Discovering Auditory Objects Through NonNegativity Constraint

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Discovering Auditory Objects Through
Non-Negativity Constraint
Paris Smaragdis SAPA 2004

• Minje Kim
• Intelligent Multimedia Lab.
• CSE, POSTECH
• Nov. 11th

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Outline

• Concepts of NMF
• Static-spectrum Object (Using NMF)
• Concepts of NMD (Non-negative Matrix
  Deconvolution)
• Time-varying Object (Using NMD)

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Concepts of NMF
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Non-negative Matrix Factorization

• Psychological and physiological evidence for
  parts-based representations in the brain.
• When NMF was first introduced it was used to
  learn parts of faces.
• It has been adopted as a very useful technique
  for linear decomposition and dimensionality
  reduction of non-negative data sets.
• Given a non-negative matrix
  , the goal is to approximate it as a product of
  two non-negative matrices and
  where , such that we
  minimize the reconstruction error of by
• Distinguished from other methods by its use of
  non-negativity constraints.

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NMF vs PCA

• V(, n) WúH(, n)

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Cost Functions Update Rules

• Euclidean distance
• Kullback-Leibler divergence

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Static-spectrum Object (Using NMF)
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Experiments (Wedding March)
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Experiments (Drums)
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Experiments (Drums)
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Concepts of NMD (Non-negative Matrix
Deconvolution)
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Non-negative Matrix Deconvolution

• NMD allows us to deal with objects that have time
  varying spectra.
• Sequence of successive spectra and its
  corresponding energy across time.
  Where
• operator is a shift operator that moves the
  columns of its argument by i spots to the right.

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Cost Function Update Rule

• We can use the already existing framework of
  NMF. Where
• Update rule
  andfirst update all Wt then update H using the
  average result of its updates from all Wt.
• If T1, then it reduces to the NMF updates.

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Experiments (Two Repeating Objects)
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Experiments (Ha-Yo, Using NMF)
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Experiments (Ha-Yo, Using NMD)
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Extraction and Reconstruction of Objects
Discussion
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Extraction and Reconstruction of Objects

• We can construct the input spectrogram using an
  arbitrary number of objects to make a selective
  approximation.
• NMF
• NMD

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Discussion

• What is an Object?
• The data provided to this algorithm needs to
  expose the individuality of each objects in a
  disassociated way.
• It seems inflexible to define objects as
  repeating spectra, but actually not.
• Real world sounds are the case with musical
  signals where repeating patterns.
• Despite the lack of statistical rigor and
  simplistic cost function, performance of NMF and
  NMD is considerably better than other
  methods(i.e. PCA, ICA) in discovering auditory
  objects.
• Strict Non-negativity constraint fits naturally
  to finding components of magnitude spectra.

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