Self-concordance and allocation for spatial power sums
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Abstract
This mathematical supplement studies the negative logarithm of one-sided product-power smoothing of arbitrary coordinatewise decreasing weights. A common real power p >= 16d gives a standard self-concordant barrier with parameter at most 2pd. For decreasing lattice inputs with at most r positive coordinates per state, p >= 32r suffices independently of ambient dimension and occupancy magnitudes. Residual Pareto tails, an exact third-cumulant identity and a sparse L3 bound supply the proof without input independence or log-concavity. Decreasing binary mixtures require power growing linearly with r for uniform convexity. Log-concave binary generating polynomials with deletion bounds admit dimension-free fixed-power statements and squared-logarithmic activity dependence. Classical damped Newton and projected-gradient methods give explicit allocation consequences.
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Research details
- Subjects
- Mathematics · Classical and Real Analysis · Optimization and Control
- Keywords
- Optimization and Control, Classical and Real Analysis, mathematical research
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
Common powers only. Decreasing continuous/lattice spatial smoothing: standard SC p>=16d; sparse active-coordinate support r: p>=32r. Ordinary-logQ-concave BINARY deletion1: absolute scale2560000 at every p>=4, parameter<=2Cpd, classical matroid premise. Deletion ratios<=M: p>=72(1+logM)^2, absolute scale360000, parameter<=3Cpd; rank-one squared-log order obstruction and literal p4/M625 curvature -192/25. Explicit-list Newton gap logm. Quartic matroid activities<=1: independence-oracle sampled projected gradient with polynomial expected arithmetic cost/high probability epsilon objective accuracy; all histories feasible. Mathematical supplement to the companion manuscript Dimension Thresholds for Concavity of Normalized Mixed Moments; the companion is a separate manuscript. The approved supplement does not release the companion.
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