We derive subordination functions for free additive and free multiplicative deconvolutions under mild moment conditions. Our results include an algorithm to calculate these subordination functions, and thus the associated Cauchy transforms, for complex numbers with imaginary part greater than a parameter depending on the measure to deconvolve. The existence of these subordination functions on such domains reduces the problem of free deconvolutions to the problem of the classical additive deconvolution with a Cauchy distribution. Thus, our results, combined with known methods for the deconvolution with a Cauchy distribution, allow us to solve the free deconvolution problem. We also present extensions of these results to the case of operator-valued deconvolutions.
Nous dérivons des fonctions de subordination pour la déconvolution libre additive et multiplicative sous des conditions de moment faibles. Nos résultats incluent un algorithme pour calculer ces fonctions de subordination, et donc les transformées de Cauchy associées, pour les nombres complexes ayant une partie imaginaire supérieure à un paramètre dépendant de la mesure à déconvoler. L’existence des fonctions de subordination sur de tels domaines réduit le problème de la déconvolution libre au problème de la déconvolution additive classique par une distribution de Cauchy. Ainsi, nos résultats, combinés à des méthodes connues de déconvolution classique par une distribution de Cauchy, nous permettent de résoudre le problème de déconvolution libre. Nous présentons également des extensions de ces résultats au cas des déconvolutions à valeur opérateur.
References
[1] M. Andersen, J. Dahl and L. Vandenberghe. CVXOPT: A Python package for convex optimization, 2013. Available at abel.ee.ucla.edu/cvxopt.[1] M. Andersen, J. Dahl and L. Vandenberghe. CVXOPT: A Python package for convex optimization, 2013. Available at abel.ee.ucla.edu/cvxopt.
[2] O. Arizmendi, I. Nechita and C. Vargas. On the asymptotic distribution of block-modified random matrices. J. Math. Phys. 57 (2016) 015216. 1332.81021 10.1063/1.4936925[2] O. Arizmendi, I. Nechita and C. Vargas. On the asymptotic distribution of block-modified random matrices. J. Math. Phys. 57 (2016) 015216. 1332.81021 10.1063/1.4936925
[3] Z. Bai, J. Chen and J. Yao. On estimation of the population spectral distribution from a high-dimensional sample covariance matrix. Aust. N. Z. J. Stat. 52 (4) (2010) 423–437. 1373.62245 10.1111/j.1467-842X.2010.00590.x[3] Z. Bai, J. Chen and J. Yao. On estimation of the population spectral distribution from a high-dimensional sample covariance matrix. Aust. N. Z. J. Stat. 52 (4) (2010) 423–437. 1373.62245 10.1111/j.1467-842X.2010.00590.x
[4] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 (5) (2005) 1643–1697. 1086.15022 10.1214/009117905000000233 euclid.aop/1127395869[4] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 (5) (2005) 1643–1697. 1086.15022 10.1214/009117905000000233 euclid.aop/1127395869
[5] S. Belinschi and H. Bercovici. A new approach to subordination results in free probability. J. Anal. Math. 101 (2007) 357–365. 1142.46030 10.1007/s11854-007-0013-1[5] S. Belinschi and H. Bercovici. A new approach to subordination results in free probability. J. Anal. Math. 101 (2007) 357–365. 1142.46030 10.1007/s11854-007-0013-1
[6] S. Belinschi, T. Mai and R. Speicher. Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem. J. Reine Angew. Math. 732 (2017) 21–53. 06801923 10.1515/crelle-2014-0138[6] S. Belinschi, T. Mai and R. Speicher. Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem. J. Reine Angew. Math. 732 (2017) 21–53. 06801923 10.1515/crelle-2014-0138
