Abstract
In this paper we present an approach to determine the smallest possible number of neurons in a layer of a neural network in such a way that the topology of the input space can be learned sufficiently well. We introduce a general procedure based on persistent homology to investigate topological invariants of the manifold on which we suspect the data set. We specify the required dimensions precisely, assuming that there is a smooth manifold on or near which the data are located. Furthermore, we require that this space is connected and has a commutative group structure in the mathematical sense. These assumptions allow us to derive a decomposition of the underlying space whose topology is well known. We use the representatives of the k-dimensional homology groups from the persistence landscape to determine an integer dimension for this decomposition. This number is the dimension of the embedding that is capable of capturing the topology of the data manifold. We derive the theory and validate it experimentally on toy data sets.
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Notes
- 1.
Invertible architectures guarantee the same differentiable structure during learning. Due to the construction of trivially invertible neural networks the embedding dimension is doubled, see [8].
References
Abadi, M., et al.: TensorFlow: large-scale machine learning on heterogeneous systems (2015). https://www.tensorflow.org/, software available from tensorflow.org
Bartlett, P., Harvey, N., Liaw, C., Mehrabian, A.: Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks. J. Mach. Learn. Res. 20, 1–17 (2019)
Boissonnat, J.D., Chazal, F., Yvinec, M.: Geometric and Topological Inference. Cambridge University Press, Cambridge (2018)
Bubenik, P.: Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16, 77–102 (2015)
Chollet, F., et al.: Keras (2015). https://keras.io
Cohen, T.S., Geiger, M., Köhler, J., Welling, M.: Spherical cnns. In: 6th International Conference on Learning Representations (2018)
Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control, Signals Syst. 5(4), 455 (1992)
Dinh, L., Sohl-Dickstein, J., Bengio, S.: Density estimation using real NVP. In: 5th International Conference on Learning Representations (2017)
Edelsbrunner, H., Harer, J.: Persistent homology - a survey. Contemp. Math. 453, 257–282 (2008)
Futagami, R., Yamada, N., Shibuya, T.: Inferring underlying manifold of data by the use of persistent homology analysis. In: 7th International Workshop on Computational Topology in Image Context, pp. 40–53 (2019)
Gruenberg, K.W.: The universal coefficient theorem in the cohomology of groups. J. London Math. Soc. 1(1), 239–241 (1968)
Deo, S.: Algebraic Topology. TRM, vol. 27. Springer, Singapore (2018). https://doi.org/10.1007/978-981-10-8734-9
Hauser, M., Gunn, S., Saab Jr., S., Ray, A.: State-space representations of deep neural networks. Neural Comput. 31(3), 538–554 (2019)
Hauser, M., Ray, A.: Principles of riemannian geometry in neural networks. Adv. Neural Inf. Process. Syst. 30, 2807–2816 (2017)
Johnson, J.: Deep, skinny neural networks are not universal approximators. In: 7th International Conference on Learning Representations (2019)
Kingma, D., Ba, J.: Adam: a method for stochastic optimization. In: 3rd International Conference on Learning Representations (2015)
Krizhevsky, A.: Learning multiple layers of features from tiny images (2009)
Lee, J.: Smooth manifolds. Introduction to Smooth Manifolds. Springer, New York (2013)
Lin, H., Jegelka, S.: Resnet with one-neuron hidden layers is a universal approximator. Adv. Neural Inf. Process. Syst. 31, 6172–6181 (2018)
Melodia, L., Lenz, R.: Persistent homology as stopping-criterion for voronoi interpolation. In: Lukić, T., Barneva, R.P., Brimkov, V.E., Čomić, L., Sladoje, N. (eds.) IWCIA 2020. LNCS, vol. 12148, pp. 29–44. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-51002-2_3
Onischtschik, A.L., Winberg, E.B., Minachin, V.: Lie Groups and Lie Algebras I. Springer (1993)
Raghu, M., Poole, B., Kleinberg, J., Ganguli, S., Dickstein, J.S.: On the expressive power of deep neural networks. In: 34th International Conference on Machine Learning, pp. 2847–2854 (2017)
Stone, M.H.: The generalized weierstrass approximation theorem. Math. Mag. 21(5), 237–254 (1948)
The GUDHI Project: GUDHI user and reference manual (2020). https://gudhi.inria.fr/doc/3.1.1/
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)
Acknowledgements
We thank Christian Holtzhausen, David Haller and Noah Becker for proofreading and anonymous reviewers for their constructive criticism and corrections. This work was partially supported by Siemens Energy AG.
Code and Data. The implementation, the data sets and experimental results can be found at: https://github.com/karhunenloeve/NTOPL.
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Melodia, L., Lenz, R. (2021). Estimate of the Neural Network Dimension Using Algebraic Topology and Lie Theory. In: Del Bimbo, A., et al. Pattern Recognition. ICPR International Workshops and Challenges. ICPR 2021. Lecture Notes in Computer Science(), vol 12665. Springer, Cham. https://doi.org/10.1007/978-3-030-68821-9_2
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