Abstract
In this paper, we propose a new type of a neural network which is inspired by Gabor systems from harmonic analysis. In this regard, we construct a class of sparsely connected neural networks utilizing the concept of time–frequency shifts, and we show that their approximation error rates can be tied to the number of modulations in the corresponding Gabor frame and to the smoothness of the input function. Furthermore, we show that such networks are easily implementable and we illustrate their performance with some numerical examples.
Similar content being viewed by others
References
LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521, 436 (2015)
Balan, R., Singh, M., Zou, D.: Lipschitz properties for deep convolutional networks. (2017) arXiv preprint arXiv:1701.05217
Shaham, U., Cloninger, A., Coifman, R.R.: Provable approximation properties for deep neural networks. Appl. Comput. Harmon. Anal. 44, 537–557 (2016)
Bolcskei, H., Grohs, P., Kutyniok, G., Petersen, P.: Optimal approximation with sparsely connected deep neural networks. SIAM J. Math. Data Sci. 1, 8–45 (2019)
Mallat, S.: Group invariant scattering. Commun. Pure Appl. Math. 65, 1331–1398 (2012)
Mallat, S.: Understanding deep convolutional networks. Philos. Trans. R. Soc. A 374, 20150203 (2016)
Wiatowski, T., Bölcskei, H.: A mathematical theory of deep convolutional neural networks for feature extraction. IEEE Trans. Inf. Theory 64, 1845–1866 (2017)
Labate, D.: A unified characterization of reproducing systems generated by a finite family. J. Geomet. Anal. 12, 469–491 (2002)
Hernández, E., Labate, D., Weiss, G.: A unified characterization of reproducing systems generated by a finite family, ii. J. Geomet. Anal. 12, 615–662 (2002)
Hernández, E., Labate, D., Weiss, G., Wilso, E.: Oversampling, quasi affine frames and wave packets. Appl. Comput. Harmon. Anal. 16, 111–147 (2004)
Labate, D., Weiss, G.: Continuous and discrete reproducing systems that arise from translations: theory and applications of composite wavelets. Four Short Courses on Harmonic Analysis, pp. 87–130. Berlin, Springer (2010)
Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2, 303–314 (1989)
Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Netw. 4, 251–257 (1991)
Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39, 930–945 (1993)
Mhaskar, H.N.: Neural networks for optimal approximation of smooth and analytic functions. Neural Comput. 8, 164–177 (1996)
Chui, C.K., Mhaskar, H.N.: Deep nets for local manifold learning. arXiv preprint arXiv:1607.07110(2016)
Czaja, W., Manning, B., Murphy, J.M., Stubbs, K.: Discrete directional Gabor frames. Appl. Comput. Harmon. Anal. 45, 1–21 (2016)
Czaja, W., Li, W.: Analysis of time–frequency scattering transforms. Appl. Comput. Harmon. Anal. 47, 149–171 (2019)
Czaja, W., Li, W.: Rotationally invariant time–frequency scattering transforms. J. Fourier Anal. Appl. 26, 4 (2020)
Grohs, P., Perekrestenko, D., Elbrächter, D., Bölcskei, H.: Deep neural network approximation theory. (2019) arXiv preprint arXiv:1901.02220
Li, Y.: Feature extraction in image processing and deep learning. Ph.D. thesis, Univeristy of Maryland, College Park (2018)
Benedetto, J.J.: Harmonic Analysis and Applications, vol. 23. CRC Press, Boca Raton (1996)
Gabor, D.: Theory of communication. J. Inst. Electr. Eng. III 93, 429–457 (1946)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Springer, Berlin (2013)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Walnut, D.F.: Continuity properties of the Gabor frame operator. J. Math. Anal. Appl. 165, 479–504 (1992)
Feichtinger, H.G., Gröchenig, K., Walnut, D.: Wilson bases and modulation spaces. Math. Nachrich. 155, 7–17 (1992)
Casazza, P.G., Christensen, O., Janssen, A.: Weyl–Heisenberg frames, translation invariant systems and the walnut representation. J. Funct. Anal. 180, 85–147 (2001)
Christensen, O., Kim, H.O., Kim, R.Y.: Regularity of dual Gabor windows. Abstract and Applied Analysis, vol. 2013, Hindawi (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Czaja, W., Li, Y. Gabor Neural Networks with Proven Approximation Properties. J Geom Anal 31, 8999–9015 (2021). https://doi.org/10.1007/s12220-020-00575-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-020-00575-z