[7] S. Belinschi, R. Speicher, J. Treilhard and C. Vargas. Operator-valued free multiplicative convolution: Analytic subordination theory and applications to random matrix theory. Int. Math. Res. Not. 14 (2015) 5933–5958. 1341.46037 10.1093/imrn/rnu114[7] S. Belinschi, R. Speicher, J. Treilhard and C. Vargas. Operator-valued free multiplicative convolution: Analytic subordination theory and applications to random matrix theory. Int. Math. Res. Not. 14 (2015) 5933–5958. 1341.46037 10.1093/imrn/rnu114
[8] S. T. Belinschi, H. Bercovici, M. Capitaine and M. Février. Outliers in the spectrum of large deformed unitarily invariant models. Ann. Probab. 45 (6A) (2017) 3571–3625. 1412.60014 10.1214/16-AOP1144 euclid.aop/1511773659[8] S. T. Belinschi, H. Bercovici, M. Capitaine and M. Février. Outliers in the spectrum of large deformed unitarily invariant models. Ann. Probab. 45 (6A) (2017) 3571–3625. 1412.60014 10.1214/16-AOP1144 euclid.aop/1511773659
[9] S. T. Belinschi, M. Popa and V. Vinnikov. Infinite divisibility and a noncommutative Boolean-to-free Bercovici–Pata bijection. J. Funct. Anal. 262 (1) (2012) 94–123. 1247.46054 10.1016/j.jfa.2011.09.006[9] S. T. Belinschi, M. Popa and V. Vinnikov. Infinite divisibility and a noncommutative Boolean-to-free Bercovici–Pata bijection. J. Funct. Anal. 262 (1) (2012) 94–123. 1247.46054 10.1016/j.jfa.2011.09.006
[10] F. Benaych-Georges. Rectangular random matrices, related convolution. Probab. Theory Related Fields 144 (3–4) (2009) 471–515. 1171.15022 10.1007/s00440-008-0152-z[10] F. Benaych-Georges. Rectangular random matrices, related convolution. Probab. Theory Related Fields 144 (3–4) (2009) 471–515. 1171.15022 10.1007/s00440-008-0152-z
[11] F. Benaych-Georges and M. Debbah. Free deconvolution: From theory to practice. In Paradigms for Biologically-Inspired Autonomic Networks and Services, 2010.[11] F. Benaych-Georges and M. Debbah. Free deconvolution: From theory to practice. In Paradigms for Biologically-Inspired Autonomic Networks and Services, 2010.
[12] H. Bercovici and D. Voiculescu. Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 (3) (1993) 733–773. 0806.46070 10.1512/iumj.1993.42.42033[12] H. Bercovici and D. Voiculescu. Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 (3) (1993) 733–773. 0806.46070 10.1512/iumj.1993.42.42033
[13] P. Biane. Processes with free increments. Math. Z. 227 (1998) 143–174. 0902.60060 10.1007/PL00004363[13] P. Biane. Processes with free increments. Math. Z. 227 (1998) 143–174. 0902.60060 10.1007/PL00004363
[14] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, 2004. 1058.90049[14] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, 2004. 1058.90049
[15] J. Bun, J. P. Bouchaud and M. Potters. Cleaning large correlation matrices: Tools from random matrix theory. Phys. Rep. 666 (2017) 1–109. 1359.15031 10.1016/j.physrep.2016.10.005[15] J. Bun, J. P. Bouchaud and M. Potters. Cleaning large correlation matrices: Tools from random matrix theory. Phys. Rep. 666 (2017) 1–109. 1359.15031 10.1016/j.physrep.2016.10.005
[16] M. Capitaine and C. Donati-Martin. Strong asymptotic freeness for Wigner and Wishart matrices. Indiana Univ. Math. J. 56 (2) (2007) 767–803. 1162.15013 10.1512/iumj.2007.56.2886[16] M. Capitaine and C. Donati-Martin. Strong asymptotic freeness for Wigner and Wishart matrices. Indiana Univ. Math. J. 56 (2) (2007) 767–803. 1162.15013 10.1512/iumj.2007.56.2886
[17] R. Couillet and M. Debbah. Random Matrix Methods for Wireless Communications. Cambridge University Press, Cambridge, 2011. 1252.94001[17] R. Couillet and M. Debbah. Random Matrix Methods for Wireless Communications. Cambridge University Press, Cambridge, 2011. 1252.94001
[18] A. Denjoy. Sur l’itération des fonctions analytiques. C. R. Acad. Sci. 182 (1926) 255–257. 52.0309.04[18] A. Denjoy. Sur l’itération des fonctions analytiques. C. R. Acad. Sci. 182 (1926) 255–257. 52.0309.04
