Add `normalize` parameter to sliced Wasserstein distance functions

[Harguna]

🚀 Feature

Add an optional normalize parameter (with companion normalize_mode) to ot.sliced_wasserstein_distance and ot.max_sliced_wasserstein_distance that standardizes inputs before computing the distance. Supports 'standard', 'minmax', and 'l2' normalization, with 'joint', 'source', or 'target' as the reference for fitting statistics. Default is normalize=None, preserving fully backward-compatible behavior.

Motivation

Sliced Wasserstein Distance projects distributions onto random 1D directions and averages the 1D Wasserstein distances. Because projection is a dot product with a random unit vector, features with larger numerical ranges dominate the projection, regardless of their actual importance.

Example: comparing two user populations on annual income ($20k–$500k) and interaction rate (0–1). If incomes are similar between the populations but interaction rates differ meaningfully, the projection will still be dominated by income and SWD reports a small distance, missing the real signal.

Two compounding problems:

1. Users often won't realize this is happening: Unlike Euclidean distance, SWD's scale-sensitivity isn't obvious from its definition. Users may interpret a small SWD as similarity, not realizing one feature drowned out the rest.

2. Manual normalization is friction and easy to get wrong: Today, users must add boilerplate before every call which adds additional steps:

   scaler = StandardScaler().fit(np.concatenate([X_s, X_t]))
   swd = ot.sliced_wasserstein_distance(scaler.transform(X_s), scaler.transform(X_t))

3. Independent normalization: A common mistake - fitting each distribution independently, which silently corrupts the distance with no warning:

   X_s_norm = StandardScaler().fit_transform(X_s)
   X_t_norm = StandardScaler().fit_transform(X_t)  # WRONG: independent fit

   This makes genuinely different distributions appear identical because each is independently re-centered to mean zero.

Pitch

Add two optional keyword arguments to both sliced_wasserstein_distance and max_sliced_wasserstein_distance:

ot.sliced_wasserstein_distance(
    X_s, X_t,
    ...,
    normalize=None,           # None | 'standard' | 'minmax' | 'l2'
    normalize_mode='joint',   # 'joint' | 'source' | 'target'
)

normalize controls the method:

• None — no normalization (default, identical to current behavior)
• 'standard' — zero mean, unit variance per feature: (x - μ) / σ
• 'minmax' — scale each feature to [0, 1]: (x - min) / (max - min)
• 'l2' — unit L2-norm per sample (row-wise): x / ‖x‖₂

normalize_mode controls which samples are used to compute the statistics (relevant for 'standard' and 'minmax'; ignored for 'l2' and None):

• 'joint' (default) — fit on concat(X_s, X_t). Preserves symmetry: SWD(X_s, X_t) == SWD(X_t, X_s). Recommended for general distribution comparison.
• 'source' — fit on X_s only. Useful for drift detection where X_s is a reference.
• 'target' — fit on X_t only. Reverse drift case.

Implementation:

• A private helper _normalize_inputs(X_s, X_t, normalize, normalize_mode, nx) performs the transformation using POT's backend abstraction (nx), so it works with NumPy, PyTorch, JAX, and TensorFlow and remains differentiable.
• Edge cases handled with warnings: zero-variance columns under 'standard', zero-range columns under 'minmax', zero-norm rows under 'l2'.
• Comprehensive tests covering backward compatibility, all method × mode combinations, edge cases, symmetry under 'joint', and at least one non-NumPy backend.
• An example script in examples/sliced-wasserstein/ demonstrating the scale-sensitivity problem and how the parameter solves it.

Additional context

I'm actively working on a skeleton PR for this and will open it shortly with the [WIP] tag to gather early feedback on the API design before completing the implementation. Happy to iterate on the proposed API based on maintainer's input.

[Harguna]

Added the Pull Request with the skeleton code here - #808. Looking forward to your feedback before completing the full implementation.