Noncommutative Christoffel-Darboux kernels

Belinschi, Serban; Magron, Victor; Vinnikov, Victor

doi:10.1090/tran/8648

Noncommutative Christoffel-Darboux kernels
HTML articles powered by AMS MathViewer

by Serban T. Belinschi, Victor Magron and Victor Vinnikov;

Trans. Amer. Math. Soc. 376 (2023), 181-230

DOI: https://doi.org/10.1090/tran/8648

Published electronically: October 7, 2022

HTML | PDF | Request permission

Abstract:

We introduce from an analytic perspective Christoffel-Darboux kernels associated to bounded, tracial noncommutative distributions. We show that properly normalized traces, respectively norms, of evaluations of such kernels on finite dimensional matrices yield classical plurisubharmonic functions as the degree tends to infinity, and show that they are comparable to certain noncommutative versions of the Siciak extremal function. We prove estimates for Siciak functions associated to free products of distributions, and use the classical theory of plurisubharmonic functions in order to propose a notion of support for noncommutative distributions. We conclude with some conjectures and numerical experiments.

References

Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge, 2010. MR 2760897
Miguel F. Anjos and Jean B. Lasserre, Introduction to semidefinite, conic and polynomial optimization, Handbook on semidefinite, conic and polynomial optimization, Internat. Ser. Oper. Res. Management Sci., vol. 166, Springer, New York, 2012, pp. 1–22. MR 2894689, DOI 10.1007/978-1-4614-0769-0_{1}
Michael Anshelevich, Monic non-commutative orthogonal polynomials, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2395–2405. MR 2390506, DOI 10.1090/S0002-9939-08-09306-4
Michael Anshelevich, Orthogonal polynomials with a resolvent-type generating function, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4125–4143. MR 2395166, DOI 10.1090/S0002-9947-08-04368-7
Michael Anshelevich, Product-type non-commutative polynomial states, Noncommutative harmonic analysis with applications to probability II, Banach Center Publ., vol. 89, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 45–59. MR 2730861, DOI 10.4064/bc89-0-2
T. Banks and T. Constantinescu, Orthogonal polynomials in several non-commuting variables. II, arXiv:math/0412528v1 [math.FA], 2004.
Sabine Burgdorf, Kristijan Cafuta, Igor Klep, and Janez Povh, The tracial moment problem and trace-optimization of polynomials, Math. Program. 137 (2013), no. 1-2, Ser. A, 557–578. MR 3010437, DOI 10.1007/s10107-011-0505-8
Sabine Burgdorf, Igor Klep, and Janez Povh, Optimization of polynomials in non-commuting variables, SpringerBriefs in Mathematics, Springer, [Cham], 2016. MR 3496028, DOI 10.1007/978-3-319-33338-0
Thomas Bloom, Orthogonal polynomials in $\mathbf C^n$, Indiana Univ. Math. J. 46 (1997), no. 2, 427–452. MR 1481598, DOI 10.1512/iumj.1997.46.1360
Thomas Bloom, Norman Levenberg, Federico Piazzon, and Franck Wielonsky, Bernstein-Markov: a survey, arXiv:1512.00739v1 [math.CV], 2015.
Joseph A. Ball, Gregory Marx, and Victor Vinnikov, Noncommutative reproducing kernel Hilbert spaces, J. Funct. Anal. 271 (2016), no. 7, 1844–1920. MR 3535321, DOI 10.1016/j.jfa.2016.06.010
Bernhard Beckermann, Mihai Putinar, Edward B. Saff, and Nikos Stylianopoulos, Perturbations of Christoffel-Darboux kernels: detection of outliers, Found. Comput. Math. 21 (2021), no. 1, 71–124. MR 4215233, DOI 10.1007/s10208-020-09458-9
Raúl E. Curto and Lawrence A. Fialkow, Flat extensions of positive moment matrices: recursively generated relations, Mem. Amer. Math. Soc. 136 (1998), no. 648, x+56. MR 1445490, DOI 10.1090/memo/0648