[19] C. J. Earle and R. S. Hamilton. A fixed point theorem for holomorphic mappings. In Proc. Sympos. Pure Math. XVI 61–65. American Mathematical Society, Providence, 1970. 0205.14702[19] C. J. Earle and R. S. Hamilton. A fixed point theorem for holomorphic mappings. In Proc. Sympos. Pure Math. XVI 61–65. American Mathematical Society, Providence, 1970. 0205.14702
[20] N. El Karoui. Spectrum estimation for large dimensional covariance matrices using random matrix theory. Ann. Statist. 36 (6) (2008) 2757–2790. 1168.62052 10.1214/07-AOS581 euclid.aos/1231165184[20] N. El Karoui. Spectrum estimation for large dimensional covariance matrices using random matrix theory. Ann. Statist. 36 (6) (2008) 2757–2790. 1168.62052 10.1214/07-AOS581 euclid.aos/1231165184
[21] C. W. Groetsch. The Theory of Tikhonov Regularization for Fredholm Equations. Boston Pitman Publication, 1984. 0545.65034[21] C. W. Groetsch. The Theory of Tikhonov Regularization for Fredholm Equations. Boston Pitman Publication, 1984. 0545.65034
[22] T. Hasebe. Monotone convolution and monotone infinite divisibility from complex analytic viewpoint. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (1) (2010) 111–131. 1198.60014 10.1142/S0219025710003973[22] T. Hasebe. Monotone convolution and monotone infinite divisibility from complex analytic viewpoint. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (1) (2010) 111–131. 1198.60014 10.1142/S0219025710003973
[23] V. Kargin. A concentration inequality and a local law for the sum of two random matrices. Probab. Theory Related Fields 154 (3–4) (2012) 677–702. 1260.60015 10.1007/s00440-011-0381-4[23] V. Kargin. A concentration inequality and a local law for the sum of two random matrices. Probab. Theory Related Fields 154 (3–4) (2012) 677–702. 1260.60015 10.1007/s00440-011-0381-4
[24] W. Kong and G. Valiant. Spectrum estimation from samples. Ann. Statist. 45 (5) (2017) 2218–2247. 06821124 10.1214/16-AOS1525 euclid.aos/1509436833[24] W. Kong and G. Valiant. Spectrum estimation from samples. Ann. Statist. 45 (5) (2017) 2218–2247. 06821124 10.1214/16-AOS1525 euclid.aos/1509436833
[25] E. Lance. Hilbert $C^{*}$-Modules: A Toolkit for Operator Algebraists. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1995.[25] E. Lance. Hilbert $C^{*}$-Modules: A Toolkit for Operator Algebraists. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1995.
[27] O. Ledoit and M. Wolf. A well-conditioned estimator for large-dimensional covariancematrices. J. Multivariate Anal. 88 (2004) 365–411. 1032.62050 10.1016/S0047-259X(03)00096-4[27] O. Ledoit and M. Wolf. A well-conditioned estimator for large-dimensional covariancematrices. J. Multivariate Anal. 88 (2004) 365–411. 1032.62050 10.1016/S0047-259X(03)00096-4
[28] O. Ledoit and M. Wolf. Nonlinear shrinkage estimation of large-dimensional covariance matrices. Ann. Statist. 40 (2) (2012) 1024–1060. 1274.62371 10.1214/12-AOS989 euclid.aos/1342625460[28] O. Ledoit and M. Wolf. Nonlinear shrinkage estimation of large-dimensional covariance matrices. Ann. Statist. 40 (2) (2012) 1024–1060. 1274.62371 10.1214/12-AOS989 euclid.aos/1342625460
[29] O. Ledoit and M. Wolf. Numerical implementation of the QuEST function. Comput. Statist. Data Anal. 115 (2017) 199–223. 06917785 10.1016/j.csda.2017.06.004[29] O. Ledoit and M. Wolf. Numerical implementation of the QuEST function. Comput. Statist. Data Anal. 115 (2017) 199–223. 06917785 10.1016/j.csda.2017.06.004
[30] W. Li and J. Yao. A local moment estimator of the spectrum of a large dimensional covariance matrix. Statist. Sinica 24 (2014) 919–936. 1285.62060[30] W. Li and J. Yao. A local moment estimator of the spectrum of a large dimensional covariance matrix. Statist. Sinica 24 (2014) 919–936. 1285.62060
