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When an object's velocity changes over time, the total distance traveled can be determined by summing small displacement intervals over short increments. This approach approximates the true distance through numerical summation and the use of integral calculus. An estimate of the total displacement can be obtained by measuring velocity at regular intervals and multiplying each value by the corresponding time step.If a runner accelerates over the first three seconds of a race, speed measurements...
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Quantifying Intermembrane Distances with Serial Image Dilations
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Matricial Wasserstein-1 Distance.

Yongxin Chen1, Tryphon T Georgiou2, Lipeng Ning3

  • 1Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY.

IEEE Control Systems Letters
|November 21, 2017
PubMed
Summary
This summary is machine-generated.

We introduce a new matrix-based Wasserstein 1-metric for density matrices and unbalanced mass transport. This computational tool extends the Earth Mover's Distance for advanced data analysis.

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Area of Science:

  • Quantum information science
  • Optimal transport theory
  • Matrix analysis

Background:

  • The Wasserstein 1-metric (W1) is a powerful tool for comparing probability distributions.
  • Extending W1 to matrix-valued data and unbalanced scenarios remains a challenge.
  • Existing methods may lack computational efficiency or applicability to quantum states.

Purpose of the Study:

  • To develop an extension of the Wasserstein 1-metric for density matrices and matrix-valued measures.
  • To incorporate an unbalanced interpretation of mass transport within this framework.
  • To provide a computationally tractable matrix analogue of the Earth Mover's Distance.

Main Methods:

  • Utilizing duality theory in optimal transport.
  • Employing a "dual of the dual" formulation for the Wasserstein 1-metric.
  • Applying the extended metric to density matrices and matrix-valued measures.

Main Results:

  • A novel extension of the Wasserstein 1-metric for density matrices is proposed.
  • The framework accommodates unbalanced mass transport scenarios.
  • The matrix analogue of the Earth Mover's Distance demonstrates computational advantages.

Conclusions:

  • The proposed matrix Wasserstein 1-metric offers a versatile and efficient tool.
  • This extension is valuable for analyzing quantum states and complex data structures.
  • The method provides a robust approach to unbalanced optimal transport problems.