We provide the first experimental results on non-synthetic datasets for the quasi-diagonal Riemannian gradient descents for neural networks introduced in [Ollivier, 2015]. These include the MNIST, SVHN, and FACE datasets as well as a previously unpublished electroencephalogram dataset. The quasi-diagonal Riemannian algorithms consistently beat simple stochastic gradient gradient descents by a varying margin. The computational overhead with respect to simple backpropagation is around a factor $2$. Perhaps more interestingly, these methods also reach their final performance quickly, thus requiring fewer training epochs and a smaller total computation time.
We also present an implementation guide to these Riemannian gradient descents for neural networks, showing how the quasi-diagonal versions can be implemented with minimal effort on top of existing routines which compute gradients.
Imperial College London, London, UK（伦敦帝国理工学院）；University Of Copenhagen, Copenhagen, Denmark（哥本哈根大学）；KTH Royal Institute of Technology, Stockholm, Sweden（瑞典皇家理工学院）
Thales Air Systems（泰雷兹空中系统公司）；Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）
Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of the Riemannian manifold of symmetric positive definite (SPD) matrices they belong to. In the context of deep networks, several architectures for these matrices have recently been proposed. In our article, we introduce a Riemannian batch normalization (batchnorm) algorithm, which generalizes the one used in Euclidean nets. This novel layer makes use of geometric oper…
In this paper, we propose a simple acceleration scheme for Riemannian gradient methods by extrapolating iterates on manifolds. We show when the iterates are generated from Riemannian gradient descent method, the accelerated scheme achieves the optimal convergence rate asymptotically and is computationally more favorable than the recently proposed Riemannian Nesterov accelerated gradient methods. Our experiments verify the practical benefit of the novel acceleration strategy.
We are interested in the challenging problem of modelling densities on Riemannian manifolds with a known symmetry group using normalising flows. This has many potential applications in physical sciences such as molecular dynamics and quantum simulations. In this work we combine ideas from implicit neural layers and optimal transport theory to propose a generalisation of existing work on exponential map flows, Implicit Riemannian Concave Potential Maps, IRCPMs. IRCPMs have some nice properties such as simplicity of incorporating symmetries and are less expensive than ODE-flows. We provide an in…
Adversarial attacks on neural networks through canonical Riemannian foliations
Ecole Nationale de l’Aviation Civile（法国国立民用航空学院）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）；UPS（保罗·萨巴蒂埃大学）
Georgia Institute of Technology（佐治亚理工学院）；IBM Research（IBM研究院）；Universidad San Pablo-CEU, CEU Universities（CEU大学圣巴勃罗校区）
This work focuses on designing low complexity hybrid tensor networks by considering trade-offs between the model complexity and practical performance. Firstly, we exploit a low-rank tensor-train deep neural network (TT-DNN) to build an end-to-end deep learning pipeline, namely LR-TT-DNN. Secondly, a hybrid model combining LR-TT-DNN with a convolutional neural network (CNN), which is denoted as CNN+(LR-TT-DNN), is set up to boost the performance. Instead of randomly assigning large TT-ranks for TT-DNN, we leverage Riemannian gradient descent to determine a TT-DNN associated with small TT-ranks.…
Although Deep Learning (DL) has achieved success in complex Artificial Intelligence (AI) tasks, it suffers from various notorious problems (e.g., feature redundancy, and vanishing or exploding gradients), since updating parameters in Euclidean space cannot fully exploit the geometric structure of the solution space. As a promising alternative solution, Riemannian-based DL uses geometric optimization to update parameters on Riemannian manifolds and can leverage the underlying geometric information. Accordingly, this article presents a comprehensive survey of applying geometric optimization in D…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）；HSE University（高等经济学院）
Geoopt is a research-oriented modular open-source package for Riemannian Optimization in PyTorch. The core of Geoopt is a standard Manifold interface that allows for the generic implementation of optimization algorithms. Geoopt supports basic Riemannian SGD as well as adaptive optimization algorithms. Geoopt also provides several algorithms and arithmetic methods for supported manifolds, which allow composing geometry-aware neural network layers that can be integrated with existing models.
