Using Transportation Distances for Measuring Melodic Similarity

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Using Transportation Distances for Measuring
Melodic Similarity

• Pichaya Tappayuthpijarn
• Qiang Wang

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Outline

• Introduction
• Melodic Similarity Measuring
• The Earth Movers Distance (EMD)
• The Proportional Transportation Distance (PTD)
• Results Conclusion

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Introduction

• Weighted notes are well suited to represent
  music.
• Measuring melodic similarity by comparing
  weighted notes.
• Using EMD or PTD to measure the similarity.

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Melodic Similarity Measuring

• Weighted Notes represented by (Time, Pitch,
  Weight). The weights only reflect the note
  durations.

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Properties of a Metric
For pseudo-metric, only properties i, iii and
iv holds.
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The Earth Movers Distance (EMD)

• measures the minimum amount of work needed to
  transform one into the other by moving weight.

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Minimize the overall cost
with constraints
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W, U are the total weights of A, B respectively,
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Example a weight flow with EMD
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The Proportional Transportation Distance (PTD)

• EMD v.s. PTD
• PTD is a modified version of EMD.
• In the case of unequal total weight, triangle
  inequality does not hold for the EMD.
• By PTD, triangle inequality holds, making
  database search more efficient by using indices.

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The Proportional Transportation Distance

• for both point set, divide every points weight
  by its point sets total weight
• then calculate the EMD for the resulting point
  sets.

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Properties of PTD

• A pseudo-metric.
• Triangle inequality holds true

abgtc
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• Gives clear distinction of groups.
• The fact that the EMD allows partial matching,
  while the PTD matches all notes, leads to a clear
  distinction of these groups by the PTD, but not
  the EMD.

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• Can recognize augmented versions of the same
  melody as similar.

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Result Conclusion

• PTD leads to the identification of about 3.9 of
  100 randomly chosen search ( for r 1).
• For r 5, we need less than 1000 calculations
  instead of 476,000.
• PTD Indexing reduces the query running time.

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Thank you!