances between respective pairs of spline functions and then clustering the spline functions on the basis of the distances. That is, the clustering generally entails defining groups of clusters that have spline functions which are close together (as defined by the distances). One clustering algorithm that can be used to perform clustering is the k means clustering algorithm. This algorithm can generate N clusters (where N is a number of clusters defined by a user in advance). If one of the clusters that is formed is empty (meaning that it contains no spline functions), a non-empty cluster can be split into two to produce the selected number of N clusters.
Blocks 706 and 708 entail, respectively, initializing the VPHMMs with the classes defined by the clustering operation (in block 704) and then retraining the VPHMMs to derive a final set of VPHMM parameters.
The following discussion provides additional details regarding the computation of distances between pairs of spline functions according to one illustrative implementation.
To begin with, the distance between two functions ƒ1 and ƒ2 given the distribution of the domain p(x) can be defined as:

d(ƒ1, ƒ2)=∫x(ƒ1(x)−ƒ2(x))2p(x)dx.   (40)

This distance is also valid for two spline functions determined by evenly distributed knots:

{y1(i)|i=1, . . . , K}, and  (41)

{y2(i)|i=1, . . . , K}.   (42)

Note that the calculation of the exact distance using equation (40) can be time-consuming. For this reason, the integration defined by equation (40) can be approximated with quantized summations:




  
    
      
        
          d
          ⁡
          
            (
            
              
                f
                1
              
              ,
              
                f
                2
              
            
            )
          
        
        =
        
          h
          ⁢
          
            
              ∑
              
                x
                =
                1
              
              K
            
            ⁢
            
              
                
                  (
                  
                    
                      y
                      1
                      
                        (
                        i
                        )
                      
                    
                    -
                    
                      y
                      2
                      
                        (
                        i
                        )
                      
                    
                  
                  )
                
                2
              
              ⁢
              
                
                  p
                  ⁡
                  
                    (
                    
                      x
                      
                        (
                        i
                        )
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




FIG. 8 graphically represents the calculation of the distance between two functions (802, 804) using the above approach. The quantized summations shown in this figure can be viewed as a histogram-based approximation of the difference between the two functions (802, 804).
Continuing with the derivation, it can be assumed that p(x)=N(x; μ,σ2) follows a Gaussian distribution determined by the mean μ and the variance σ2. As such, the above equation can be rewritten as:




  
    
      
        
          d
          ⁡
          
            (
            
              
                f
                1
              
              ,
              
                f
                2
              
            
            )
          
        
        =
        
          h
          ⁢
          
            
              ∑
              
                x
                =
                1
              
              K
            
            ⁢
            
              
                
                  (
                  
                    
                      y
                      1
                      
                        (
                        i
                        )
                      
                    
                    -
                    
                      y
                      2
                      
                        (
                        i
                        )
                      
                    
                  
                  )
                
                2
              
              ⁢
              
                1
                
                  σ
                  ⁢
                  
                    
                      2
                      ⁢
                      π
                    
                  
                
              
              ⁢
              
                
                  exp
                  (
                  
                    -
                    
                      
                        
                          (
                          
                            
                              x
                              
                                (
                                i
                                )
                              
                            
                            -
                            μ
                          
                          )
                        
                        2
                      
                      
                        2
                        ⁢
                        
                          σ
                          2
                        
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    
      
        (
        44
        )
      
    
  




Note that parameters h, μ, and σ are the same for all the splines to be clustered, and:




  
    
      
        
          h
          =
          
            
              2
              ⁢
              ασ
            
            
              K
              -
              1
            
          
        
        ,
      
    
    
      
        (
        45
        )
      
    
  
  
    
      
        
          
            x
            
              (
              i
              )
            
          
          =
          
            
              x
              
                (
                1
                )
              
            
            +
            
              
                (
                
                  i
                  -
                  1
                
                )
              
              ⁢
              h
            
          
        
        ,
        and
      
    
    
      
        (
        46
        )
      
    
  
  
    
      
        μ
        =
        
          
            x
            
              (
              1
              )
            
          
          +
          
            
              
                K
                -
                1
              
              2
            
            ⁢
            
              h
              .
            
