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

Restricted Boltzmann Machines: Introduction and Review

Author(s): Guido Montúfar
Published in: Information Geometry and Its Applications - On the Occasion of Shun-ichi Amari's 80th Birthday, IGAIA IV Liblice, Czech Republic, June 2016, Issue 252, 2018, Page(s) 75-115, ISBN 978-3-319-97797-3
Publisher: Springer International Publishing
DOI: 10.1007/978-3-319-97798-0_4

A Top-Down Approach to Attain Decentralized Multi-agents

Author(s): Alex Tong Lin, Guido Montufar, Stanley Osher
Published in: Handbook of Reinforcement Learning and Control, Issue Studies in Systems, Decision and Control book series (SSDC, volume 325), 2021, Page(s) 419-431
Publisher: Springer International Publishing

Stochastic Feedforward Neural Networks: Universal Approximation

Author(s): Merkh, Thomas; Montúfar, Guido
Published in: Mathematical Aspects of Deep Learning, 2022, Page(s) 267–313, ISBN 9781009025096
Publisher: Cambridge University Press
DOI: 10.1017/9781009025096.007

Optimal Transport to a Variety

Author(s): Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar, Bernd Sturmfels, Lorenzo Venturello
Published in: Mathematical Aspects of Computer and Information Sciences - 8th International Conference, MACIS 2019, Gebze, Turkey, November 13–15, 2019, Revised Selected Papers, Issue 11989, 2020, Page(s) 364-381, ISBN 978-3-030-43119-8
Publisher: Springer International Publishing
DOI: 10.1007/978-3-030-43120-4_29

Affine Natural Proximal Learning

Author(s): Wuchen Li, Alex Tong Lin, Guido Montúfar
Published in: Geometric Science of Information - 4th International Conference, GSI 2019, Toulouse, France, August 27–29, 2019, Proceedings, Issue 11712, 2019, Page(s) 705-714, ISBN 978-3-030-26979-1
Publisher: Springer International Publishing
DOI: 10.1007/978-3-030-26980-7_73

Ricci curvature for parametric statistics via optimal transport

Author(s): Wuchen Li, Guido Montúfar
Published in: Information Geometry, Issue 3/1, 2020, Page(s) 89-117, ISSN 2511-2481
Publisher: Springer
DOI: 10.1007/s41884-020-00026-2

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

Author(s): Jin, Hui; Montúfar, Guido
Published in: Journal of Machine Learning Research, Issue volume 24, number 137, 2023, Page(s) 1--97, ISSN 1533-7928
Publisher: JMLR

Mixtures and products in two graphical models

Author(s): Anna Seigal, Guido Montufar
Published in: Journal of Algebraic Statistics, Issue 9/1, 2018, Page(s) 1-20, ISSN 1309-3452
Publisher: Journal of Algebraic Statistics
DOI: 10.18409/jas.v9i1.90

Wasserstein distance to independence models

Author(s): Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar, Bernd Sturmfels, Lorenzo Venturello
Published in: Journal of Symbolic Computation, Issue 104, 2021, Page(s) 855-873, ISSN 0747-7171
Publisher: Academic Press
DOI: 10.1016/j.jsc.2020.10.005

Factorized mutual information maximization

Author(s): Thomas Merkh, Guido Montúfar
Published in: Kybernetika, Issue Kybernetika 56 no. 5, 2020, Page(s) 948-978, ISSN 0023-5954
Publisher: Akademie Ved Ceske Republiky
DOI: 10.14736/kyb-2020-5-0948

Geometry and convergence of natural policy gradient methods

Author(s): Johannes Müller; Guido Montúfar
Published in: Information Geometry, Issue Volume 7, 2023, Page(s) 485–523, ISSN 2511-249X
Publisher: Springer
DOI: 10.1007/s41884-023-00106-z

Algebraic optimization of sequential decision problems

Author(s): Mareike Dressler; Marina Garrote-López; Guido Montúfar; Johannes Müller; Kemal Rose
Published in: Journal of Symbolic Computation, Issue Volume 121, 2023, Page(s) 102241, ISSN 0747-7171
Publisher: Academic Press
DOI: 10.1016/j.jsc.2023.102241

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

Author(s): Guido Montúfar, Yue Ren, Leon Zhang
Published in: SIAM Journal on Applied Algebra and Geometry, Issue Volume 6, Number 4, 2022, Page(s) 618-649, ISSN 2470-6566
Publisher: SIAM
DOI: 10.1137/21m1413699

Distributed Learning via Filtered Hyperinterpolation on Manifolds

Author(s): Guido Montúfar, Yu Guang Wang
Published in: Foundations of Computational Mathematics, 2021, ISSN 1615-3375
Publisher: Springer Verlag
DOI: 10.1007/s10208-021-09529-5

Invariance properties of the natural gradient in overparametrised systems

Author(s): Jesse van Oostrum; Johannes Müller; Nihat Ay
Published in: Information Geometry, Issue Volume 6, 2022, Page(s) 51–67, ISSN 2511-249X
Publisher: Springer
DOI: 10.1007/s41884-022-00067-9

Natural gradient via optimal transport

Author(s): Wuchen Li, Guido Montúfar
Published in: Information Geometry, Issue 1/2, 2018, Page(s) 181-214, ISSN 2511-2481
Publisher: Springer Nature
DOI: 10.1007/s41884-018-0015-3

Geometry of Linear Convolutional Networks

Author(s): Kathlén Kohn, Thomas Merkh, Guido Montúfar, Matthew Trager
Published in: SIAM Journal on Applied Algebra and Geometry, Issue Volume 6, Number 3, 2022, Page(s) 368-406, ISSN 2470-6566
Publisher: SIAM
DOI: 10.1137/21m1441183

Continuity and additivity properties of information decompositions

Author(s): Johannes Rauh; Pradeep Kr. Banerjee; Eckehard Olbrich; Guido Montúfar; Jürgen Jost
Published in: International Journal of Approximate Reasoning, Issue Volume 161, 2023, Page(s) 108979, ISSN 0888-613X
Publisher: Elsevier BV
DOI: 10.1016/j.ijar.2023.108979

Decimated Framelet System on Graphs and Fast G-Framelet Transforms

Author(s): Zheng, X.; Zhou, B.; Wang, Y. G.; Xiaosheng ZHUANG
Published in: Journal of Machine Learning Research, Issue Volume 23, Number 18, 2022, Page(s) 1--68, ISSN 1533-7928
Publisher: JMLR
DOI: 10.48550/arxiv.2012.06922

Training Wasserstein GANs without gradient penalties

Author(s): Dohyun Kwon, Yeoneung Kim, Guido Montúfar, Insoon Yang
Published in: arxiv, 2021
Publisher: arxiv

Enumeration of max-pooling responses with generalized permutohedra

Author(s): Escobar, Laura; Gallardo, Patricio; González-Anaya, Javier; González, José L.; Montúfar, Guido; Morales, Alejandro H.
Published in: arxiv, Issue 2022, 2022
Publisher: arxiv
DOI: 10.48550/arxiv.2209.14978

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

Author(s): Quynh Nguyen, Pierre Brechet, Marco Mondelli
Published in: arxiv, 2021
Publisher: arxiv

Pull-back Geometry of Persistent Homology Encodings

Author(s): Liang, Shuang; Turkeš, Renata; Li, Jiayi; Otter, Nina; Montúfar, Guido
Published in: arxiv, 2023
Publisher: arxiv
DOI: 10.48550/arxiv.2310.07073

How Well Do WGANs Estimate the Wasserstein Metric?

Author(s): Mallasto, Anton; Montúfar, Guido; Gerolin, Augusto
Published in: arxiv, 2019
Publisher: arxiv

Supermodular Rank: Set Function Decomposition and Optimization

Author(s): Sonthalia, Rishi; Seigal, Anna; Montufar, Guido
Published in: arxiv, Issue 2023, 2023
Publisher: arxiv
DOI: 10.48550/arxiv.2305.14632

Mildly Overparameterized ReLU Networks Have a Favorable Loss Landscape

Author(s): Karhadkar, Kedar; Murray, Michael; Tseran, Hanna; Montúfar, Guido
Published in: arxiv, 2023
Publisher: arxiv
DOI: 10.48550/arxiv.2305.19510

Function Space and Critical Points of Linear Convolutional Networks

Author(s): Kohn, Kathlén; Montúfar, Guido; Shahverdi, Vahid; Trager, Matthew
Published in: arxiv, Issue 2023, 2023
Publisher: arxiv
DOI: 10.48550/arxiv.2304.05752

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

Author(s): Karhadkar, Kedar; Banerjee, Pradeep Kr.; Montúfar, Guido
Published in: International Conference on Learning Representations, Issue 11, 2023
Publisher: OpenReview
DOI: 10.48550/arxiv.2210.11790

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

Author(s): Pierre Bréchet, Katerina Papagiannouli, Jing An, Guido Montufar
Published in: Proceedings of the 40th International Conference on Machine Learning, Issue PMLR 202, 2023, Page(s) 3106-3147, ISSN 2640-3498
Publisher: PMLR

Implicit Bias of MSE Gradient Optimization in Underparameterized Neural Networks

Author(s): Bowman, Benjamin; Montufar, Guido
Published in: International Conference on Learning Representations, Issue 2022, 2022
Publisher: OpenReview
DOI: 10.48550/arxiv.2201.04738

Error Estimates for the Deep Ritz Method with Boundary Penalty

Author(s): Müller, Johannes; Zeinhofer, Marius
Published in: Proceedings of Mathematical and Scientific Machine Learning, Issue 2022, 2022, Page(s) 215-230, ISSN 2640-3498
Publisher: PMLR
DOI: 10.48550/arxiv.2103.01007

How Framelets Enhance Graph Neural Networks

Author(s): Xuebin Zheng, Bingxin Zhou, Junbin Gao, Yuguang Wang, Pietro Lió, Ming Li, Guido Montufar
Published in: Proceedings of the 38th International Conference on Machine Learning, Issue PMLR 139, 2021, Page(s) 12761-12771
Publisher: PMLR

Task-agnostic constraining in average reward POMDPs

Author(s): Montúfar, Guido ; Rauh, Johannes and Ay, Nihat
Published in: Task-Agnostic Reinforcement Learning Workshop at ICLR 2019, 2019
Publisher: GitHub

On the expected complexity of maxout networks

Author(s): Hanna Tseran, Guido Montúfar
Published in: Advances in Neural Information Processing Systems, Issue 34, 2021, Page(s) 28995--29008
Publisher: Curran Associates, Inc.

On the effectiveness of persistent homology

Author(s): Turkeš, Renata; Montúfar, Guido; Otter, Nina
Published in: Advances in Neural Information Processing Systems, Issue 35, 2022, Page(s) 35432--35448, ISBN 9781713871088
Publisher: 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

Author(s): Quynh Nguyen
Published in: Proceedings of the 38th International Conference on Machine Learning, Issue PMLR 139, 2021, Page(s) 8056-8062
Publisher: PMLR

Optimization Theory for ReLU Neural Networks Trained with Normalization Layers

Author(s): Dukler, Yonatan; Gu, Quanquan; Montúfar, Guido
Published in: Proceedings of the 37th International Conference on Machine Learning, Issue volume 119, 2020, Page(s) 2751-2760
Publisher: PMLR

The Variational Deficiency Bottleneck

Author(s): Pradeep Kr. Banerjee, Guido Montufar
Published in: 2020 International Joint Conference on Neural Networks (IJCNN), 2020, Page(s) 1-8, ISBN 978-1-7281-6926-2
Publisher: IEEE
DOI: 10.1109/ijcnn48605.2020.9206900

Haar Graph Pooling

Author(s): Yu Guang Wang, Ming Li, Zheng Ma, Guido Montufar, Xiaosheng Zhuang, Yanan Fan
Published in: Proceedings of the 37th International Conference on Machine Learning, Issue PMLR 119, 2020, Page(s) 9952-9962
Publisher: PMLR

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

Author(s): Quynh Nguyen, Marco Mondelli, Guido Montufar
Published in: Proceedings of the 38th International Conference on Machine Learning, Issue PMLR 139, 2021, Page(s) 8119-8129
Publisher: PMLR

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

Author(s): Müller, Johannes; Montúfar, Guido
Published in: RLDM The Multi-disciplinary Conference on Reinforcement Learning and Decision Making, Issue 2022, 2022, Page(s) 435-439
Publisher: RLDM
DOI: 10.48550/arxiv.2205.14098

Weisfeiler and Lehman Go Cellular: CW Networks

Author(s): Cristian Bodnar, Fabrizio Frasca, Nina Otter, Yu Guang Wang, Pietro Liò, Guido Montúfar, Michael Bronstein
Published in: Advances in Neural Information Processing Systems, Issue 34, 2021, Page(s) 2625--2640
Publisher: Curran Associates, Inc.

Can neural networks learn persistent homology features?

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

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

Author(s): Bowman, Benjamin; Montufar, Guido
Published in: Advances in Neural Information Processing Systems, Issue 35, 2022, Page(s) 30362--30377, ISBN 9781713871088
Publisher: Curran Associates, Inc.
DOI: 10.48550/arxiv.2206.02927

Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks

Author(s): Cristian Bodnar, Fabrizio Frasca, Yuguang Wang, Nina Otter, Guido F Montufar, Pietro Lió, Michael Bronstein
Published in: Proceedings of the 38th International Conference on Machine Learning, Issue PMLR 139, 2021, Page(s) 1026-1037
Publisher: PMLR

Computing the Unique Information

Author(s): Pradeep Kr. Banerjee, Johannes Rauh, Guido Montufar
Published in: 2018 IEEE International Symposium on Information Theory (ISIT), 2018, Page(s) 141-145, ISBN 978-1-5386-4781-3
Publisher: IEEE
DOI: 10.1109/isit.2018.8437757

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

Author(s): Müller, Johannes; Montúfar, Guido
Published in: International Conference on Learning Representations, Issue 2022, 2022
Publisher: OpenReview
DOI: 10.48550/arxiv.2110.07409

Wasserstein of Wasserstein loss for learning generative models

Author(s): Yonatan Dukler, Wuchen Li, Alex Tong Lin, Guido Montúfar
Published in: Proceedings of the 36th international conference on machine learning, Issue PMLR 97, 2019, Page(s) 1716-1725
Publisher: JMLR, Inc. and Microtome Publishing

Wasserstein diffusion Tikhonov regularization

Author(s): Lin, Alex Tong ; Dukler, Yonatan ; Li, Wuchen and Montúfar, Guido
Published in: NeurIPS 2019 Workshop on optimal transport and machine learning, 2019
Publisher: Arxiv

A continuity result for optimal memoryless planning in POMDPs

Author(s): Ay, Nihat ; Rauh, Johannes and Montúfar, Guido
Published in: RLDM, 2019
Publisher: RLDM

Wasserstein proximal of GANs

Author(s): Alex Tong Lin, Wuchen Li, Stanley Osher, Guido Montúfar
Published in: International Conference on Geometric Science of Information GSI 2021, Issue Lecture Notes in Computer Science book series (LNCS, volume 12829), 2018, Page(s) 524-533, ISBN 978-3-030-80209-7
Publisher: Springer International Publishing
DOI: 10.1007/978-3-030-80209-7_57

PAC-Bayes and Information Complexity

Author(s): Pradeep Kr. Banerjee, Guido Montufar
Published in: ICLR 2021 Workshop Neural Compression: From Information Theory to Applications, 2021
Publisher: OpenReview

When Are Solutions Connected in Deep Networks?

Author(s): Nguyen, Quynh; Brechet, Pierre; Mondelli, Marco
Published in: Advances in Neural Information Processing Systems, Issue 34, 2021, Page(s) 20956--20969, ISBN 9781713845393
Publisher: Curran Associates, Inc.
DOI: 10.48550/arxiv.2102.09671

Weisfeiler and Lehman go Topological: Message Passing Simplicial Networks

Author(s): Cristian Bodnar, Fabrizio Frasca, Yu Guang Wang, Nina Otter, Guido Montufar, Pietro Lio, Michael Bronstein
Published in: ICLR 2021 Workshop on Geometrical and Topological Representation Learning, 2021
Publisher: OpenReview

Characterizing the Spectrum of the NTK via a Power Series Expansion

Author(s): Murray, Michael; Jin, Hui; Bowman, Benjamin; Montufar, Guido
Published in: International Conference on Learning Representations, Issue 11, 2023
Publisher: OpenReview
DOI: 10.48550/arxiv.2211.07844

Achieving High Accuracy with PINNs via Energy Natural Gradients

Author(s): Müller, Johannes; Zeinhofer, Marius
Published in: Proceedings of the 40th International Conference on Machine Learning, Issue 40, 2023, Page(s) 25471--25485
Publisher: PMLR
DOI: 10.48550/arxiv.2302.13163

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

Author(s): Jin, Hui; Banerjee, Pradeep Kr.; Mont��far, Guido
Published in: International Conference on Learning Representations, Issue 2022, 2022
Publisher: OpenReview
DOI: 10.48550/arxiv.2110.12231

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

Author(s): Pradeep Kr. Banerjee; Kedar Karhadkar; Yu Guang Wang; Uri Alon; Guido Montufar
Published in: 58th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Issue 58, 2022, Page(s) 1-8
Publisher: IEEE
DOI: 10.1109/allerton49937.2022.9929363

Decentralized Multi-Agents by Imitation of a Centralized Controller

Author(s): Alex Lin, Mark Debord, Gary Hewer, Katia Estabridi, Guido Montufar, Stanley Osher
Published in: 2nd Annual Conference on Mathematical and Scientific Machine Learning, Issue PMLR vol 145, 2021
Publisher: PMLR

Expected Gradients of Maxout Networks and Consequences to Parameter Initialization

Author(s): Tseran, Hanna; Montúfar, Guido
Published in: Proceedings of the 40th International Conference on Machine Learning, Issue PMLR 202, 2023, Page(s) 34491-34532
Publisher: PMLR
DOI: 10.48550/arxiv.2301.06956

Kernelized Wasserstein Natural Gradient

Author(s): Arbel, Michael; Gretton, Arthur; Li, Wuchen; Montufar, Guido
Published in: International Conference on Learning Representations, Issue 2020, 2020, Page(s) 31
Publisher: ICLR

Information Complexity and Generalization Bounds

Author(s): Pradeep Kr. Banerjee, Guido Montufar
Published in: 2021 IEEE International Symposium on Information Theory (ISIT), 2021, Page(s) 676-681, ISBN 978-1-5386-8209-8
Publisher: IEEE
DOI: 10.1109/isit45174.2021.9517960

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