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Sliced Wasserstein Distance (SWD)

Key: swd · Family: distribution · Better: lower ↓ · Per-lead: yes

SWD asks: how much work would it take to move the synthetic distribution onto the real one? That transport cost (the Wasserstein distance) is hard to compute in many dimensions but easy in one. SWD therefore projects the data onto many random lines and averages the 1-D costs.

Try it

The purple line is one projection direction. The histograms show where the points land on that line. The bottom chart shows the 1-D distance W₁ for every angle; SWD is its average.

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What to notice:

  • Shift along x: projections along x (0°) show the gap, while projections near 90° do not. One direction can miss a difference, so many directions are averaged.
  • Same marginals, wrong correlation: the x and y histograms alone look identical, yet projections at 45° and 135° disagree. Slicing at many angles catches differences in how features co-vary.

How it works

For each lead:

  1. Describe every recording with the 7-feature descriptor (mean, standard deviation, RMS, maximum |x|, energy, zero-crossing rate, lag-1 autocorrelation).
  2. Draw n_projections (default 200) random unit directions with a fixed seed.
  3. Project both sets onto each direction and compute the exact 1-D Wasserstein-1 distance by sorting the projected values and averaging the absolute differences of the matching quantiles.
  4. Average over directions.

The math

\[ \mathrm{SWD}(P, Q) = \frac{1}{K}\sum_{k=1}^{K} W_1\!\left(\theta_k^\top P,\ \theta_k^\top Q\right), \qquad W_1(a,b) = \frac{1}{n}\sum_{i=1}^{n} \left| a_{(i)} - b_{(i)} \right| \]

where \(\theta_k\) are random unit vectors and \(a_{(i)}\) are the sorted projected values. In 1-D the optimal transport plan simply matches the i-th smallest value with the i-th smallest value, which is why sorting is enough.

Reading the score

  • 0 means every projection matches. The score is in feature units: comparable across models on the same data, not across data sets.
  • SWD makes no Gaussian assumption and is cheap: \(O(K\,n\log n)\).
  • With few projections in high dimensions, some differences can be missed; 200 directions is plenty for 7 features.

Use it with ECGEN-Eval

ecgen-eval eval --real real.npy --synthetic synth.npy -m swd --output report.html

from ecgen_eval import load_ecg_npy, AVAILABLE_METRICS

real = load_ecg_npy("real.npy", fs=500)
synth = load_ecg_npy("synth.npy", fs=500)

swd = AVAILABLE_METRICS["swd"](n_projections=200, seed=42)
result = swd.compute(real.data, synth.data, lead_names=real.lead_names)
print(result.score, result.per_lead_scores)

Parameters are in the Metric Reference.

Learn more

  • Rabin et al., Wasserstein Barycenter and its Application to Texture Mixing, 2011.
  • Kolouri et al., Generalized Sliced Wasserstein Distances, NeurIPS 2019.