[31] H. Maassen. Addition of freely independent random variables. J. Funct. Anal. 106 (1992) 409–438. 0784.46047 10.1016/0022-1236(92)90055-N[31] H. Maassen. Addition of freely independent random variables. J. Funct. Anal. 106 (1992) 409–438. 0784.46047 10.1016/0022-1236(92)90055-N
[32] V. Marchenko and L. Pastur. Distribution of eigenvalues of some sets of random matrices. Math. USSR, Sb. 1 (1967) 457–486. 0162.22501 10.1070/SM1967v001n04ABEH001994[32] V. Marchenko and L. Pastur. Distribution of eigenvalues of some sets of random matrices. Math. USSR, Sb. 1 (1967) 457–486. 0162.22501 10.1070/SM1967v001n04ABEH001994
[33] X. Mestre. Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates. IEEE Trans. Inf. Theory 54 (11) (2008) 5113–5129. 1318.62191 10.1109/TIT.2008.929938[33] X. Mestre. Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates. IEEE Trans. Inf. Theory 54 (11) (2008) 5113–5129. 1318.62191 10.1109/TIT.2008.929938
[34] J. Mingo and R. Speicher. Free Probability and Random Matrices. Fields Institute Monographs. Amer. Math. Soc., Providence, RI, 2017. 1387.60005[34] J. Mingo and R. Speicher. Free Probability and Random Matrices. Fields Institute Monographs. Amer. Math. Soc., Providence, RI, 2017. 1387.60005
[35] A. Nica, D. Shlyaktenko and R. Speicher. Operator-valued distributions I: Characterizations of freeness. Int. Math. Res. Not. 29 (2002) 1509–1538. 1007.46052 10.1155/S1073792802201038[35] A. Nica, D. Shlyaktenko and R. Speicher. Operator-valued distributions I: Characterizations of freeness. Int. Math. Res. Not. 29 (2002) 1509–1538. 1007.46052 10.1155/S1073792802201038
[36] A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability. LMS Lecture Note Series 335. Cambridge University Press, Cambridge, 2006. 1133.60003[36] A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability. LMS Lecture Note Series 335. Cambridge University Press, Cambridge, 2006. 1133.60003
[37] J. Pennington, S. Schoenholz and S. Ganguli. The emergence of spectral universality in deep networks. In Proceedings of Machine Learning Research, 84 International Conference on Artificial Intelligence and Statistics, 2018.[37] J. Pennington, S. Schoenholz and S. Ganguli. The emergence of spectral universality in deep networks. In Proceedings of Machine Learning Research, 84 International Conference on Artificial Intelligence and Statistics, 2018.
[38] M. Popa and V. Vinnikov. Non-commutative functions and the non-commutative free Lévy–Hincin formula. Adv. Math. 236 (2013) 131–157. 1270.46060 10.1016/j.aim.2012.12.013[38] M. Popa and V. Vinnikov. Non-commutative functions and the non-commutative free Lévy–Hincin formula. Adv. Math. 236 (2013) 131–157. 1270.46060 10.1016/j.aim.2012.12.013
[39] N. R. Rao, J. A. Mingo, R. Speicher and A. Edelman. Statistical eigen-inference from large Wishart matrices. Ann. Statist. 36 (6) (2008) 2850–2885. 1168.62056 10.1214/07-AOS583 euclid.aos/1231165187[39] N. R. Rao, J. A. Mingo, R. Speicher and A. Edelman. Statistical eigen-inference from large Wishart matrices. Ann. Statist. 36 (6) (2008) 2850–2885. 1168.62056 10.1214/07-AOS583 euclid.aos/1231165187
[40] Ø. Ryan and M. Debbah. Free deconvolution for signal processing applications. In Proceedings of IEEE International Symposium of Information Theory (ISIT’ 07) 1846–1850. Nice, France, 2007.[40] Ø. Ryan and M. Debbah. Free deconvolution for signal processing applications. In Proceedings of IEEE International Symposium of Information Theory (ISIT’ 07) 1846–1850. Nice, France, 2007.
[41] Ø. Ryan and M. Debbah. Multiplicative free convolution and information-plus-Noise type matrices. Preprint. Available at arXiv:math/0702342.[41] Ø. Ryan and M. Debbah. Multiplicative free convolution and information-plus-Noise type matrices. Preprint. Available at arXiv:math/0702342.