          
        
      
    
    
      
        (
        47
        )
      
    
  




The distance calculation can thus be further simplified as:




  
    
      
        
          
            
              
                
                  
                    d
                    ⁡
                    
                      (
                      
                        
                          f
                          1
                        
                        ,
                        
                          f
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                  ∝
                  
                    
                      ∑
                      
                        x
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      
                        
                          (
                          
                            
                              y
                              1
                              
                                (
                                i
                                )
                              
                            
                            -
                            
                              y
                              2
                              
                                (
                                i
                                )
                              
                            
                          
                          )
                        
                        2
                      
                      ⁢
                      
                        exp
                        (
                        
                          -
                          
                            
                              
                                (
                                
                                  
                                    x
                                    
                                      (
                                      i
                                      )
                                    
                                  
                                  -
                                  μ
                                
                                )
                              
                              2
                            
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                        )
                      
                    
                  
                
                ,
              
              ⁢
              
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  =
                  
                    
                      ∑
                      
                        x
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      
                        
                          (
                          
                            
                              y
                              1
                              
                                (
                                i
                                )
                              
                            
                            -
                            
                              y
                              2
                              
                                (
                                i
                                )
                              
                            
                          
                          )
                        
                        2
                      
                      ⁢
                      
                        exp
                        (
                        
                          -
                          
                            
                              
                                
                                  h
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    i
                                    -
                                    1
                                    -
                                    
                                      
                                        K
                                        -
                                        1
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                              2
                            
                            
                              2
                              ⁢
                              
                                σ
                                2
                              
                            
                          
                        
                        )
                      
                    
                  
                
                ,
              
              ⁢
              
                
              
            
          
        
        
          
            
              
              ⁢
              
                =
                
                  
                    ∑
                    
                      x
                      =
                      1
                    
                    K
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            y
                            1
                            
                              (
                              i
                              )
                            
                          
                          -
                          
                            y
                            2
                            
                              (
                              i
                              )
                            
                          
                        
                        )
                      
                      2
                    
                    ⁢
                    
                      
                        exp
                        (
                        
                          -
                          
                            
                              2
                              ⁢
                              
                                
                                  
                                    α
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      i
                                      -
                                      
                                        
                                          K
                                          +
                                          1
                                        
                                        2
                                      
                                    
                                    )
                                  
                                
                                2
                              
                            
                            
                              
                                (
                                
                                  K
                                  -
                                  1
                                
                                )
                              
                              2
                            
                          
                        
                        )
                      
                      .
                    
                  
                
              
            
          
        
      
    
    
      
        (
        48
        )
      
    
  




