The Influence Function of Transport-based Quantiles
arXiv:2607.19080
2026
Architecture
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper identifies a structural instability of multivariate transport quantiles: unlike scalar quantiles, their pointwise influence function has a pole when a contaminated sample lies near the queried transport location, with magnitude proportional to \(\|z-\mathbf F_P(x_0)\|^{-(d-1)}\). This means empirical transport-quantile features can have infinite variance even when the underlying distribution is well behaved, making naive pointwise use in neural modules statistically and numerically fragile. A direct neural-network transfer is to replace pointwise transport-quantile queries by latent-space mollified queries, which integrate out the pole while preserving the global transport representation.
Ideas from this paper
Unverified
2026
Use a transport map \(Q_\theta\) from a fixed latent reference distribution to a data distribution, but expose only its locally averaged version \(\bar Q_{\theta,\sigma}(z)=\mathbb E_{u\sim K_\sigma(\cdot-z)}Q_\theta(u)\). Latent-space mollification integrates the pole-type influence singularity instead of allowing one training sample near \(Q_\theta(z)\) to dominate the quantile feature or its gradient.
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