Time Coupled Diffusion Maps- An online approach

Ev Zisselman

Content

• Accounting for Changing Dynamics
• The proposed approach
• Numerical example
• Online extension – general case
• The proposed solution- Perturbation theory
• Perturbation theory- next steps

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Content

• Accounting for Changing Dynamics
• The proposed approach
• Numerical example
• Online extension – general case
• The proposed solution- Perturbation theory
• Perturbation theory- next steps

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Accounting for Changing Dynamics in manifold

• Current graph Laplacian methods assume stationary process, hence a static intrinsic manifold.
• However, in many applications accounting for changing in the intrinsic manifold may offer an advantage.
• Examples for such applications are:
• Video segmentation
• Vortices in the atmosphere or in ocean

flows�

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Content

• Accounting for Changing Dynamics
• The proposed approach
• Numerical example
• Online extension – general case
• The proposed solution- Perturbation theory
• Perturbation theory- next steps

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The proposed approach

The proposed approach

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The proposed approach

The proposed approach

The proposed approach

Content

• Accounting for Changing Dynamics
• The proposed approach
• Numerical example
• Online extension – general case
• The proposed solution- Perturbation theory
• Perturbation theory- next steps

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Numerical example

Numerical example – other methods

Numerical example

1. Standard diffusion map:

• At time t=1 it separates between the red and the blue points
• At time t=5 there is no such a separation- all the points are uniformly spread on a unit disk
• Also at time t=9 it separates between the top points and the bottom points

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Numerical example

2. Concatenated data Diffusion map:

• At time t=1 we get the same separation as before
• At time t=5 we get basically nothing interesting – on average the points are spread out over the disc
• Notice that at t=9 also provides no separation

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Numerical example

Time coupled diffusion map:

• At time t=1 we get the same separation as before
• At time t=5 we still get the information of the past
• At time t=9 it clearly captures the foure events: the four lines in the embedding represent the four classes in the data
• The time couple diffusion map aggregates information over time

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Numerical example

• First two coordinate of the embedding:

The figure shows a clear separation in the case of time coupled DM, as opposed to lack of separation in the case of concatenated data DM

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Content

• Accounting for Changing Dynamics
• The proposed approach
• Numerical example
• Online extension – general case
• The proposed solution- Perturbation theory
• Perturbation theory- next steps

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Online extension- general case

Content

• Accounting for Changing Dynamics
• The proposed approach
• Numerical example
• Online extension – general case
• The proposed solution- Perturbation theory
• Perturbation theory- next steps

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Perturbation theory

Perturbation theory

Perturbation theory

Perturbation theory- eigenvalue approximation

Perturbation theory- eigenvalue approximation

Content

• Accounting for Changing Dynamics
• The proposed approach
• Numerical example
• Online extension – general case
• The proposed solution- Perturbation theory
• Perturbation theory- next steps

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Perturbation theory- next steps

Questions