An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning

doi:10.2307/1968861

J. H. C. Whitehead, On C 1 C^{1} -complexes , Ann. of Math. (2) 41 (1940) 809–824 · 1940

Earlier work this paper cites.

L. Kantorovitch, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942) 199–201

1942

Earlier work this paper cites.

doi:10.1002/j.1538-7305.1948.tb01338.x

C. E. Shannon, A mathematical theory of communication , Bell System Tech. J. 27 (1948) 379–423, 623–656 · 1948

Earlier work this paper cites.

A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, in: Séminaire Bourbaki, Vol. 2, Soc. Math. France, Paris, 1954, pp. Exp. No. 69, 193–200

1954

Earlier work this paper cites.

doi:10.2307/1993204

J. Milnor, On spaces having the homotopy type of a CW {\rm CW} -complex , Trans. Amer. Math. Soc. 90 (1959) 272–280 · 1959

Earlier work this paper cites.

doi:10.4064/sm-24-2-113-190

A.-P. Calderón, Intermediate spaces and interpolation, the complex method , Studia Math. 24 (1964) 113–190 · 1964

Earlier work this paper cites.

doi:10.2307/2033897

C. Todd, Stone-Weierstrass theorems for the strict topology , Proc. Amer. Math. Soc. 16 (1965) 654–659 · 1965

Earlier work this paper cites.

E. H. Spanier, Algebraic topology, Springer-Verlag, New York, 1995, corrected reprint of the 1966 original

1966

Earlier work this paper cites.

R. D. Anderson, On topological infinite deficiency , Michigan Math. J. 14 (1967) 365–383. URL http://projecteuclid.org/euclid.mmj/1028999787

1967

Earlier work this paper cites.

J. a. B. Prolla, Bishop’s generalized Stone-Weierstrass theorem for weighted spaces, Memórias de Matemática da Universidade Federal do Rio de Janeiro, No. 8. [Mathematical Reports of the Federal University of Rio de Janeiro, No. 8], Universidade Federal do Rio de Janeiro, Instituto de Matemática, Rio de Janeiro, 1972

1972

Earlier work this paper cites.

J.-L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972, translated from the French by P. Kenneth

1972

Earlier work this paper cites.

O. Lehto, Quasiconformal mappings in the plane, in: Lectures on quasiconformal mappings, Springer, 1975, pp. 1–43. Dept. Math., Univ. Maryland, Lecture Note, No. 14

1975

Earlier work this paper cites.

C. a. Bessaga, A. Peł czyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975

1975

Earlier work this paper cites.

M. W. Hirsch, Differential topology, Vol. 33 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1994, corrected reprint of the 1976 original

1976

Earlier work this paper cites.

T. Kato, Perturbation theory for linear operators, 2nd Edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976

1976

Earlier work this paper cites.

J. Bergh, J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976

1976

Earlier work this paper cites.

H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of l ​ _ ​ 2 l\_2 -manifolds, Fundamenta Mathematicae 101 (2) (1978) 93–110

1978

Earlier work this paper cites.

H. Triebel, Interpolation theory, function spaces, differential operators, Vol. 18 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978

1978

Earlier work this paper cites.

V. J. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen. 14 (2) (1980) 25–34

1980

Earlier work this paper cites.

doi:10.1016/0021-9045(82)90060-0

W. A. Dahmen, Adaptive approximation by multivariate smooth splines , J. Approx. Theory 36 (2) (1982) 119–140 · 1982

Earlier work this paper cites.

M. Gromov, Filling Riemannian manifolds , J. Differential Geom. 18 (1) (1983) 1–147. URL http://projecteuclid.org/euclid.jdg/1214509283

1983

Earlier work this paper cites.

doi:10.1016/0021-9045(84)90119-9

B. Wood, Uniform approximation by linear combinations of Bernstein-type polynomials , J. Approx. Theory 41 (1) (1984) 51–55 · 1984

Earlier work this paper cites.

doi:10.1007/978-3-642-69894-1

A. Pinkus, n n -widths in approximation theory , Vol. 7 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1985 · 1985

Earlier work this paper cites.

I. Lasiecka, J.-L. Lions, R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65 (2) (1986) 149–192

1986

Earlier work this paper cites.

doi:10.1016/0021-9045(87)90109-2

D. D. Pence, Further asymptotic properties of best approximation by splines , J. Approx. Theory 49 (1) (1987) 1–17 · 1987

Earlier work this paper cites.

N. I. Fisher, T. Lewis, B. J. J. Embleton, Statistical analysis of spherical data, Cambridge University Press, Cambridge, 1993, revised reprint of the 1987 original

1987

Earlier work this paper cites.

