Finite-rank kernel realization and completion

James Tian

doi:10.33205/cma.1942507

Finite-rank kernel realization and completion

Abstract

We consider when local finite-rank positive definite kernels come from a single global finite-rank kernel. The first part treats the case where the global kernel is fixed and shows that the missing mixed blocks control when a global rank-$r$ realization exists and how rank grows when it does not. The second part treats the completion problem, where only local kernels are given. In that setting, rank-preserving completion becomes a unitary patching problem on overlap generated subspaces. Forests always patch, full overlaps are governed by cycle conditions for the induced unitaries, and partial overlaps lead to finite-dimensional subspace compatibility conditions. We also give a rank bound for finite completions obtained by joining two completed pieces along their common feature subspace.

Keywords

References

N. Aronszajn: Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337–404.
A. Aydın, A. Gheondea: Probability error bounds for approximation of functions in reproducing kernel Hilbert spaces, J. Funct. Spaces, 15 (2021), Article ID: 6617774.
F. Bach: Sharp analysis of low-rank kernel matrix approximations, ArXiv, https://arxiv.org/abs/1208.2015
M. Bakonyi, H. Woerdeman: Matrix completions, moments, and sums of Hermitian squares, Princeton University Press, Princeton, New Jersey (2011).
W. Barrett, C. Johnson and P. Tarazaga: The real positive definite completion problem for a simple cycle Linear Algebra Appl., 192 (1993), 3–31.
W. Barrett, C. Johnson and M. Lundquist: Determinantal formulae for matrix completions associated with chordal graphs, Linear Algebra Appl., 121 (1989), 265-289.
M. Booth, P. Hackney, B. Harris, C. Johnson, M. Lay, L. Mitchell, S. Narayan, A. Pascoe, K. Steinmetz, B. Sutton and W. Wang: On the minimum rank among positive semidefinite matrices with a given graph, SIAM J. Matrix Anal. Appl., 30 (2008), 731–740.
D. Cichoń, J. Stochel and F. Szafraniec: Extending positive definiteness, Trans. Amer. Math. Soc., 363 (2011), 545–577.

D. Cichoń, J. Stochel and F. Szafraniec: Riesz-Haviland criterion for incomplete data, J. Math. Anal. Appl., 380 (2011), 94–104.
J. Dancis: Positive semidefinite completions of partial Hermitian matrices, Linear Algebra Appl., 175 (1992), 97–114.
M. E.-Nagy, M. Laurent and A. Varvitsiotis: Complexity of the positive semidefinite matrix completion problem with a rank constraint, Discrete Geometry And Optimization, 69 (2013), 105–120.
M. Ghaemi, M. Moslehian and Q. Xu: Kolmogorov decomposition of conditionally completely positive definite kernels, Positivity, 25 (2021), 515–530.
S. Gribling, D. Laat and M. Laurent: Matrices with high completely positive semidefinite rank, Linear Algebra Appl., 513 (2017), 122–148.
R. Grone, C. Johnson, E. Sá and H. Wolkowicz: Positive definite completions of partial Hermitian matrices, Linear Algebra Appl., 58 (1984), 109–124.
J. Helton, D. Lam and H.Woerdeman: Sparsity patterns with high rank extremal positive semidefinite matrices, SIAM J. Matrix Anal. Appl., 15 (1994), 299–312.
T. Hofmann, B. Schölkopf and A. Smola: Kernel methods in machine learning Ann. Statist., 36 (2008), 1171–1220.
Z. Jabłoński, J. Stochel and F. Szafraniec: Unitary propagation of operator data, Proc. Edinb. Math. Soc., 50 (3) (2007), 689–699.
C. Johnson: Matrix completion problems: a survey, Matrix Theory And Applications (Phoenix, AZ, 1989), 40 (1990), 171–198.
M. Journée, F. Bach, P. Absil and R. Sepulchre: Low-rank optimization on the cone of positive semidefinite matrices, SIAM J. Optim., 20 (2010), 2327–2351.
S. Kim, M. Kojima, M. Mevissen and M. Yamashita: Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion, Math. Program., 129 (2011), 33–68.
M. Laurent, A. Varvitsiotis: A new graph parameter related to bounded rank positive semidefinite matrix completions. Math. Program., 145 (2014), 291–325.
M. Moslehian: Conditionally positive definite kernels in Hilbert $C^*$-modules, Positivity. 21 (2017), 1161–1172.
G. Murphy: Positive definite kernels and Hilbert $C^∗$-modules, Proc. Edinburgh Math. Soc. (2), 40 (1997), 367–374.
B. Sturmfels, C. Uhler: Multivariate Gaussian, semidefinite matrix completion, and convex algebraic geometry, Ann. Inst. Statist. Math., 62 (2010), 603–638.
F. Szafraniec: The reproducing kernel Hilbert space and its multiplication operators, Complex Analysis And Related Topics (Cuernavaca, 1996), 114 (2000), 253–263.
F. Szafraniec: Composition in reproducing kernel Hilbert spaces à Rebours, Operator And Matrix Theory, Function Spaces, And Applications, 295 (2024), 365–384.
F. Szafraniec: Interpolation and domination by positive definite kernels, Complex Analysis—fifth Romanian-Finnish Seminar, Part 2 (Bucharest, 1981), 1014 (1983), 291–295.
K. Waghmare, V. Panaretos: The completion of covariance kernels, Ann. Statist., 50 (2022), 3281–3306.
K.Waghmare, V. Panaretos: The positive-definite completion problem, Trans. Amer. Math. Soc., 377 (2024), 6549–6594.

Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

James Tian ^*
0000-0003-4996-1227
United States

Early Pub Date

June 30, 2026

Publication Date

-

Submission Date

May 2, 2026

Acceptance Date

June 27, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.33205/cma.1942507

IZ

https://izlik.org/JA78HT33UZ

Cite

RIS / Bibtex

APA

Tian, J. (2026). Finite-rank kernel realization and completion. Constructive Mathematical Analysis, Advanced Online Publication. https://doi.org/10.33205/cma.1942507

AMA

1.Tian J. Finite-rank kernel realization and completion. CMA. 2026;(Advanced Online Publication). doi:10.33205/cma.1942507

Chicago

Tian, James. 2026. “Finite-Rank Kernel Realization and Completion”. Constructive Mathematical Analysis, no. Advanced Online Publication. https://doi.org/10.33205/cma.1942507.

EndNote

Tian J (June 1, 2026) Finite-rank kernel realization and completion. Constructive Mathematical Analysis Advanced Online Publication

IEEE

[1]J. Tian, “Finite-rank kernel realization and completion”, CMA, no. Advanced Online Publication, June 2026, doi: 10.33205/cma.1942507.

ISNAD

Tian, James. “Finite-Rank Kernel Realization and Completion”. Constructive Mathematical Analysis. Advanced Online Publication (June 1, 2026). https://doi.org/10.33205/cma.1942507.

JAMA

1.Tian J. Finite-rank kernel realization and completion. CMA. 2026. doi:10.33205/cma.1942507.

MLA

Tian, James. “Finite-Rank Kernel Realization and Completion”. Constructive Mathematical Analysis, no. Advanced Online Publication, June 2026, doi:10.33205/cma.1942507.

Vancouver

1.James Tian. Finite-rank kernel realization and completion. CMA. 2026 Jun. 1;(Advanced Online Publication). doi:10.33205/cma.1942507