We study how the topology of a data set $M = M_a \cup M_b \subseteq \mathbb{R}^d$, representing two classes $a$ and $b$ in a binary classification problem, changes as it passes through the layers of a well-trained neural network, i.e., with perfect accuracy on training set and near-zero generalization error ($\approx 0.01\%$). The goal is to shed light on two mysteries in deep neural networks: (i) a nonsmooth activation function like ReLU outperforms a smooth one like hyperbolic tangent; (ii) successful neural network architectures rely on having many layers, even though a shallow network can approximate any function arbitrary well. We performed extensive experiments on the persistent homology of a wide range of point cloud data sets, both real and simulated. The results consistently demonstrate the following: (1) Neural networks operate by changing topology, transforming a topologically complicated data set into a topologically simple one as it passes through the layers. No matter how complicated the topology of $M$ we begin with, when passed through a well-trained neural network $f : \mathbb{R}^d \to \mathbb{R}^p$, there is a vast reduction in the Betti numbers of both components $M_a$ and $M_b$; in fact they nearly always reduce to their lowest possible values: $β_k\bigl(f(M_i)\bigr) = 0$ for $k \ge 1$ and $β_0\bigl(f(M_i)\bigr) = 1$, $i =a, b$. Furthermore, (2) the reduction in Betti numbers is significantly faster for ReLU activation than hyperbolic tangent activation as the former defines nonhomeomorphic maps that change topology, whereas the latter defines homeomorphic maps that preserve topology. Lastly, (3) shallow and deep networks transform data sets differently -- a shallow network operates mainly through changing geometry and changes topology only in its final layers, a deep one spreads topological changes more evenly across all layers.
Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of Massachusetts（马萨诸塞大学）
In established network architectures, shortcut connections are often used to take the outputs of earlier layers as additional inputs to later layers. Despite the extraordinary effectiveness of shortcuts, there remain open questions on the mechanism and characteristics. For example, why are shortcuts powerful? Why do shortcuts generalize well? In this paper, we investigate the expressivity and generalizability of a novel sparse shortcut topology. First, we demonstrate that this topology can empower a one-neuron-wide deep network to approximate any univariate continuous function. Then, we presen…
The learnability of different neural architectures can be characterized directly by computable measures of data complexity. In this paper, we reframe the problem of architecture selection as understanding how data determines the most expressive and generalizable architectures suited to that data, beyond inductive bias. After suggesting algebraic topology as a measure for data complexity, we show that the power of a network to express the topological complexity of a dataset in its decision region is a strictly limiting factor in its ability to generalize. We then provide the first empirical cha…
Deep Echo State Networks (DeepESNs) recently extended the applicability of Reservoir Computing (RC) methods towards the field of deep learning. In this paper we study the impact of constrained reservoir topologies in the architectural design of deep reservoirs, through numerical experiments on several RC benchmarks. The major outcome of our investigation is to show the remarkable effect, in terms of predictive performance gain, achieved by the synergy between a deep reservoir construction and a structured organization of the recurrent units in each layer. Our results also indicate that a parti…
Duke University（杜克大学）；Peking University（北京大学）；University of Nevada, Reno（内华达大学雷诺分校）；Tsinghua University（清华大学）；ECE Department, Duke University（杜克大学电气与计算机工程系）；Institute of Computational Linguistics, Peking University（北京大学计算语言学研究所）；CSE Department, University of Nevada, Reno（内华达大学雷诺分校计算机科学与工程系）；Department of Electronic Engineering, Tsinghua University（清华大学电子工程系）
Resource is an important constraint when deploying Deep Neural Networks (DNNs) on mobile and edge devices. Existing works commonly adopt the cell-based search approach, which limits the flexibility of network patterns in learned cell structures. Moreover, due to the topology-agnostic nature of existing works, including both cell-based and node-based approaches, the search process is time consuming and the performance of found architecture may be sub-optimal. To address these problems, we propose AutoShrink, a topology-aware Neural Architecture Search(NAS) for searching efficient building block…
There is significant recent interest to parallelize deep learning algorithms in order to handle the enormous growth in data and model sizes. While most advances focus on model parallelization and engaging multiple computing agents via using a central parameter server, aspect of data parallelization along with decentralized computation has not been explored sufficiently. In this context, this paper presents a new consensus-based distributed SGD (CDSGD) (and its momentum variant, CDMSGD) algorithm for collaborative deep learning over fixed topology networks that enables data parallelization as w…