The parameters associated with a class (or with an individual instance of a spline function if sharing is not used) can be represented in an economical form. More specifically, note that a K-knot cubic spline uses 2K parameters, that is, K parameters for x(i) and another K parameters for y(i). The number of parameters can be greatly reduced by using a spline function with evenly distributed x(j), as set forth in equation (10).
Based on these assumption, a spline function can be represented by storing only y(i)and {K,x(1),x(K)}. Further note that only one {K,x(1),x(K)} is needed for each dimension of the conditioning parameter. Thus, the average number of parameters needed for each spline function is close to K.
FIG. 9 shows how the classes formed by clustering can be used to simplify the expression of spline functions. That figure shows a data store 902 that provide sets of parameters associated with respective classes. A mapping module 904 can receive reference information (e.g., ω(i,l,d) ) which defines a class of spline functions. The mapping module 904 can map the reference information to the parameters associated with that class, as obtained from the data store 902. This aspect of the recognition system 100 omits the need for an instance of the spline function to maintain a local copy of the spline parameters which define the spline function.
G. Representative Processing Functionality
FIG. 10 sets forth illustrative electrical data processing functionality 1000 (simply “processing functionality” below) that can be used to implement any aspect of the functions described above. With reference to FIGS. 1 and 2, for instance, the type of processing functionality 1000 shown in FIG. 10 can be used to implement any aspect of the recognition system 100. In one case, the processing functionality 1000 may correspond to any type of computing device.
The processing functionality 1000 can include volatile and non-volatile memory, such as RAM 1002 and ROM 1004, as well as one or more processing devices 1006. The processing functionality 1000 also optionally includes various media devices 1008, such as a hard disk module, an optical disk module, and so forth. The processing functionality 1000 can perform various operations identified above when the processing device(s) 1006 executes instructions that are maintained by memory (e.g., RAM 1002, ROM 1004, or elsewhere). More generally, instructions and other information can be stored on any computer-readable medium 1010, including, but not limited to, static memory storage devices, magnetic storage devices, optical storage devices, and so on. The term “computer-readable medium also encompasses plural storage devices. The term computer-readable medium also encompasses signals transmitted from a first location to a second location, e.g., via wire, cable, wireless transmission, etc.
The processing functionality 1000 also includes an input/output module 1012 for receiving various inputs from a user (via input modules 1014), and for providing various outputs to the user (via output modules). One particular output mechanism may include a presentation module 1016 and an associated graphical user interface (GUI) 1018. The processing functionality 1000 can also include one or more network interfaces 1020 for exchanging data with other devices via one or more communication conduits 1022. One or more communication buses 1024 communicatively couple the above-described components together.
In closing, the description may have described various concepts in the context of illustrative challenges or problems. This manner of explication does not constitute an admission that others have appreciated and/or articulated the challenges or problems in the manner specified herein.
More generally, although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as example forms of implementing the claims.

## Claims

1. A speech recognition system, comprising:
a speech recognition module for using variable-parameter hidden Markov models (VPHMMs) to recognize speech based on extracted speech features and at least one environmental conditioning parameter; and
a store in computer memory for storing at least one shared set of parameters that represents a class of piecewise functions for use by the VPHMMs,
each of the VPHMMs using a plurality of mixture components L in modeling state observations, and
a particular mixture component l of the plurality of mixture components L having at least one mixture component parameter that varies as a function of said at least one environmental conditioning parameter, as approximated by using a piecewise function that includes a plurality of connected segments,
the particular mixture component l having a mean parameter μ and a variance parameter Σ that are functions of said at least one environmental conditioning parameter ζ, as represented by:




  
    
      
        
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      =
      
        
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                  i
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    ,
    and
  




  
    
      
        
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where i identifies a state, d identifies a dimension, ζ refers to said at least one environmental conditioning parameter, r refers to utterance, t refers to frame, ξ refers to a spline function which serves as the piecewise function, μi,l,d(0) and Σi,l,d(0) are Gaussian-component-specific mean and variance parts, and μi,l,d(1). . . , μi,l,d(K) and Σi,l,d(1). . . , Σi,l,d(K) are spline knots that are shareable across different mixture components,
the particular mixture component l making reference to said at least one shared set of parameters in the store to constitute the piecewise function.

2. The speech recognition system of claim 1, wherein said at least one environmental conditioning parameter is based on one or more of:
a signal-to-noise-ratio (SNR);
a power of noise in the environment;
a level of noise in the environment;
an accent of a speaker;
a specific style of speaking by the speaker, or
a rate of speech.

3. The speech recognition system of claim 1, wherein the class of piecewise functions corresponds to a class of spline functions.

4. The speech recognition system of claim 3, wherein the spline functions are cubic spline functions.

5. The speech recognition system of claim 1, wherein the piecewise function for the particular mixture component l is represented by: K, x(1),x(K), and yi, where K is a number of knots in the piecewise function, x(1) and x(K) are first and last x-dimension positions of the knots in the piecewise function, and yi are y-dimension positions of the knots in the piecewise function.

