arXiv subjects

Rajeev Muthu

Publications and source records attributed to Rajeev Muthu.

3 recordsLinked to original sources

arXiv2026

Finite-sample certification and operating envelopes for spectral clustering and graph centrality

Spectral clustering and node rankings are commonly reported from one observed network without a finite-sample statement of what the observation supports. We develop a certification protocol that either returns a coverage-guaranteed set or explicitly returns ``no nontrivial certificate.'' For an inhomogeneous Bernoulli graph, a matrix-Bernstein quantile with all numerical constants and its ambient-dimension factor retained is combined with a one-sided spectral-gap certificate. The resulting Grassmann ball is valid at finite $n$, but is reported as informative only when its radius is below the diameter of the Grassmannian. We propagate the ball through a certificate-bearing approximate $k$-means map under declared population separation and minimum-cluster envelopes, derive simultaneous bands and an observed-gap certificate for degree centrality, and give a corrected normalized-Katz extension. A $12$-cell simulation study with $1{,}000$ graphs per cell maps the difference between coverage and usefulness. The submitted $n=200$ block-model example is shown to be necessarily vacuous after the dimension factor is restored; in the benchmark $p=0.30,q=0.10$, the subspace radius first falls below one at $n=\ExactRadiusThreshold$, whereas the mean-square clustering certificate remains unavailable until $n=\HammingThreshold$. An unequal-block example produces a genuine centrality certificate, while an analysis of the Zachary karate-club network correctly declines to certify despite $97.1\%$ agreement with the observed factions. These results separate algorithmic success, coverage validity and inferential informativeness.

math.ST↗

arXiv2026

Quantitative fixed-point theorems with verifiable hypotheses: rates and stability

Let $(X,\dist)$ be a complete metric space and let $C\subseteq X$ be a closed invariant set. We study fixed points of maps $T\colon C\to C$ governed by a \emph{verifiable} contractive modulus. The modulus is encoded by a contractive gauge $\omega$ and a certified constant $\kappa=\sup_{0<r\le R}\omega(r)/r<1$ on a computable working radius $R$. From this datum we derive explicit a priori bounds $\dist(x_n,x^\ast)\le \Phi(n;\kappa,\delta_0)$ for Picard iterates, a residual-to-error estimate, and a quantitative data dependence bound $\dist(x^\ast,y^\ast)\le (1-\kappa)^{-1}\sup_{x\in C}\dist(Tx,Sx)$. We further treat inexact evaluations $\dist(\tilde x_{n+1},T\tilde x_n)\le \eta_n$ and obtain certified resilience bounds with the same stability factor. The framework applies to Hammerstein--Volterra integral equations and to boundary value problems via Green operators, where kernel bounds yield certified convergence rates.

math.DS↗

arXiv2025

Finite diagonalizable torsors and Kummer cohomology over semiring schemes

Classical descent treats a finite diagonalizable torsor as a higher-rank locally free object. Over semirings that route is incomplete: fpqc-local freeness is not known to imply Zariski-local freeness in arbitrary rank. We show that diagonalizable symmetry bypasses this obstruction. The coordinate semialgebra splits into character pieces, each piece descends in rank one, and the torsor identity makes their multiplication maps invertible. This gives a natural equivalence between fpqc $D_X(Q)$-torsors and Picard-strong $Q$-gradings, and represents every torsor by a finite locally free morphism of rank $|Q|$. The same viewpoint proves that finite-index monomial maps of split tori are finite locally free torsors and yields matrix Kummer classification sequences. A concrete example is the nontrivial Boolean torsor $\Spec\mathbb B[v^{\pm1}]\to\Spec\mathbb B[u^{\pm1}]$, $u\mapsto v^m$, whose unit-power classes contribute $\mathbb Z/m\mathbb Z$ to degree-one cohomology. A two-term resolution of the character group then gives a presentation-independent universal-coefficient filtration for diagonalizable fpqc cohomology. This formalism organizes, but does not compute, $\mathbf G_{\mathrm m}$-cohomology. In degree two it identifies the Kummer boundary of a line bundle with its root gerbe. On projective space over the Boolean or real tropical semifield all $\mu_m$-torsors vanish, whereas the root gerbe of $\mathcal O(1)$ has exact order $m$. Thus torsors and gerbes retain information that can disappear under ordinary ring completion.

math.RA↗