correct information imbalance normalization

[SpatLyu]

This PR corrects a critical normalization error in the Information Imbalance calculation pipeline. The previous implementation used Npred as the normalization denominator, which caused II values to exceed the theoretical upper bound of 1 whenever Nlib > Npred. The corrected implementation uses Nlib (the library size) as the normalization base, ensuring that II values are strictly bounded within $[0, 1]$.

Background

Information Imbalance measures how well the neighborhood structure in space A (e.g., X) predicts the neighborhood structure in space B (e.g., Y):

$$\Delta(A \to B) = \frac{2}{N} \cdot \overline{r^B}$$

Under the null hypothesis of independence between A and B, ranks follow a uniform distribution with expected value $N/2$, yielding $\Delta = 1$. When A perfectly predicts B, ranks concentrate near 1, yielding $\Delta \to 0$.

Problem

The previous implementation used:

return 2.0 / static_cast<double>(Npred) * mean_rank;

Since ranks are computed within the library space (size $N_{lib}$), the expected rank under independence is $N_{lib}/2$, not $N_{pred}/2$. When $N_{lib} > N_{pred}$ (common in cross-validation or out-of-sample settings), the computed II value becomes:

$$\Delta_{incorrect} = \frac{N_{lib}}{N_{pred}} > 1$$

This violates the theoretical bound $\Delta \in [0, 1]$ and renders downstream metrics (e.g., Information Imbalance Gain) meaningless.

Mathematical Verification

For the corrected formula, the Information Imbalance is computed as:

$$\Delta(X \to Y) = \frac{2}{N_{lib} \cdot N_{pred} \cdot k} \sum_{i=1}^{N_{pred}} \sum_{j \in \text{NN}_k^X(i)} r^Y_{ij}$$

where $NN_k^X(i)$ denotes the set of $k$ nearest neighbours of prediction point $i$ in the X space, and $r^Y_{ij}$ is the rank of library point $j$ in the Y space with respect to point $i$. Only the ranks corresponding to the selected $k$ nearest neighbours in X are included in the summation.

This is algebraically equivalent to:

$$\Delta = \frac{2}{N_{lib}} \times \overline{r^B}$$

where $\overline{r^B} = \frac{1}{N_{pred}} \sum_{i=1}^{N_{pred}} \left( \frac{1}{k} \sum_{j \in NN_k^X(i)} r^Y_{ij} \right)$ is the mean conditional rank across all prediction points.

For the corrected formula:

Scenario	Expected Conditional Rank	II Value
Perfect prediction (k=1)	1	$2/N_{lib} \to 0$
Complete independence	$N_{lib}/2$	$2/N_{lib} \cdot N_{lib}/2 = 1$
Partial association	between	$(0, 1)$