6. The speech recognition system of claim 1, wherein the piecewise function for the particular mixture component l is represented in a form in which a Gaussian-specific part is separated from a shared part.

7. The speech recognition system of claim 1, wherein the VPHMMs are produced by discriminative training.

8. One or more computer storage devices for storing computer-executable instructions, which when executed by one or more processing devices perform steps, comprising:
using variable-parameter hidden Markov models (VPHMMs) to recognize speech based on extracted speech features and at least one environmental conditioning parameter; and
using a plurality of Gaussian mixture components L in modeling state observations, a particular mixture component l of the plurality of Gaussian mixture components L having a mean parameter μ and a variance parameter Σ that each vary as a function of the at least one environmental conditioning parameter, as approximated by a piecewise function that includes a plurality of connected segments,
each of the mean parameter μ and the variance parameter Σ being represented by




  
    
      
        
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      =
      
        
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        +
        
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where i identifies a state, d identifies a dimension, ζ refers to said at least one environmental conditioning parameter, r refers to utterance, t refers to frame, ξ refers to a spline function which serves as the piecewise function, μi,l,d(0) and Σi,l,d(0) are Gaussian-component-specific mean and variance parts of the particular mixture component l, and μi,l,d(1). . . , μi,l,d(K) and Σi,l,d(1). . . , Σi,l,d(K) are spline knots for the particular mixture component l that are shareable across different mixture components.

9. The one or more computer storage devices of claim 8, wherein the piecewise function is a spline function, and the plurality of connected segments correspond to a plurality of spline segments.

10. The one or more computer storage devices of claim 9, wherein the spline function is a cubic spline function.

11. The one or more computer storage devices of claim 9, wherein the spline function is represented in a store as: K, x(1), x(K), and y(i), where K is a number of knots in the spline function, x(1) and x(K) are first and last x-dimension positions of the knots in the spline function, respectively, and yi are y-dimension positions of the knots in the spline function.

12. The one or more computer storage devices of claim 9, wherein the spline function references a set of spline parameters that are shared by plural spline functions in a class.

13. A method for recognizing speech comprising:
receiving a speech signal comprising informational content and noise;
extracting features associated with the informational content of the speech signal;
determining at least one environmental conditioning parameter that impacts speech recognition; and
determining, by a processor, the informational content using a plurality of Gaussian mixture components L in modeling state observations and a particular mixture component l of the plurality of mixture components L having a mean parameter μ and a variance parameter Σ represented by:




  
    
      
        
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where i identifies a state, d identifies a dimension, ζ refers to said at least one environmental conditioning parameter, r refers to utterance, t refers to frame, ξ refers to a spline function which serves as a piecewise function for the particular mixture component l, μi,l,d(0) and Σi,l,d(0) are Gaussian-component-specific mean and variance parts of the particular mixture component l, and μi,l,d(1). . . , μi,l,d(K) and Σi,l,d(1). . . , Σi,l,d(K) are spline knots for particular mixture component l that are shareable across different mixture components.

14. The method of claim 13, wherein the environmental conditioning parameter is based on one or more of:
a signal-to-noise-ratio (SNR);
a power of noise in the environment;
a level of noise in the environment;
an accent of a speaker;
a specific style of speaking by the speaker, or
a rate of speech.

15. The method of claim 13, wherein determining the informational content uses variable-parameter hidden Markov models (VPHMMs) produced by discriminative training.

16. The method of claim 15, wherein the variable hidden Markov models (VPHMMs) produced by the discriminative training comprising clustering spline functions into a plurality of classes.

17. The method of claim 13, wherein the piecewise function for the particular mixture component l is represented by: K, x(1), x(K), and yi, where K is a number of knots in the piecewise function, x(1) and X(K) are first and last x-dimension positions of the knots in the piecewise function, and yi are y-dimension positions of the knots in the piecewise function.

18. The method of claim 13, wherein the spline function is a cubic spline function.

19. The method of claim 13, wherein the spline function references a set of spline parameters that are shared by plural spline functions in a class.