Kristijan Cafuta, Igor Klep, and Janez Povh, Constrained polynomial optimization problems with noncommuting variables, SIAM J. Optim. 22 (2012), no. 2, 363–383. MR 2968858, DOI 10.1137/110830733
Tiberiu Constantinescu, Orthogonal polynomials in several variables. I, arXiv:math/0205333v1 [math.FA], 2002.
Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 155, Cambridge University Press, Cambridge, 2014. MR 3289583, DOI 10.1017/CBO9781107786134
Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
Edward G. Effros and Soren Winkler, Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems, J. Funct. Anal. 144 (1997), no. 1, 117–152. MR 1430718, DOI 10.1006/jfan.1996.2958
A. Guionnet and D. Shlyakhtenko, Free monotone transport, Invent. Math. 197 (2014), no. 3, 613–661. MR 3251831, DOI 10.1007/s00222-013-0493-9
Vincent Guedj and Ahmed Zeriahi, Degenerate complex Monge-Ampère equations, EMS Tracts in Mathematics, vol. 26, European Mathematical Society (EMS), Zürich, 2017. MR 3617346, DOI 10.4171/167
J. William Helton, “Positive” noncommutative polynomials are sums of squares, Ann. of Math. (2) 156 (2002), no. 2, 675–694. MR 1933721, DOI 10.2307/3597203
Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
Didier Henrion and Jean-Bernard Lasserre, Detecting global optimality and extracting solutions in GloptiPoly, Positive polynomials in control, Lect. Notes Control Inf. Sci., vol. 312, Springer, Berlin, 2005, pp. 293–310. MR 2123528, DOI 10.1007/10997703_{1}5
J. William Helton, Jean B. Lasserre, and Mihai Putinar, Measures with zeros in the inverse of their moment matrix, Ann. Probab. 36 (2008), no. 4, 1453–1471. MR 2435855, DOI 10.1214/07-AOP365
J. William Helton and Scott A. McCullough, A Positivstellensatz for non-commutative polynomials, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3721–3737. MR 2055751, DOI 10.1090/S0002-9947-04-03433-6
J. W. Helton, R. L. Miller, and M. Stankus, NCAlgebra: a mathematica package for doing non commuting algebra, ncalg@ucsd.edu, 1996.
Maciej Klimek, Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1150978
Igor Klep, Victor Magron, and Janez Povh, Sparse noncommutative polynomial optimization, Math. Program. (2021), 1–41.
Igor Klep, Victor Magron, and Jurij Volčič, Optimization over trace polynomials, Ann. Henri Poincaré 23 (2022), no. 1, 67–100. MR 4361871, DOI 10.1007/s00023-021-01095-4
Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, Foundations of free noncommutative function theory, Mathematical Surveys and Monographs, vol. 199, American Mathematical Society, Providence, RI, 2014. MR 3244229, DOI 10.1090/surv/199
Jean B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11 (2000/01), no. 3, 796–817. MR 1814045, DOI 10.1137/S1052623400366802
Monique Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging applications of algebraic geometry, IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 157–270. MR 2500468, DOI 10.1007/978-0-387-09686-5_{7}
Jean B. Lasserre and Edouard Pauwels, The empirical Christoffel function with applications in data analysis, Adv. Comput. Math. 45 (2019), no. 3, 1439–1468. MR 3955724, DOI 10.1007/s10444-019-09673-1
Scott McCullough, Factorization of operator-valued polynomials in several non-commuting variables, Linear Algebra Appl. 326 (2001), no. 1-3, 193–203. MR 1815959, DOI 10.1016/S0024-3795(00)00285-8
Victor Magron, Marcelo Forets, and Didier Henrion, Semidefinite approximations of invariant measures for polynomial systems, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), no. 12, 6745–6770. MR 4026902, DOI 10.3934/dcdsb.2019165
Attila Máté and Paul G. Nevai, Bernstein’s inequality in $L^{p}$ for $0<p<1$ and $(C,\,1)$ bounds for orthogonal polynomials, Ann. of Math. (2) 111 (1980), no. 1, 145–154. MR 558399, DOI 10.2307/1971219
