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We build a rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators.
Theory and application of digital signal processing
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A practical guide to splines , volume 27
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Structural shape optimization with geometric description and adaptive mesh refinement
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Leo Breiman · 1993
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Convex functions
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Autoencoders, minimum description length and helmholtz free energy
Geoffrey E Hinton and Richard S Zemel · 1994
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A minimum description length framework for unsupervised learning
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Neural networks for pattern recognition
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Factorial hidden markov models
Zoubin Ghahramani and Michael I Jordan · 1996
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Flat minima
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The mnist database of handwritten digits
Yann LeCun · 1998
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Accurate piecewise linear continuous approximations to one-dimensional curves: Error estimates and algorithms
Hiroaki Nishikawa · 1998
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A Wavelet Tour of Signal Processing
Stéphane Mallat · 1999
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Bayesian tree-structured image modeling using wavelet-domain hidden markov models
J. K. Romberg, H. Choi, and Richard G Baraniuk · 1999
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Fast texture synthesis using tree-structured vector quantization
Li-Yi Wei and Marc Levoy · 2000
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Orthogonal matched filter detection
Yonina C Eldar and Alan V Oppenheim · 2001
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Convex optimization
Stephen Boyd and Lieven Vandenberghe · 2004
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Framewise phoneme classification with bidirectional lstm networks
Alex Graves and Jürgen Schmidhuber · 2005
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Extracting and composing robust features with denoising autoencoders
Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol · 2008
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S Jayaraman, S Esakkirajan, and T Veerakumar · 2009
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Convex piecewise-linear fitting
Alessandro Magnani and Stephen P Boyd · 2009
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Data clustering: 50 years beyond k-means
Anil K Jain · 2010
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Adaptive subgradient methods for online learning and stochastic optimization
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Matthew D Zeiler and Rob Fergus · 2014
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An exploration of softmax alternatives belonging to the spherical loss family
Alexandre de Brébisson and Pascal Vincent · 2015
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Batch normalization: accelerating deep network training by reducing internal covariate shift
S. Ioffe and C. Szegedy · 2015
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Training very deep networks
Rupesh K Srivastava, Klaus Greff, and Jürgen Schmidhuber · 2015
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Deep learning and the information bottleneck principle
Naftali Tishby and Noga Zaslavsky · 2015
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Vector quantization and signal compression , volume 159
Allen Gersho and Robert M Gray · 2012
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Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude
Tijmen Tieleman and Geoffrey Hinton · 2012
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Adadelta: An adaptive learning rate method
Matthew D Zeiler · 2012
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Provable bounds for learning some deep representations
S. Arora, A. Bhaskara, R. Ge, and T. Ma · 2013
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Advances in optimizing recurrent networks
Yoshua Bengio, Nicolas Boulanger-Lewandowski, and Razvan Pascanu · 2013
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Joan Bruna and Stéphane Mallat · 2013
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On the expressive power of deep learning: A tensor analysis
Nadav Cohen, Or Sharir, and Amnon Shashua · 2016
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Dropout as a bayesian approximation: Representing model uncertainty in deep learning
Yarin Gal and Zoubin Ghahramani · 2016
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Deep residual learning for image recognition
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Adversarial examples in the physical world
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A probabilistic framework for deep learning
Ankit B Patel, Tan Nguyen, and Richard G Baraniuk · 2016
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Visual representations: Defining properties and deep approximations
S. Soatto and A. Chiuso · 2016
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Resnet in resnet: generalizing residual architectures
Sasha Targ, Diogo Almeida, and Kevin Lyman · 2016
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Residual networks behave like ensembles of relatively shallow networks
Andreas Veit, Michael J Wilber, and Serge Belongie · 2016
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Understanding deep learning requires rethinking generalization
Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals · 2016
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Approximating continuous functions by relu nets of minimal width
Boris Hanin and Mark Sellke · 2017
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Depth creates no bad local minima
Haihao Lu and Kenji Kawaguchi · 2017
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Convolutional neural networks analyzed via convolutional sparse coding
Vardan Papyan, Yaniv Romano, and Michael Elad · 2017
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Piecewise convexity of artificial neural networks
Blaine Rister and Daniel L Rubin · 2017
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Exponentially vanishing sub-optimal local minima in multilayer neural networks
Daniel Soudry and Elad Hoffer · 2017
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Inception-v4, inception-resnet and the impact of residual connections on learning
Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alexander A Alemi · 2017
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A spline theory of deep networks
Randall Balestriero and Richard Baraniuk · 2018
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