Ground Metrics

The ground metric defines the cost of moving a unit mass from one location to another. By default SDOT uses the squared Euclidean distance (Norm2), but many other metrics are supported.

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Default: Squared Euclidean (Norm2)

from sdot import SumOfDiracs, SplineGrid, distance

# Uses Norm2 by default — no need to specify
d = distance( SumOfDiracs( positions ), SplineGrid( values ) )

The cost between point $x$ and Dirac $y_i$ is $\parallel x-y_i{\parallel }^2$. This gives the standard Wasserstein-2 distance $W_2$.

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Periodic Metrics (coming soon)

Coming soon

Useful for periodic domains (torus topology), e.g., textures or angle spaces.

from sdot.metrics import PeriodicMetric, Norm2

metric = PeriodicMetric( Norm2(), period = [ 1.0, 1.0 ] )  # [0,1]² torus

plan = optimal_transport_plan( f, g, metric = metric )

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Entropic Regularization (coming soon)

Coming soon

Adds an entropy penalty to the transport cost, smoothing the plan. Useful for high-dimensional problems or when a soft assignment is preferable.

from sdot.metrics import Entropy

metric = Entropy( epsilon = 0.01 )   # regularization strength

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Radial Kernel Combinations (coming soon)

Coming soon

Custom ground metrics built from radial functions:

from sdot.metrics import Norm2, Polynomial, pos_part

# Cost = pos_part( w - r² )  where r = ‖x - y‖
metric = pos_part( "w - r**2" )

# Polynomial cost
metric = Polynomial( degree = 4 )