We study projection-free methods for constrained Riemannian optimization. In particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize RFW to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein …
The minimax optimization over Riemannian manifolds (possibly nonconvex constraints) has been actively applied to solve many problems, such as robust dimensionality reduction and deep neural networks with orthogonal weights (Stiefel manifold). Although many optimization algorithms for minimax problems have been developed in the Euclidean setting, it is difficult to convert them into Riemannian cases, and algorithms for nonconvex minimax problems with nonconvex constraints are even rare. On the other hand, to address the big data challenges, decentralized (serverless) training techniques have re…
The Chinese University of Hong Kong（香港中文大学）；Peking University（北京大学）；Beijing International Center for Mathematical Research（北京国际数学研究中心）
This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By introducing the notion of Fisher information matrix in the manifold setting, we propose a novel Riemannian natural gradient method, which can be viewed as a natural extension of the natural gradient method from the Euclidean setting to the manifold setting. We establish the almost-sure global convergence of our proposed method under standard assumptions. Mo…
Natural Riemannian gradient for learning functional tensor networks
Institute of Mathematics, University of Augsburg（奥格斯堡大学数学研究所）；Department of Mathematics, Technical University of Munich（慕尼黑工业大学数学系）；Institute of Mathematics & Centre for Advanced Analytics and Predictive Sciences, University of Augsburg（奥格斯堡大学数学研究所与高级分析与预测科学中心）
University of California, Berkeley（加州大学伯克利分校）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Texas, Austin（德克萨斯大学奥斯汀分校）；CREST - ENSAE（法国国家统计与经济管理学院经济与统计研究中心）；Google Brain（谷歌大脑）
Projection robust Wasserstein (PRW) distance, or Wasserstein projection pursuit (WPP), is a robust variant of the Wasserstein distance. Recent work suggests that this quantity is more robust than the standard Wasserstein distance, in particular when comparing probability measures in high-dimensions. However, it is ruled out for practical application because the optimization model is essentially non-convex and non-smooth which makes the computation intractable. Our contribution in this paper is to revisit the original motivation behind WPP/PRW, but take the hard route of showing that, despite i…
Manifold optimization appears in a wide variety of computational problems in the applied sciences. In recent statistical methodologies such as sufficient dimension reduction and regression envelopes, estimation relies on the optimization of likelihood functions over spaces of matrices such as the Stiefel or Grassmann manifolds. Recently, Huang, Absil, Gallivan, and Hand (2016) have introduced the library ROPTLIB, which provides a framework and state of the art algorithms to optimize real-valued objective functions over commonly used matrix-valued Riemannian manifolds. This article presents Man…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；University of Pittsburgh（匹兹堡大学）
In the paper, we study a class of useful minimax problems on Riemanian manifolds and propose a class of effective Riemanian gradient-based methods to solve these minimax problems. Specifically, we propose an effective Riemannian gradient descent ascent (RGDA) algorithm for the deterministic minimax optimization. Moreover, we prove that our RGDA has a sample complexity of $O(κ^2ε^{-2})$ for finding an $ε$-stationary solution of the Geodesically-Nonconvex Strongly-Concave (GNSC) minimax problems, where $κ$ denotes the condition number. At the same time, we present an effective Riemannian stochas…
National Special Education Resource Center for Children with Autism, Zhejiang Normal University（浙江师范大学国家孤独症儿童特殊教育资源中心）；Institute of Cyber-Systems and Control, Zhejiang University（浙江大学 cyber 系统与控制研究所）；School of Computer Science and Technology, Zhejiang Normal University（浙江师范大学计算机科学与技术学院）
Recurrent neural networks are powerful models for sequential data, able to represent complex dependencies in the sequence that simpler models such as hidden Markov models cannot handle. Yet they are notoriously hard to train. Here we introduce a training procedure using a gradient ascent in a Riemannian metric: this produces an algorithm independent from design choices such as the encoding of parameters and unit activities. This metric gradient ascent is designed to have an algorithmic cost close to backpropagation through time for sparsely connected networks. We use this procedure on gated le…
Università di Milano（米兰大学）；Gruppo Nazionale Calcolo Scientifico, INDAM（意大利国家数学研究所国家科学计算组）；Università degli Studi di Milano（米兰大学）；Gruppo Nazionale per la Fisica Matematica, INDAM（意大利国家数学研究所国家数学物理组）
Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced trough pullbacks on the other manifolds of the …
We describe four algorithms for neural network training, each adapted to different scalability constraints. These algorithms are mathematically principled and invariant under a number of transformations in data and network representation, from which performance is thus independent. These algorithms are obtained from the setting of differential geometry, and are based on either the natural gradient using the Fisher information matrix, or on Hessian methods, scaled down in a specific way to allow for scalability while keeping some of their key mathematical properties.
Emergent Riemannian geometry over learning discrete computations on continuous manifolds
School of Informatics University of Edinburgh（信息学院爱丁堡大学）；Département d’Etudes Cognitives Ecole Normale Supérieure PSL（认知研究部巴黎高等师范学院PSL）；Gastsby Unit University College London（加斯特比单位伦敦大学学院）；Data Science Center Ecole Normale Supérieure PSL（数据科学中心巴黎高等师范学院PSL）
