arXiv subjects

Sujit Sakharam Damase

Publications and source records attributed to Sujit Sakharam Damase.

5 recordsLinked to original sources

arXiv2026

On positive definite thresholding of correlation matrices

Standard thresholding techniques for correlation matrices often destroy positive semidefiniteness. We investigate the construction of positive definite functions that vanish on specific sets $K \subseteq [-1,1)$, ensuring that the thresholded matrix remains a valid correlation matrix. We establish existence results, define a criterion for faithfulness based on the linear coefficient of the normalized Gegenbauer expansion in analogy with Delsarte's method in coding theory, and provide bounds for thresholding at single points and pairs of points. We prove that for correlation matrices of rank $n$, any soft-thresholding operator that preserves positive semidefiniteness necessarily induces a geometric collapse of the feature space, as quantified by an $\mathcal{O}(1/n)$ bound on the faithfulness constant. Such demonstrates that geometrically unbiased soft-thresholding limits the recoverable signal.

math.ST↗

arXiv2026

Complete discrete Schoenberg-Delsarte theory for homogeneous spaces

We develop a theory of partially defined complete positivity preservers, extending Schoenberg's classical characterization to functions defined only on discrete subsets or constrained domains. We frame the extension problem through the theory of completely positive maps on operator systems -- we characterize general partially defined completely positive definite functions on general homogeneous spaces. We apply our interpolation to constrained packing problems and Delsarte theory, where one uses positive definite functions on homogeneous spaces to obtain bounds on various packing problems. We prove the specific positive definite function witnesses that a code is sharp for constrained angle codes must be from polynomials.

math.FA↗

arXiv2025

Numerical radius and $\ell_p$ operator norm of Kronecker products and Schur powers: inequalities and equalities

Suppose $A=[a_{ij}]\in \mathcal{M}_n(\mathbb{C})$ is a complex $n \times n$ matrix and $B\in \mathcal{B}(\mathcal{H})$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$. We show that $w(A\otimes B)\leq w(C),$ where $w(\cdot)$ denotes the numerical radius and $C=[c_{ij}]$ with $c_{ij}= w\left(\begin{bmatrix} 0& a_{ij}\\ a_{ji}&0 \end{bmatrix} \otimes B\right).$ This refines Holbrook's classical bound $w(A\otimes B)\leq w(A) \|B\|$ [J. Reine Angew. Math. 1969], when all entries of $A$ are non-negative. If moreover $a_{ii}\neq 0$ $ \forall i$, we prove that $w(A\otimes B)= w(A) \|B\|$ if and only if $w(B)=\|B\|.$ We then extend these and other results to the more general setting of semi-Hilbertian spaces induced by a positive operator. In the reverse direction, we also specialize these results to Kronecker products and hence to Schur/entrywise products, of matrices: (1)(a) We first provide an alternate proof (using $w(A)$) of a result of Goldberg-Zwas [Linear Algebra Appl. 1974] that if the spectral norm of $A$ equals its spectral radius, then each Jordan block for each maximum-modulus eigenvalue must be $1 \times 1$ ("partial diagonalizability"). (b) Using our approach, we further show given $m \geq 1$ that $w(A^{\circ m})\leq w^m(A)$ - we also characterize when equality holds here. (2) We provide upper and lower bounds for the $\ell_p$ operator norm and the numerical radius of $A\otimes B$ for all $A \in \mathcal{M}_n(\mathbb{C})$, which become equal when restricted to doubly stochastic matrices $A$. Finally, using these bounds we obtain an improved estimation for the roots of an arbitrary complex polynomial.

math.FA↗

arXiv2025

A remark on characterizing inner product spaces via strong three-point homogeneity

We show that a normed linear space is isometrically isomorphic to an inner product space if and only if it is a strongly $n$-point homogeneous metric space for any (or every) $n \geqslant 3$. The counterpart for $n=2$ is the Banach-Mazur problem.

math.FA↗

arXiv2024

Multivariate transforms of total positivity

Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified the post-composition operators that preserve TP/TN kernels of each specified order. We explain how to extend this from preservers to transforms, and from one to several variables. Namely, given arbitrary nonempty totally ordered sets $X,Y$, we characterize the transforms that send each tuple of kernels on $X \times Y$ that are TP/TN of orders $k_1, \dots, k_p$, to a TP/TN kernel of order $l$, for arbitrary positive integers (or infinite) $k_j$ and $l$. An interesting feature is that to preserve TP (or TN) of order $2$, the preservers are products of individual power (or Heaviside) functions in each variable; but for all higher orders, the preservers are powers in a single variable. We also classify the multivariate transforms of symmetric TP/TN kernels; in this case it is the preservers of TP/TN of order 3 that are multivariate products of power functions, and of order 4 that are individual powers. The proofs use generalized Vandermonde kernels, Hankel kernels, (strictly totally positive) Polya frequency functions, and a kernel studied recently but tracing back to works of Schoenberg [Ann. of Math. 1955] and Karlin [Trans. Amer. Math. Soc. 1964].

math.FA↗