[Repo Assist] perf(gcm): vectorise _remove_constant_columns and optimise cross-covariance in kernel independence test

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Summary

Two complementary performance improvements to dowhy/gcm/independence_test/kernel.py, both on the hot path of every approx_kernel_based and kernel_based independence test call.

1. Vectorise _remove_constant_columns

Before – per-column np.unique loop: O(p · n log n)

return X[:, [np.unique(X[:, i]).shape[0] > 1 for i in range(X.shape[1])]]

After – single vectorised comparison: O(p · n)

return X[:, ~np.all(X == X[0], axis=0)]

np.all(X == X[0], axis=0) compares all rows against the first row simultaneously in one NumPy broadcast, avoiding the per-column sort inside np.unique. The semantics are identical for all valid inputs (NaN values are rejected downstream at the existing check on line 88–89, so the NaN/NaN inequality edge case never reaches this function in practice).

_remove_constant_columns is called 2–3 times per independence test invocation; with large feature matrices the savings compound.

2. Optimise _estimate_column_wise_covariances

Before – full (dx+dy) × (dx+dy) covariance via np.cov: O(n·(dx+dy)2)

return np.cov(X, Y, rowvar=False)[: X.shape[1], -Y.shape[1] :]

np.cov(X, Y) stacks both matrices and computes the complete (dx+dy)-square covariance matrix, then discards the auto-covariance blocks and returns only the (dx, dy) cross-covariance slice.

After – direct cross-covariance: O(n·dx·dy)

n = X.shape[0]
X_c = X - X.mean(axis=0)
Y_c = Y - Y.mean(axis=0)
return X_c.T @ Y_c / (n - 1)
```

Result is **numerically identical** to the original.  For the default `num_random_features=50` this cuts the covariance work by ~4× (from `1002` to `502` multiplications per call).

`_estimate_column_wise_covariances` is called once per permutation inside `_rit` / `_rcit` (default `num_permutations=100`), plus 3 extra times in `_rcit` for `cov_zz`, `cov_xz`, `cov_zy`.

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## Test Status

```
tests/gcm/independence_test/test_kernel.py  22 passed in 17.26s ✅

All 22 existing kernel independence-test unit tests (categorical, continuous, conditional, unconditional, determinism, constant-column edge cases) pass unchanged.

Format check: black – no changes needed; isort – no changes needed.
Lint: flake8 – no new errors introduced (pre-existing E501 in docstrings at lines 32–41 are unchanged).

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Pull request overview

Optimizes hot paths in GCM kernel independence tests.

Changes:

• Vectorizes constant-column detection.
• Computes cross-covariance directly.
• Adds empty-input handling.

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