Attila Máté, Paul Nevai, and Vilmos Totik, Szegő’s extremum problem on the unit circle, Ann. of Math. (2) 134 (1991), no. 2, 433–453. MR 1127481, DOI 10.2307/2944352
Swann Marx, Edouard Pauwels, Tillmann Weisser, Didier Henrion, and Jean Bernard Lasserre, Semi-algebraic approximation using Christoffel-Darboux kernel, Constr. Approx. 54 (2021), no. 3, 391–429. MR 4342619, DOI 10.1007/s00365-021-09535-4
James A. Mingo and Roland Speicher, Free probability and random matrices, Fields Institute Monographs, vol. 35, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017. MR 3585560, DOI 10.1007/978-1-4939-6942-5
Paul Nevai, Géza Freud, orthogonal polynomials and Christoffel functions. A case study, J. Approx. Theory 48 (1986), no. 1, 3–167. MR 862231, DOI 10.1016/0021-9045(86)90016-X
Miguel Navascués, Stefano Pironio, and Antonio Acín, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations, New J. Phys. 10 (2008), no. 7, 073013.
Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
Edouard Pauwels, Francis Bach, and Jean-Philippe Vert, Relating leverage scores and density using regularized Christoffel functions, Advances in Neural Information Processing Systems, 2018, pp. 1663–1672.
Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR 2006539, DOI 10.1017/CBO9781107360235
Edouard Pauwels and Jean B. Lasserre, Sorting out typicality with the inverse moment matrix SOS polynomial, Advances in Neural Information Processing Systems, 2016, pp. 190–198.
S. Pironio, M. Navascués, and A. Acín, Convergent relaxations of polynomial optimization problems with noncommuting variables, SIAM J. Optim. 20 (2010), no. 5, 2157–2180. MR 2650843, DOI 10.1137/090760155
Edouard Pauwels, Mihai Putinar, and Jean-Bernard Lasserre, Data analysis from empirical moments and the Christoffel function, Found. Comput. Math. 21 (2021), no. 1, 243–273. MR 4215235, DOI 10.1007/s10208-020-09451-2
Mihai Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (1993), no. 3, 969–984. MR 1254128, DOI 10.1512/iumj.1993.42.42045
Barry Simon, The Christoffel-Darboux kernel, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 295–335. MR 2500498, DOI 10.1090/pspum/079/2500498
Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778, DOI 10.1007/978-3-662-03329-6
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728, DOI 10.1007/978-1-4612-6188-9
A. F. Timan, Theory of approximation of functions of a real variable, Dover Publications, Inc., New York, 1994. Translated from the Russian by J. Berry; Translation edited and with a preface by J. Cossar; Reprint of the 1963 English translation. MR 1262128
Mai Trang Vu, François Bachoc, and Edouard Pauwels, Rate of convergence for geometric inference based on the empirical Christoffel function, Preprint, arXiv:1910.14458, 2019.
Dan Voiculescu, Symmetries of some reduced free product $C^\ast$-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588. MR 799593, DOI 10.1007/BFb0074909
Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math. 132 (1998), no. 1, 189–227. MR 1618636, DOI 10.1007/s002220050222
Dan Voiculescu, The coalgebra of the free difference quotient and free probability, Internat. Math. Res. Notices 2 (2000), 79–106. MR 1744647, DOI 10.1155/S1073792800000064
Jie Wang and Victor Magron, Exploiting term sparsity in noncommutative polynomial optimization, Comput. Optim. Appl. 80 (2021), no. 2, 483–521. MR 4317480, DOI 10.1007/s10589-021-00301-7

AMS Website Down

Transactions of the American Mathematical Society

Noncommutative Christoffel-Darboux kernels
HTML articles powered by AMS MathViewer

Abstract: