Deep Learning Theory: Geometric Analysis of Capacity, Optimization, and Generalization for Improving Learning in Deep Neural Networks

Restricted Boltzmann Machines: Introduction and Review

Autori: Guido Montúfar
Pubblicato in: Information Geometry and Its Applications - On the Occasion of Shun-ichi Amari's 80th Birthday, IGAIA IV Liblice, Czech Republic, June 2016, Numero 252, 2018, Pagina/e 75-115, ISBN 978-3-319-97797-3
Editore: Springer International Publishing
DOI: 10.1007/978-3-319-97798-0_4

A Top-Down Approach to Attain Decentralized Multi-agents

Autori: Alex Tong Lin, Guido Montufar, Stanley Osher
Pubblicato in: Handbook of Reinforcement Learning and Control, Numero Studies in Systems, Decision and Control book series (SSDC, volume 325), 2021, Pagina/e 419-431
Editore: Springer International Publishing

Stochastic Feedforward Neural Networks: Universal Approximation

Autori: Merkh, Thomas; Montúfar, Guido
Pubblicato in: Mathematical Aspects of Deep Learning, 2022, Pagina/e 267–313, ISBN 9781009025096
Editore: Cambridge University Press
DOI: 10.1017/9781009025096.007

Optimal Transport to a Variety

Autori: Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar, Bernd Sturmfels, Lorenzo Venturello
Pubblicato in: Mathematical Aspects of Computer and Information Sciences - 8th International Conference, MACIS 2019, Gebze, Turkey, November 13–15, 2019, Revised Selected Papers, Numero 11989, 2020, Pagina/e 364-381, ISBN 978-3-030-43119-8
Editore: Springer International Publishing
DOI: 10.1007/978-3-030-43120-4_29

Affine Natural Proximal Learning

Autori: Wuchen Li, Alex Tong Lin, Guido Montúfar
Pubblicato in: Geometric Science of Information - 4th International Conference, GSI 2019, Toulouse, France, August 27–29, 2019, Proceedings, Numero 11712, 2019, Pagina/e 705-714, ISBN 978-3-030-26979-1
Editore: Springer International Publishing
DOI: 10.1007/978-3-030-26980-7_73

Ricci curvature for parametric statistics via optimal transport

Autori: Wuchen Li, Guido Montúfar
Pubblicato in: Information Geometry, Numero 3/1, 2020, Pagina/e 89-117, ISSN 2511-2481
Editore: Springer
DOI: 10.1007/s41884-020-00026-2

Implicit bias of gradient descent for mean squared error regression with wide neural networks

Autori: Jin, Hui; Montúfar, Guido
Pubblicato in: Journal of Machine Learning Research, Numero volume 24, number 137, 2023, Pagina/e 1--97, ISSN 1533-7928
Editore: JMLR

Mixtures and products in two graphical models

Autori: Anna Seigal, Guido Montufar
Pubblicato in: Journal of Algebraic Statistics, Numero 9/1, 2018, Pagina/e 1-20, ISSN 1309-3452
Editore: Journal of Algebraic Statistics
DOI: 10.18409/jas.v9i1.90

Wasserstein distance to independence models

Autori: Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar, Bernd Sturmfels, Lorenzo Venturello
Pubblicato in: Journal of Symbolic Computation, Numero 104, 2021, Pagina/e 855-873, ISSN 0747-7171
Editore: Academic Press
DOI: 10.1016/j.jsc.2020.10.005

Factorized mutual information maximization

Autori: Thomas Merkh, Guido Montúfar
Pubblicato in: Kybernetika, Numero Kybernetika 56 no. 5, 2020, Pagina/e 948-978, ISSN 0023-5954
Editore: Akademie Ved Ceske Republiky
DOI: 10.14736/kyb-2020-5-0948

Geometry and convergence of natural policy gradient methods

Autori: Johannes Müller; Guido Montúfar
Pubblicato in: Information Geometry, Numero Volume 7, 2023, Pagina/e 485–523, ISSN 2511-249X
Editore: Springer
DOI: 10.1007/s41884-023-00106-z

Algebraic optimization of sequential decision problems

Autori: Mareike Dressler; Marina Garrote-López; Guido Montúfar; Johannes Müller; Kemal Rose
Pubblicato in: Journal of Symbolic Computation, Numero Volume 121, 2023, Pagina/e 102241, ISSN 0747-7171
Editore: Academic Press
DOI: 10.1016/j.jsc.2023.102241

Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums

Autori: Guido Montúfar, Yue Ren, Leon Zhang
Pubblicato in: SIAM Journal on Applied Algebra and Geometry, Numero Volume 6, Number 4, 2022, Pagina/e 618-649, ISSN 2470-6566
Editore: SIAM
DOI: 10.1137/21m1413699

Distributed Learning via Filtered Hyperinterpolation on Manifolds

Autori: Guido Montúfar, Yu Guang Wang
Pubblicato in: Foundations of Computational Mathematics, 2021, ISSN 1615-3375
Editore: Springer Verlag
DOI: 10.1007/s10208-021-09529-5

Invariance properties of the natural gradient in overparametrised systems

Autori: Jesse van Oostrum; Johannes Müller; Nihat Ay
Pubblicato in: Information Geometry, Numero Volume 6, 2022, Pagina/e 51–67, ISSN 2511-249X
Editore: Springer
DOI: 10.1007/s41884-022-00067-9

Natural gradient via optimal transport

Autori: Wuchen Li, Guido Montúfar
Pubblicato in: Information Geometry, Numero 1/2, 2018, Pagina/e 181-214, ISSN 2511-2481
Editore: Springer Nature
DOI: 10.1007/s41884-018-0015-3

Geometry of Linear Convolutional Networks

Autori: Kathlén Kohn, Thomas Merkh, Guido Montúfar, Matthew Trager
Pubblicato in: SIAM Journal on Applied Algebra and Geometry, Numero Volume 6, Number 3, 2022, Pagina/e 368-406, ISSN 2470-6566
Editore: SIAM
DOI: 10.1137/21m1441183

Continuity and additivity properties of information decompositions

Autori: Johannes Rauh; Pradeep Kr. Banerjee; Eckehard Olbrich; Guido Montúfar; Jürgen Jost
Pubblicato in: International Journal of Approximate Reasoning, Numero Volume 161, 2023, Pagina/e 108979, ISSN 0888-613X
Editore: Elsevier BV
DOI: 10.1016/j.ijar.2023.108979

Decimated Framelet System on Graphs and Fast G-Framelet Transforms

Autori: Zheng, X.; Zhou, B.; Wang, Y. G.; Xiaosheng ZHUANG
Pubblicato in: Journal of Machine Learning Research, Numero Volume 23, Number 18, 2022, Pagina/e 1--68, ISSN 1533-7928
Editore: JMLR
DOI: 10.48550/arxiv.2012.06922

Training Wasserstein GANs without gradient penalties

Autori: Dohyun Kwon, Yeoneung Kim, Guido Montúfar, Insoon Yang
Pubblicato in: arxiv, 2021
Editore: arxiv

Enumeration of max-pooling responses with generalized permutohedra

Autori: Escobar, Laura; Gallardo, Patricio; González-Anaya, Javier; González, José L.; Montúfar, Guido; Morales, Alejandro H.
Pubblicato in: arxiv, Numero 2022, 2022
Editore: arxiv
DOI: 10.48550/arxiv.2209.14978

On Connectivity of Solutions in Deep Learning: The Role of Over-parameterization and Feature Quality

Autori: Quynh Nguyen, Pierre Brechet, Marco Mondelli
Pubblicato in: arxiv, 2021
Editore: arxiv

Pull-back Geometry of Persistent Homology Encodings

Autori: Liang, Shuang; Turkeš, Renata; Li, Jiayi; Otter, Nina; Montúfar, Guido
Pubblicato in: arxiv, 2023
Editore: arxiv
DOI: 10.48550/arxiv.2310.07073

How Well Do WGANs Estimate the Wasserstein Metric?

Autori: Mallasto, Anton; Montúfar, Guido; Gerolin, Augusto
Pubblicato in: arxiv, 2019
Editore: arxiv

Supermodular Rank: Set Function Decomposition and Optimization

Autori: Sonthalia, Rishi; Seigal, Anna; Montufar, Guido
Pubblicato in: arxiv, Numero 2023, 2023
Editore: arxiv
DOI: 10.48550/arxiv.2305.14632

Mildly Overparameterized ReLU Networks Have a Favorable Loss Landscape

Autori: Karhadkar, Kedar; Murray, Michael; Tseran, Hanna; Montúfar, Guido
Pubblicato in: arxiv, 2023
Editore: arxiv
DOI: 10.48550/arxiv.2305.19510

Function Space and Critical Points of Linear Convolutional Networks

Autori: Kohn, Kathlén; Montúfar, Guido; Shahverdi, Vahid; Trager, Matthew
Pubblicato in: arxiv, Numero 2023, 2023
Editore: arxiv
DOI: 10.48550/arxiv.2304.05752

FoSR: First-order spectral rewiring for addressing oversquashing in GNNs

Autori: Karhadkar, Kedar; Banerjee, Pradeep Kr.; Montúfar, Guido
Pubblicato in: International Conference on Learning Representations, Numero 11, 2023
Editore: OpenReview
DOI: 10.48550/arxiv.2210.11790

Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss

Autori: Pierre Bréchet, Katerina Papagiannouli, Jing An, Guido Montufar
Pubblicato in: Proceedings of the 40th International Conference on Machine Learning, Numero PMLR 202, 2023, Pagina/e 3106-3147, ISSN 2640-3498
Editore: PMLR

Implicit Bias of MSE Gradient Optimization in Underparameterized Neural Networks

Autori: Bowman, Benjamin; Montufar, Guido
Pubblicato in: International Conference on Learning Representations, Numero 2022, 2022
Editore: OpenReview
DOI: 10.48550/arxiv.2201.04738

Error Estimates for the Deep Ritz Method with Boundary Penalty

Autori: Müller, Johannes; Zeinhofer, Marius
Pubblicato in: Proceedings of Mathematical and Scientific Machine Learning, Numero 2022, 2022, Pagina/e 215-230, ISSN 2640-3498
Editore: PMLR
DOI: 10.48550/arxiv.2103.01007

How Framelets Enhance Graph Neural Networks

Autori: Xuebin Zheng, Bingxin Zhou, Junbin Gao, Yuguang Wang, Pietro Lió, Ming Li, Guido Montufar
Pubblicato in: Proceedings of the 38th International Conference on Machine Learning, Numero PMLR 139, 2021, Pagina/e 12761-12771
Editore: PMLR

Task-agnostic constraining in average reward POMDPs

Autori: Montúfar, Guido ; Rauh, Johannes and Ay, Nihat
Pubblicato in: Task-Agnostic Reinforcement Learning Workshop at ICLR 2019, 2019
Editore: GitHub

On the expected complexity of maxout networks

Autori: Hanna Tseran, Guido Montúfar
Pubblicato in: Advances in Neural Information Processing Systems, Numero 34, 2021, Pagina/e 28995--29008
Editore: Curran Associates, Inc.

On the effectiveness of persistent homology

Autori: Turkeš, Renata; Montúfar, Guido; Otter, Nina
Pubblicato in: Advances in Neural Information Processing Systems, Numero 35, 2022, Pagina/e 35432--35448, ISBN 9781713871088
Editore: Curran Associates, Inc.
DOI: 10.48550/arxiv.2206.10551

On the Proof of Global Convergence of Gradient Descent for Deep ReLU Networks with Linear Widths

Autori: Quynh Nguyen
Pubblicato in: Proceedings of the 38th International Conference on Machine Learning, Numero PMLR 139, 2021, Pagina/e 8056-8062
Editore: PMLR

Optimization Theory for ReLU Neural Networks Trained with Normalization Layers

Autori: Dukler, Yonatan; Gu, Quanquan; Montúfar, Guido
Pubblicato in: Proceedings of the 37th International Conference on Machine Learning, Numero volume 119, 2020, Pagina/e 2751-2760
Editore: PMLR

The Variational Deficiency Bottleneck

Autori: Pradeep Kr. Banerjee, Guido Montufar
Pubblicato in: 2020 International Joint Conference on Neural Networks (IJCNN), 2020, Pagina/e 1-8, ISBN 978-1-7281-6926-2
Editore: IEEE
DOI: 10.1109/ijcnn48605.2020.9206900

Haar Graph Pooling

Autori: Yu Guang Wang, Ming Li, Zheng Ma, Guido Montufar, Xiaosheng Zhuang, Yanan Fan
Pubblicato in: Proceedings of the 37th International Conference on Machine Learning, Numero PMLR 119, 2020, Pagina/e 9952-9962
Editore: PMLR

Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks

Autori: Quynh Nguyen, Marco Mondelli, Guido Montufar
Pubblicato in: Proceedings of the 38th International Conference on Machine Learning, Numero PMLR 139, 2021, Pagina/e 8119-8129
Editore: PMLR

Solving infinite-horizon POMDPs with memoryless stochastic policies in state-action space

Autori: Müller, Johannes; Montúfar, Guido
Pubblicato in: RLDM The Multi-disciplinary Conference on Reinforcement Learning and Decision Making, Numero 2022, 2022, Pagina/e 435-439
Editore: RLDM
DOI: 10.48550/arxiv.2205.14098

Weisfeiler and Lehman Go Cellular: CW Networks

Autori: Cristian Bodnar, Fabrizio Frasca, Nina Otter, Yu Guang Wang, Pietro Liò, Guido Montúfar, Michael Bronstein
Pubblicato in: Advances in Neural Information Processing Systems, Numero 34, 2021, Pagina/e 2625--2640
Editore: Curran Associates, Inc.

Can neural networks learn persistent homology features?

Autori: Guido Montúfar, Nina Otter, Yuguang Wang
Pubblicato in: Topological Data Analysis and Beyond Workshop at the 34th Conference on Neural Information Processing Systems (NeurIPS 2020), Vancouver, Canada, 2020
Editore: Openreview

Spectral Bias Outside the Training Set for Deep Networks in the Kernel Regime

Autori: Bowman, Benjamin; Montufar, Guido
Pubblicato in: Advances in Neural Information Processing Systems, Numero 35, 2022, Pagina/e 30362--30377, ISBN 9781713871088
Editore: Curran Associates, Inc.
DOI: 10.48550/arxiv.2206.02927

Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks

Autori: Cristian Bodnar, Fabrizio Frasca, Yuguang Wang, Nina Otter, Guido F Montufar, Pietro Lió, Michael Bronstein
Pubblicato in: Proceedings of the 38th International Conference on Machine Learning, Numero PMLR 139, 2021, Pagina/e 1026-1037
Editore: PMLR

Computing the Unique Information

Autori: Pradeep Kr. Banerjee, Johannes Rauh, Guido Montufar
Pubblicato in: 2018 IEEE International Symposium on Information Theory (ISIT), 2018, Pagina/e 141-145, ISBN 978-1-5386-4781-3
Editore: IEEE
DOI: 10.1109/isit.2018.8437757

The Geometry of Memoryless Stochastic Policy Optimization in Infinite-Horizon POMDPs

Autori: Müller, Johannes; Montúfar, Guido
Pubblicato in: International Conference on Learning Representations, Numero 2022, 2022
Editore: OpenReview
DOI: 10.48550/arxiv.2110.07409

Wasserstein of Wasserstein loss for learning generative models

Autori: Yonatan Dukler, Wuchen Li, Alex Tong Lin, Guido Montúfar
Pubblicato in: Proceedings of the 36th international conference on machine learning, Numero PMLR 97, 2019, Pagina/e 1716-1725
Editore: JMLR, Inc. and Microtome Publishing

Wasserstein diffusion Tikhonov regularization

Autori: Lin, Alex Tong ; Dukler, Yonatan ; Li, Wuchen and Montúfar, Guido
Pubblicato in: NeurIPS 2019 Workshop on optimal transport and machine learning, 2019
Editore: Arxiv

A continuity result for optimal memoryless planning in POMDPs

Autori: Ay, Nihat ; Rauh, Johannes and Montúfar, Guido
Pubblicato in: RLDM, 2019
Editore: RLDM

Wasserstein proximal of GANs

Autori: Alex Tong Lin, Wuchen Li, Stanley Osher, Guido Montúfar
Pubblicato in: International Conference on Geometric Science of Information GSI 2021, Numero Lecture Notes in Computer Science book series (LNCS, volume 12829), 2018, Pagina/e 524-533, ISBN 978-3-030-80209-7
Editore: Springer International Publishing
DOI: 10.1007/978-3-030-80209-7_57

PAC-Bayes and Information Complexity

Autori: Pradeep Kr. Banerjee, Guido Montufar
Pubblicato in: ICLR 2021 Workshop Neural Compression: From Information Theory to Applications, 2021
Editore: OpenReview

When Are Solutions Connected in Deep Networks?

Autori: Nguyen, Quynh; Brechet, Pierre; Mondelli, Marco
Pubblicato in: Advances in Neural Information Processing Systems, Numero 34, 2021, Pagina/e 20956--20969, ISBN 9781713845393
Editore: Curran Associates, Inc.
DOI: 10.48550/arxiv.2102.09671

Weisfeiler and Lehman go Topological: Message Passing Simplicial Networks

Autori: Cristian Bodnar, Fabrizio Frasca, Yu Guang Wang, Nina Otter, Guido Montufar, Pietro Lio, Michael Bronstein
Pubblicato in: ICLR 2021 Workshop on Geometrical and Topological Representation Learning, 2021
Editore: OpenReview

Characterizing the Spectrum of the NTK via a Power Series Expansion

Autori: Murray, Michael; Jin, Hui; Bowman, Benjamin; Montufar, Guido
Pubblicato in: International Conference on Learning Representations, Numero 11, 2023
Editore: OpenReview
DOI: 10.48550/arxiv.2211.07844

Achieving High Accuracy with PINNs via Energy Natural Gradients

Autori: Müller, Johannes; Zeinhofer, Marius
Pubblicato in: Proceedings of the 40th International Conference on Machine Learning, Numero 40, 2023, Pagina/e 25471--25485
Editore: PMLR
DOI: 10.48550/arxiv.2302.13163

Learning curves for Gaussian process regression with power-law priors and targets

Autori: Jin, Hui; Banerjee, Pradeep Kr.; Mont��far, Guido
Pubblicato in: International Conference on Learning Representations, Numero 2022, 2022
Editore: OpenReview
DOI: 10.48550/arxiv.2110.12231

Oversquashing in GNNs through the lens of information contraction and graph expansion

Autori: Pradeep Kr. Banerjee; Kedar Karhadkar; Yu Guang Wang; Uri Alon; Guido Montufar
Pubblicato in: 58th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Numero 58, 2022, Pagina/e 1-8
Editore: IEEE
DOI: 10.1109/allerton49937.2022.9929363

Decentralized Multi-Agents by Imitation of a Centralized Controller

Autori: Alex Lin, Mark Debord, Gary Hewer, Katia Estabridi, Guido Montufar, Stanley Osher
Pubblicato in: 2nd Annual Conference on Mathematical and Scientific Machine Learning, Numero PMLR vol 145, 2021
Editore: PMLR

Expected Gradients of Maxout Networks and Consequences to Parameter Initialization

Autori: Tseran, Hanna; Montúfar, Guido
Pubblicato in: Proceedings of the 40th International Conference on Machine Learning, Numero PMLR 202, 2023, Pagina/e 34491-34532
Editore: PMLR
DOI: 10.48550/arxiv.2301.06956

Kernelized Wasserstein Natural Gradient

Autori: Arbel, Michael; Gretton, Arthur; Li, Wuchen; Montufar, Guido
Pubblicato in: International Conference on Learning Representations, Numero 2020, 2020, Pagina/e 31
Editore: ICLR

Information Complexity and Generalization Bounds

Autori: Pradeep Kr. Banerjee, Guido Montufar
Pubblicato in: 2021 IEEE International Symposium on Information Theory (ISIT), 2021, Pagina/e 676-681, ISBN 978-1-5386-8209-8
Editore: IEEE
DOI: 10.1109/isit45174.2021.9517960

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