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We survey current developments in the approximation theory of sequence modelling in machine learning.
Theory of Functionals and of Integral and Integro-differential Equations
Vito Volterra · 1930
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A brief survey on sequence classification
Zhengzheng Xing, Jian Pei, and Eamonn Keogh · 1931
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Dimension of metric spaces and Hilbert’s problem 13
Phillip A. Ostrand · 1936
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Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks
Tianping Chen and Hong Chen · 1941
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Learning long-term dependencies with gradient descent is difficult
Y. Bengio, P. Simard, and P. Frasconi · 1941
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Analytical Foundations of Volterra Series
Stephen Boyd, L. O. Chua, and C. A. Desoer · 1984
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Fading memory and the problem of approximating nonlinear operators with Volterra series
S. Boyd and L. Chua · 1985
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Learning representations by back-propagating errors
David E. Rumelhart, Geoffrey E. Hinton, and Ronald J. Williams · 1986
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Approximation by superpositions of a sigmoidal function
George Cybenko · 1989
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Neural Nets As Systems Models And Controllers
Eduardo Sontag · 1992
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Neural net approximation
Andrew R. Barron · 1992
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Approximation of dynamical systems by continuous time recurrent neural networks
Ken-ichi Funahashi and Yuichi Nakamura · 1993
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Universal approximation bounds for superpositions of a sigmoidal function
Andrew R. Barron · 1993
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A sharp Bernstein-type inequality for exponential sums
Tamás Erdélyi and Peter Borwein · 1996
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Long Short-Term Memory
Sepp Hochreiter and Jürgen Schmidhuber · 1997
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Nonlinear approximation
Ronald A DeVore · 1998
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Neural network approximation of continuous functionals and continuous functions on compactifications
M. B. Stinchcombe · 1999
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Modeling of continuous time dynamical systems with input by recurrent neural networks
T.W.S. Chow and Xiao-Dong Li · 2000
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An introduction to the proper orthogonal decomposition
Anindya Chatterjee · 2000
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Approximation of Functions
George G. Lorentz · 2005
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Approximation of dynamical time-variant systems by continuous-time recurrent neural networks
Xiao-Dong Li, J.K.L. Ho, and T.W.S. Chow · 2005
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Recurrent Neural Networks Are Universal Approximators
Anton Maximilian Schäfer and Hans Georg Zimmermann · 2006
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Modelling Financial Time Series
Stephen J. Taylor · 2008
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Tensor Decompositions and Applications
Tamara G. Kolda and Brett W. Bader · 2009
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A Wavelet Tour of Signal Processing: The Sparse Way
S. G. Mallat · 2009
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Fourier Analysis: An Introduction , volume 1
Elias M. Stein and Rami Shakarchi · 2011
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Theory of Approximation
N. I. Achieser · 2013
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Recurrent Continuous Translation Models
Nal Kalchbrenner and Phil Blunsom · 2013
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Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation
Kyunghyun Cho, Bart van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio · 2014
Cited alongside, same era.
Sequence to sequence learning with neural networks
Ilya Sutskever, Oriol Vinyals, and Quoc V. Le · 2014
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Document Modeling with Gated Recurrent Neural Network for Sentiment Classification
Duyu Tang, Bing Qin, and Ting Liu · 2015
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Neural Machine Translation by Jointly Learning to Align and Translate
Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio · 2016
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On the Relationship between Self-Attention and Convolutional Layers
Jean-Baptiste Cordonnier, Andreas Loukas, and Martin Jaggi · 2020
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Understanding Generalization in Recurrent Neural Networks
Zhuozhuo Tu, Fengxiang He, and Dacheng Tao · 2020
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Neural controlled differential equations for irregular time series
Patrick Kidger, James Morrill, James Foster, and Terry Lyons · 2020
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Neural operator: Learning maps between function spaces
Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar · 2021
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Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis · 2021
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Wavenet: A generative model for raw audio
Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu · 2016
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Attention is All you Need
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin · 2017
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ImageNet classification with deep convolutional neural networks
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton · 2017
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Temporal Convolutional Networks for Action Segmentation and Detection
Colin Lea, Michael D. Flynn, Rene Vidal, Austin Reiter, and Gregory D. Hager · 2017
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Reservoir Computing Universality With Stochastic Inputs
Lukas Gonon and Juan-Pablo Ortega · 2018
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An empirical evaluation of generic convolutional and recurrent networks for sequence modeling
Shaojie Bai, J. Zico Kolter, and Vladlen Koltun · 2018
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Fading memory echo state networks are universal
Lukas Gonon and Juan-Pablo Ortega · 2021
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Approximation Bounds for Random Neural Networks and Reservoir Systems
Lukas Gonon, Lyudmila Grigoryeva, and Juan-Pablo Ortega · 2021
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Approximation Theory of Convolutional Architectures for Time Series Modelling
Haotian Jiang, Zhong Li, and Qianxiao Li · 2021
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Attention is not all you need: Pure attention loses rank doubly exponentially with depth
Yihe Dong, Jean-Baptiste Cordonnier, and Andreas Loukas · 2021
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On the Provable Generalization of Recurrent Neural Networks
Lifu Wang, Bo Shen, Bo Hu, and Xing Cao · 2021
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Neural sdes as infinite-dimensional gans
Patrick Kidger, James Foster, Xuechen Li, and Terry J. Lyons · 2021
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Neural stochastic PDEs: Resolution-invariant learning of continuous spatiotemporal dynamics
Cristopher Salvi, Maud Lemercier, and Andris Gerasimovics · 2021
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Neural rough differential equations for long time series
James Morrill, Cristopher Salvi, Patrick Kidger, and James Foster · 2021
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Chaotic Hedging with Iterated Integrals and Neural Networks, September 2022
Ariel Neufeld and Philipp Schmocker · 2022
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Universal approximation theorems for continuous functions of c\‘adl\‘ag paths and L\’evy-type signature models, August 2022
Christa Cuchiero, Francesca Primavera, and Sara Svaluto-Ferro · 2022
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Minimal Width for Universal Property of Deep RNN
Geonho Hwang and Myungjoo Kang · 2022
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Learnability of convolutional neural networks for infinite dimensional input via mixed and anisotropic smoothness
Sho Okumoto and Taiji Suzuki · 2022
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On the Universal Approximation Property of Deep Fully Convolutional Neural Networks
Ting Lin, Zuowei Shen, and Qianxiao Li · 2022
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Universal Approximations of Invariant Maps by Neural Networks
Dmitry Yarotsky · 2022
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Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer, December 2022
Beatrice Acciaio, Anastasis Kratsios, and Gudmund Pammer · 2022
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Luca Galimberti, Giulia Livieri, and Anastasis Kratsios · 2022
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Neural operator with regularity structure for modeling dynamics driven by spdes
Peiyan Hu, Qi Meng, Bingguang Chen, Shiqi Gong, Yue Wang, Wei Chen, Rongchan Zhu, Zhi-Ming Ma, and Tie-Yan Liu · 2022
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On the choice of interpolation scheme for neural CDEs
James Morrill, Patrick Kidger, Lingyi Yang, and Terry Lyons · 2022
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Neural networks in Fréchet spaces
Fred Espen Benth, Nils Detering, and Luca Galimberti · 2023
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Neural SDEs for Conditional Time Series Generation and the Signature-Wasserstein-1 metric
Pere Díaz Lozano, Toni Lozano Bagén, and Josep Vives · 2023
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