Research Article · Preprint Mathematics
Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
We study how numerical errors in symmetric kernel scaling affect the spatial score obtained from its smooth kernel extension. A full square-root-mass residual gives a sharp kernel-energy bound for every positive candidate scaling, including singular positive semidefinite Gram matrices. For Gaussian kernels this yields computable, scale-invariant pointwise score certificates. For iid samples from a strictly positive C^3 density f on T^d, we prove a simultaneous nonlinear score bound and determine the exact local modulus in a shrinking-bandwidth regime. Its leading constant is C_d/√(f(x)). When √(n)ε^(d/4+1/2)→∞, residual o(n^(-1/2)) suffices for negligible score error at that normalization; positive sample-dependent candidates with residual asymptotic to c/√(n) attain a nonzero normalized error. The sampling condition nε^d/log n→∞ is sufficient. A reproducible solver experiment illustrates the computable certificate on actual iterates. The results concern numerical accuracy.
Keywords entropic self-transport · matrix scaling · residual error bounds · score estimation · empirical sharpness
Scope of the work
Sharp residual-to-energy and computable score-error certificates, including singular PSD Gram matrices, and the exact local empirical score modulus in the stated shrinking-bandwidth regime.
Reproduction notes
The paper has 14 pages. The editable-source archive contains 25 files. Compile main.tex with its figures. Use NumPy 2.3.5 to run code/solver_demo.py and the isolated code/reproduce_diagnostics.py workflow; optional plot rendering uses ReportLab. The diagnostics are finite floating-point computations.
The manuscript
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Cite this edition
Scott Schubr. (2026). Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/residual-certificates-empirical-sharpness-entropic-self-transport-scores/v/2
BibTeX
@techreport{luminary_residual_certificates_empirical_sharpness_entropic_self_transport_scores_v2,
title = {Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/residual-certificates-empirical-sharpness-entropic-self-transport-scores/v/2}
}Loading discussion…