H. Busemann, B. B. Phadke, Spaces with distinguished geodesics, Vol. 108 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1987

1987

Earlier work this paper cites.

doi:10.1016/0167-6377(88)90030-2

H. Konno, T. Kuno, Best piecewise constant approximation of a function of single variable , Oper. Res. Lett. 7 (4) (1988) 205–210 · 1988

Earlier work this paper cites.

doi:10.1080/00036818808839730

G. Alessandrini, Stable determination of conductivity by boundary measurements , Appl. Anal. 27 (1-3) (1988) 153–172 · 1988

Earlier work this paper cites.

K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural Netw. 2 (5) (1989) 359–366

1989

Earlier work this paper cites.

G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems 2 (4) (1989) 303–314

1989

Earlier work this paper cites.

A. M. Avakyan, Best approximation of classes of functions with a bounded first derivative by piecewise constants in the one-dimensional and two-dimensional cases, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 25 (3) (1990) 215–235, 309

1990

Earlier work this paper cites.

J. Cheriyan, T. Hagerup, A randomized maximum-flow algorithm, Tech. rep., Cornell University Operations Research and Industrial Engineering (1991)

1991

Earlier work this paper cites.

B. Jawerth, M. Milman, Wavelets and best approximation in Besov spaces, in: Interpolation spaces and related topics (Haifa, 1990), Vol. 5 of Israel Math. Conf. Proc., Bar-Ilan Univ., Ramat Gan, 1992, pp. 107–112

1992

Earlier work this paper cites.

R. A. DeVore, G. G. Lorentz, Constructive approximation, Vol. 303 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1993

1993

Earlier work this paper cites.

B. Bru, H. Heinich, J.-C. Lootgieter, Distances de Lévy et extensions des théoremès de la limite centrale et de Glivenko-Cantelli, Publ. Inst. Statist. Univ. Paris 37 (3-4) (1993) 29–42

1993

Earlier work this paper cites.

doi:10.1007/BF02071385

R. A. DeVore, G. Kyriazis, D. Leviatan, V. M. Tikhomirov, Wavelet compression and nonlinear n n -widths , Adv. Comput. Math. 1 (2) (1993) 197–214 · 1993

Earlier work this paper cites.

T. Chen, H. Chen, Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems, IEEE Transactions on Neural Networks 6 (4) (1995) 911–917

1995

Earlier work this paper cites.

doi:10.1006/jfan.1997.3188

P. Stefanov, G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media , J. Funct. Anal. 154 (2) (1998) 330–358 · 1997

Earlier work this paper cites.

T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310

1998

Earlier work this paper cites.

doi:10.1016/S0040-9383(98)00059-7

L. Lechuga, A. Murillo, Complexity in rational homotopy , Topology 39 (1) (2000) 89–94 · 2000

Earlier work this paper cites.

doi:10.1007/BFb0103945

S. Graf, H. Luschgy, Foundations of quantization for probability distributions , Vol. 1730 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000 · 2000

Earlier work this paper cites.

doi:10.1090/coll/048

Y. Benyamini, J. Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1 , Vol. 48 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2000 · 2000

Earlier work this paper cites.

J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001

2001

Earlier work this paper cites.

doi:10.1007/PL00000526

J. Latschev, Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold , Arch. Math. (Basel) 77 (6) (2001) 522–528 · 2001

Earlier work this paper cites.

J. van Mill, The infinite-dimensional topology of function spaces, Vol. 64 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 2001

2001

Earlier work this paper cites.

doi:10.1088/0266-5611/17/5/313

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation , Inverse Problems 17 (5) (2001) 1435–1444 · 2001

Earlier work this paper cites.

doi:10.1201/9781420036220

A. Katchalov, Y. Kurylev, M. Lassas, Inverse boundary spectral problems , Vol. 123 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001 · 2001

Earlier work this paper cites.

doi:10.1201/9781420036220

A. Katchalov, Y. Kurylev, M. Lassas, Inverse boundary spectral problems , Vol. 123 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001 · 2001

Earlier work this paper cites.

doi:10.1090/gsm/033

D. Burago, Y. Burago, S. Ivanov, A course in metric geometry , Vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001 · 2001

Earlier work this paper cites.

R. Wilson, Four colors suffice, Princeton Science Library, Princeton University Press, Princeton, NJ, 2014, how the map problem was solved, Revised color edition of the 2002 original, with a new foreword by Ian Stewart

2002

Earlier work this paper cites.

