Abstract
We introduce a family of bounded, multiscale distances on any space equipped with an operator semigroup. In many examples, these distances are equivalent to a snowflake of the natural distance on the space. Under weak regularity assumptions on the kernels defining the semigroup, we derive simple characterizations of the Hölder–Lipschitz norm and its dual with respect to these distances. As the dual norm of the difference of two probability measures is the Earth Mover’s Distance (EMD) between these measures, our characterizations give simple formulas for a metric equivalent to EMD. We extend these results to the mixed Hölder–Lipschitz norm and its dual on the product of spaces, each of which is equipped with its own semigroup. Additionally, we derive an approximation theorem for mixed Lipschitz functions in this setting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003)
Bui, H.Q., Duong, X.T., Yan, L.: Calderóon reproducing formulas and new Besov spaces associated with operators. Adv. Math. 229(4), 2449–2502 (2012)
Butzer, P.L., Berens, H.: Semi-groups of Operators and Approximation. Springer, Berlin (1967)
Charikar, M.S.: Similarity estimation techniques from rounding algorithms. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 380–388 (2002)
Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press, Orlando (1984)
Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006)
Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Nadler, B., Warner, F., Zucker, S.W.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc. Natl. Acad. Sci. USA 102(21), 7426–7431 (2005)
Coifman, R.R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21(1), 53–94 (2006)
Dachkovski, S.: Anisotropic function spaces and related semi-linear hypoelliptic equations. Math. Nachr. 248(1), 40–61 (2003)
Do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)
Donoho, D.L., Johnstone, I.M.: Neo-classical minimax problems, thresholding and adaptive function estimation. Bernoulli 2(1), 39–62 (1996)
Dudley, R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002)
Farkas, W.: Atomic and subatomic decompositions in anisotropic function spaces. Math. Nachr. 209, 83–113 (2000)
Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley, New York (1999)
Goldberg, M.J., Kim, S.: Some remarks on diffusion distances. J. Appl. Math. 2010, 464815 (2010)
Goldberg, M.J., Kim, S.: An efficient tree-based computation of a met ric comparable to a natural diffusion distance. Appl. Comput. Harmon. Anal. 33(2), 261–281 (2012)
Grigor’yan, A., Liu, L.: Heat kernel and Lipschitz-Besov spaces. Forum Math. (2014)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)
Indyk, P., Thaper, N.: Fast image retrieval via embeddings. In: 3rd International Workshop on Statistical and Computational Theories of Vision (at ICCV) (2003)
Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(1), 153–201 (1986)
Little, A.V., Jung, Y.M., Maggioni, M.: Multiscale estimation of intrinsic dimensionality of data sets. In: AAAI fall symposium: manifold learning and its applications (2009)
Marinai, S., Miotti, B., Soda, G.: Using earth mover’s distance in the bag-of- visual-words model for mathematical symbol retrieval. In: 2011 International Conference on Document Analysis and Recognition, pp. 1309–1313 (2011)
Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)
Neumann, M.H.: Multivariate wavelet thresholding in anisotropic function spaces. Stat. Sin. 10(2), 399–432 (2000)
Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval. Int. J. Comput. Vis. 40(2), 99–121 (2000)
Sandler, R., Lindenbaum, M.: Nonnegative matrix factorization with earth mover’s distance metric for image analysis. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1590–1602 (2011)
Schmeisser, H., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester (1987)
Shirdhonkar, S., Jacobs, D.W.: Approximate earth mover’s distance in linear time. In: IEEE conference on computer vision and pattern recognition, pp. 1–8 (2008)
Stein, E.M.: Topics in Harmonic Analysis, Related to the Littlewood-Paley Theory. Princeton University Press, Princeton (1970)
Temlyakov, V.N.: Approximation of Periodic Functions. Computational Mathematics and Analysis Series. Nova Science Publishers Inc, New York (1993)
Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)
Wan, X.: A novel document similarity measure based on earth mover’s distance. Inf. Sci. 177(18), 3718–3730 (2007)
Zolotarev, V.M.: One-Dimensional Stable Distributions. American Mathematical Society, Providence (1986)
Acknowledgments
The authors thank Maxim Goldberg for many insightful and stimulating discussions throughout the development of this work. We also thank the anonymous reviewers for their helpful comments. Ronald Coifman was supported by NSF Grant No. 1309858.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Massimo Fornasier.
Rights and permissions
About this article
Cite this article
Leeb, W., Coifman, R. Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups, with Applications to Earth Mover’s Distance. J Fourier Anal Appl 22, 910–953 (2016). https://doi.org/10.1007/s00041-015-9439-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-015-9439-5