Positivity sets of hinge functions

Josef Schicho; Ayush Kumar Tewari; Audie Warren

Home/Archive/arXiv

arXiv · 2503.13512

Positivity sets of hinge functions

Josef Schicho·Ayush Kumar Tewari·Audie Warren

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we investigate which subsets of the real plane are realisable as the set of points on which a one-layer ReLU neural network takes a positive value. In the case of cones we give a full characterisation of such sets. Furthermore, we give a necessary condition for any subset of $\mathbb R^d$. We give various examples of such one-layer neural networks.

Explore related subjects

stat.ML cs.DM cs.LG cs.SC math.CO math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Josef Schicho, Ayush Kumar Tewari, Audie Warren. 2025-03-14. Positivity sets of hinge functions. https://arxiv.org/abs/2503.13512

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Conditional Distributional Treatment Effects: Doubly Robust Estimation and Testing

Beyond conditional average treatment effects, treatments may impact the entire outcome distribution in covariate-dependent ways, for example, by altering the variance or tail risks for specific subpopulations. We propose a novel estimand to capture such conditional distributional treatment effects, and develop a doubly robust estimator that is minimax optimal in the local asymptotic sense. Using this, we develop a test for the global homogeneity of conditional potential outcome distributions that accommodates discrepancies beyond the maximum mean discrepancy (MMD), has provably valid type 1 error, and is consistent against fixed alternatives---the first test, to our knowledge, with such guarantees in this setting. We then provide a test that aggregates evidence across a grid of kernel-bandwidth choices. Furthermore, we derive exact closed-form expressions for two natural discrepancies (including the MMD), and provide a computationally efficient, permutation-free algorithm for our test.

stat.ML↗

arXiv2026

Chaos Is a LADDER: Domain Generalization Beyond Invariance via Reweighting

Domain generalization (DG) aims to learn from multiple source domains and generalize to unseen target domains. Most DG methods pursue invariance: they seek a causal representation whose prediction rule is invariant across domains. This principle is effective when the causal mechanism is stable, but becomes restrictive when the domain itself modulates how causal content maps to the response. In this case, directly feeding domain style into the predictor can create misleading shortcuts, since style does not by itself cause the response. Yet the apparent chaos of multiple styles can become a ladder: style can locate the unseen target domain among source domains and guide which domain-dependent prediction rules should be trusted. We propose \emph{Latent Adaptive Domain Disentanglement and Environment Reweighting} (LADDER), a fixed-model DG pipeline that learns causal/style representations, freezes the encoders, fits source-specific classifiers, and uses an unlabeled target-domain covariate set only at inference to compute weights over these fixed classifiers, with no target labels or model-state updates. We establish theoretical guarantees for source reweighting and validate LADDER on simulations, FMoW, and a location-grouped iWildCam protocol, with gains in overall and group-averaged accuracy.

stat.ML↗

arXiv2026

Density-Ratio Rescoring for Imbalanced Classification Using Raking Duals and Classifier Scores

Density-Ratio Rescoring (DRR) augments a classifier trained at the original class prior with a survey-raking dual score. Raking reweights the majority sample to match minority feature moments within a tolerance. DRR marginally standardizes the dual and base scores and combines them with a fixed weight of one half, using the fitted dual directly for prediction without resampling or refitting the base classifier. Under exact population matching and a correctly specified log-linear tilt model, the dual equals the log density ratio up to an additive constant. A class-separation analysis characterizes the signal strength and correlation conditions under which fusion improves separation under common within-class covariance. On 24 tabular benchmarks, evaluated over 30 trials and five base learners, DRR at the D=128 random-feature setting improves average precision over the standardized base on every dataset, with a mean gain of 0.034. It exceeds the shared-dual raking-and-relabeling resampler on 22 of 24 datasets, with a mean gain of $0.092$, and on all eight one-versus-rest tasks of a shared gene-expression cohort. These results demonstrate the effectiveness of using raking duals as reusable scores for improving rare-class ranking while retaining classifiers trained at the original prior.

stat.ML↗