[42] D. Shlyaktenko. Random Gaussian band matrices and freeness with amalgamation. Int. Math. Res. Not. 20 (1996) 1013–1025. 0872.15018 10.1155/S1073792896000633[42] D. Shlyaktenko. Random Gaussian band matrices and freeness with amalgamation. Int. Math. Res. Not. 20 (1996) 1013–1025. 0872.15018 10.1155/S1073792896000633
[43] R. Speicher. Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298 (1994) 611–628. 0791.06010 10.1007/BF01459754[43] R. Speicher. Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298 (1994) 611–628. 0791.06010 10.1007/BF01459754
[44] R. Speicher. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Amer. Math. Soc. 132 (627) (1998) x+88 pp. 0935.46056[44] R. Speicher. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Amer. Math. Soc. 132 (627) (1998) x+88 pp. 0935.46056
[45] W. Tarnowski, P. Warchol, S. Jastrzebski, J. Tabor and M. Nowak. Dynamical isometry is achieved in residual networks in a universal way for any activation function. In Proceedings of Machine Learning Research, 89 the 22nd International Conference on Artificial Intelligence and Statistics, 2019.[45] W. Tarnowski, P. Warchol, S. Jastrzebski, J. Tabor and M. Nowak. Dynamical isometry is achieved in residual networks in a universal way for any activation function. In Proceedings of Machine Learning Research, 89 the 22nd International Conference on Artificial Intelligence and Statistics, 2019.
[46] D. Voiculescu. Symmetries of some reduced free product $C^{*}$-algebras. In Operator Algebras and Their Connections with Topology and Ergodic Theory (Busteni, 1983) 556–588. Lecture Notes in Math. 1132. Springer, Berlin, 1985.[46] D. Voiculescu. Symmetries of some reduced free product $C^{*}$-algebras. In Operator Algebras and Their Connections with Topology and Ergodic Theory (Busteni, 1983) 556–588. Lecture Notes in Math. 1132. Springer, Berlin, 1985.
[47] D. Voiculescu. Addition of certain non-commutative random variables. J. Funct. Anal. 66 (1986) 323–346. 0651.46063 10.1016/0022-1236(86)90062-5[47] D. Voiculescu. Addition of certain non-commutative random variables. J. Funct. Anal. 66 (1986) 323–346. 0651.46063 10.1016/0022-1236(86)90062-5
[48] D. Voiculescu. Multiplication of certain non-commutative random variables. J. Oper. Theory 18 (1987) 223–235. 0662.46069[48] D. Voiculescu. Multiplication of certain non-commutative random variables. J. Oper. Theory 18 (1987) 223–235. 0662.46069
[49] D. Voiculescu. Limit laws for random matrices and free products. Invent. Math. 104 (1991) 201–220. 0736.60007 10.1007/BF01245072[49] D. Voiculescu. Limit laws for random matrices and free products. Invent. Math. 104 (1991) 201–220. 0736.60007 10.1007/BF01245072
[50] D. Voiculescu. The analogues of entropy and of Fisher’s information measure in free probability theory. I. Comm. Math. Phys. 155 (1) (1993) 71–92. 0781.60006 10.1007/BF02100050 euclid.cmp/1104253200[50] D. Voiculescu. The analogues of entropy and of Fisher’s information measure in free probability theory. I. Comm. Math. Phys. 155 (1) (1993) 71–92. 0781.60006 10.1007/BF02100050 euclid.cmp/1104253200
[51] D. Voiculescu. Operations on certain non-commutative operator-valued random variables. Astérisque 232 (1995) 243–275. 0839.46060[51] D. Voiculescu. Operations on certain non-commutative operator-valued random variables. Astérisque 232 (1995) 243–275. 0839.46060
[52] D. Voiculescu. The coalgebra of the free difference quotient and free probability. Int. Math. Res. Not. 2 (2000) 79–106. 0952.46038 10.1155/S1073792800000064[52] D. Voiculescu. The coalgebra of the free difference quotient and free probability. Int. Math. Res. Not. 2 (2000) 79–106. 0952.46038 10.1155/S1073792800000064
[53] E. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. 67 (1958) 325–327. 0085.13203 10.2307/1970008[53] E. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. 67 (1958) 325–327. 0085.13203 10.2307/1970008
[54] J. D. Williams. Analytic function theory for operator-valued free probability. J. Reine Angew. Math. 729 (2017) 119–149. 1381.46059[54] J. D. Williams. Analytic function theory for operator-valued free probability. J. Reine Angew. Math. 729 (2017) 119–149. 1381.46059
[55] J. Wolff. Sur l’itération des fonctions holomorphes dans une région, et dont les valeurs appartiennent a cette région. C. R. Acad. Sci. 182 (1926) 42–43.[55] J. Wolff. Sur l’itération des fonctions holomorphes dans une région, et dont les valeurs appartiennent a cette région. C. R. Acad. Sci. 182 (1926) 42–43.
