Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores
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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.
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Research details
- Subjects
- Mathematics · Probability and Statistics
- Keywords
- entropic self-transport, matrix scaling, residual error bounds, score estimation, empirical sharpness
- Manuscript date
- 2026-10-18
- Reader reuse
- No additional permission granted
Scope
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.
Disclosures
AI tools assisted mathematical development, computation and editing. The author is responsible for the final manuscript. Independent AI reviews are not external human peer review or proof-assistant verification. Company affiliation is retained from the manuscript.
Reproduction
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.