M. Balasubramanian, E. L. Schwartz, The isomap algorithm and topological stability, Science 295 (5552) (2002) 7–7

2002

Earlier work this paper cites.

doi:10.1016/S0166-8641(02)00340-1

A. Garvín, L. Lechuga, The computation of the Betti numbers of an elliptic space is a NP-hard problem , Topology Appl. 131 (3) (2003) 235–238 · 2003

Earlier work this paper cites.

G. Godefroy, N. J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (1) (2003) 121–141, dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday

2003

Earlier work this paper cites.

doi:10.1090/conm/338/06080

K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature , in: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Vol. 338 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2003, pp. 357–390 · 2003

Earlier work this paper cites.

doi:10.4064/sm159-1-6

G. Godefroy, N. J. Kalton, Lipschitz-free Banach spaces , Studia Math. 159 (1) (2003) 121–141, dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday · 2003

Earlier work this paper cites.

doi:10.1112/S0025579300014480

A. Naor, A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between L p L_{p} spaces , Mathematika 48 (1-2) (2001) 253–271 (2003) · 2003

Earlier work this paper cites.

doi:10.1016/j.jat.2004.03.004

B. Della Vecchia, G. Mastroianni, J. Szabados, Weighted approximation of functions on the real line by Bernstein polynomials , J. Approx. Theory 127 (2) (2004) 223–239 · 2004

Earlier work this paper cites.

doi:10.1007/BF02772227

J. Galindo, M. Sanchis, Stone-Weierstrass theorems for group-valued functions , Israel J. Math. 141 (2004) 341–354 · 2004

Earlier work this paper cites.

N. J. Kalton, Spaces of Lipschitz and Hölder functions and their applications, Collect. Math. 55 (2) (2004) 171–217

2004

Earlier work this paper cites.

D. B. Bazarkhanov, On the best N N -term approximations by wavelets in a mixed norm, Mat. Zh. 4 (1(11)) (2004) 17–20

2004

Earlier work this paper cites.

P. T. Fletcher, C. Lu, S. M. Pizer, S. Joshi, Principal geodesic analysis for the study of nonlinear statistics of shape, IEEE transactions on medical imaging 23 (8) (2004) 995–1005

2004

Earlier work this paper cites.

doi:10.1215/S0012-7094-04-12332-2

P. Stefanov, G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity , Duke Math. J. 123 (3) (2004) 445–467 · 2004

Earlier work this paper cites.

doi:10.1109/TIT.2005.862083

E. J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , IEEE Trans. Inform. Theory 52 (2) (2006) 489–509 · 2005

Earlier work this paper cites.

doi:10.1016/j.jat.2005.05.004

V. Timofte, Stone-Weierstrass theorems revisited , J. Approx. Theory 136 (1) (2005) 45–59 · 2005

Cited alongside, same era.

doi:10.1016/j.matpur.2005.02.004

M. Bellassoued, M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation , J. Math. Pures Appl. (9) 85 (2) (2006) 193–224 · 2005

Cited alongside, same era.

doi:10.1155/IMRN.2005.1047

P. Stefanov, G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map , Int. Math. Res. Not. (17) (2005) 1047–1061 · 2005

Cited alongside, same era.

doi:10.1007/BF02785360

G. Lancien, B. Randrianantoanina, On the extension of Hölder maps with values in spaces of continuous functions , Israel J. Math. 147 (2005) 75–92 · 2005

Cited alongside, same era.

X. Pennec, P. Fillard, N. Ayache, A riemannian framework for tensor computing, International Journal of computer vision 66 (1) (2006) 41–66

2006

Cited alongside, same era.

doi:10.1090/proc/13811

P. Hajł asz, The ( n + 1 ) (n+1) -Lipschitz homotopy group of the Heisenberg group ℍ n \mathbb{H}^{n} , Proc. Amer. Math. Soc. 146 (3) (2018) 1305–1308 · 2018

Later among the works it cites.

N. Weaver, Lipschitz algebras, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018, second edition of [ MR1832645]

2018

Later among the works it cites.

G. Basso, Fixed point and lipschitz extension theorems for barycentric metric spaces, Ph.D. thesis, ETH Zurich (2019)

2019

Later among the works it cites.

doi:10.33205/cma.481186

S. G. Gal, S. Trifa, Quantitative estimates for L p L^{p} -approximation by Bernstein-Kantorovich-Choquet polynomials with respect to distorted Lebesgue measures , Constr. Math. Anal. 2 (1) (2019) 15–21 · 2019

Later among the works it cites.

F. J. Király, H. Oberhauser, Kernels for sequentially ordered data, J. Mach. Learn. Res. 20 (2019) Paper No. 31, 45

2019

Later among the works it cites.