Feature Space Gaussianization

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Feature Space Gaussianization

• George Saon, Satya Dharanipragada and Dan Povey
• IBM T.J. Watson Rearch Center, Yorktown Heights,
  NY,10598
• ICASSP 2004
• ?? 2004/06/18

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Gaussianization Transformation

• Definition
• X, Y are both random variables, Y T(X)
• If random variable Y T(X) is normally
  distributed, then T is so-called Gaussianization
  Transformation
• Problem
• In general, its difficult to find the joint
  transformation
• Makes the simplifying assumption that the
  dimensions are statistically independent

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Gaussianization Transformation

• Problem
• This can be recast as finding n independent
  mappings
• For clarity
• we will drop the superscripts, and deal with only
  1-D
• Notation

4
Gaussianization Transformation

• Correspondingly, let Fx be the CDF of X
• The aim
• Finding such a transformation which minimize the
  KL divergence between

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Gaussianization Transformation

• Then px and py are related through the following
  equation
• The second part represents the absolute value of
  the Jacobian of the transformation
• the determinant of a matrix of derivatives
• For 1-D, this is simply the derivative

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Jacobian
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Gaussianization Transformation

• Its known that the divergence is minimized when
  the two distributions are point-wise the same
• In order to find , it need to solve the
  differential
• while applying the rule ,and assume
  that

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Gaussianization Transformation

• Now, we can get
• Since the CDF of X is not available, we can
  approximate it with the following

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Gaussianization Transformation

• The final form of Gaussianization transform is

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Gaussianization Transformation
11
Histogram Normalization

• For each feature space dimension (training set)
• For each condition and each
  feature space dimension (testing set)

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Histogram Normalization
13
HTN V.S GT

• ??
• Both are non-linear transformation
• ??
• PDF, integration
• ??
• Cost