Institute of Material Systems Modeling, Helmholtz-Zentrum Hereon（亥姆霍兹重离子研究中心材料系统建模研究所）；Institute for Continuum and Material Mechanics, Hamburg University of Technology（汉堡工业大学连续介质与材料力学研究所）
Representation Topology Divergence: A Method for Comparing Neural Network Representations
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；CNRS（法国国家科学研究中心）；Université Paris Cité（巴黎西岱大学）；Artificial Intelligence Research Institute (AIRI)（人工智能研究院（AIRI））
Consistent machine learning for topology optimization with microstructure-dependent neural network material models
Delft University of Technology（代尔夫特理工大学）；Princeton University（普林斯顿大学）；Princeton Materials Institute (PMI), Princeton University（普林斯顿材料研究所（PMI），普林斯顿大学）；Brown University（布朗大学）
Golisano College of Computing and Information Sciences（戈利萨诺计算机与信息科学学院）；Rochester Institute of Technology（罗切斯特理工学院）
This work introduces a novel, nature-inspired neural architecture search (NAS) algorithm based on ant colony optimization, Continuous Ant-based Neural Topology Search (CANTS), which utilizes synthetic ants that move over a continuous search space based on the density and distribution of pheromones, is strongly inspired by how ants move in the real world. The paths taken by the ant agents through the search space are utilized to construct artificial neural networks (ANNs). This continuous search space allows CANTS to automate the design of ANNs of any size, removing a key limitation inherent to…
ETH Zurich（苏黎世联邦理工学院）；SIB Swiss Institute of Bioinformatics（瑞士生物信息学研究所）
While many approaches to make neural networks more fathomable have been proposed, they are restricted to interrogating the network with input data. Measures for characterizing and monitoring structural properties, however, have not been developed. In this work, we propose neural persistence, a complexity measure for neural network architectures based on topological data analysis on weighted stratified graphs. To demonstrate the usefulness of our approach, we show that neural persistence reflects best practices developed in the deep learning community such as dropout and batch normalization. Mo…
Stanford University（斯坦福大学）；UnboxAI（昂博克斯人工智能公司）；Queen Mary University of London（伦敦玛丽女王大学）
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set filtrations and edge-based filtrations. We present three novel applications: the topological layer can (i) regularize data reconstruction or the weights of machine learning models, (ii) construct a loss on the output of a deep generative network to incorporate topological priors, and (iii) perform topological adversarial attacks on deep networks trained wit…
This paper proposes a deep Convolutional Neural Network(CNN) with strong generalization ability for structural topology optimization. The architecture of the neural network is made up of encoding and decoding parts, which provide down- and up-sampling operations. In addition, a popular technique, namely U-Net, was adopted to improve the performance of the proposed neural network. The input of the neural network is a well-designed tensor with each channel includes different information for the problem, and the output is the layout of the optimal structure. To train the neural network, a large d…
In this study, a novel topology optimization approach based on conditional Wasserstein generative adversarial networks (CWGAN) is developed to replicate the conventional topology optimization algorithms in an extremely computationally inexpensive way. CWGAN consists of a generator and a discriminator, both of which are deep convolutional neural networks (CNN). The limited samples of data, quasi-optimal planar structures, needed for training purposes are generated using the conventional topology optimization algorithms. With CWGANs, the topology optimization conditions can be set to a required …
Understanding how neural networks learn remains one of the central challenges in machine learning research. From random at the start of training, the weights of a neural network evolve in such a way as to be able to perform a variety of tasks, like classifying images. Here we study the emergence of structure in the weights by applying methods from topological data analysis. We train simple feedforward neural networks on the MNIST dataset and monitor the evolution of the weights. When initialized to zero, the weights follow trajectories that branch off recurrently, thus generating trees that de…
Université de Montréal（蒙特利尔大学）；ETH Zurich（苏黎世联邦理工学院）
Recent advances in implicit neural representations show great promise when it comes to generating numerical solutions to partial differential equations. Compared to conventional alternatives, such representations employ parameterized neural networks to define, in a mesh-free manner, signals that are highly-detailed, continuous, and fully differentiable. In this work, we present a novel machine learning approach for topology optimization -- an important class of inverse problems with high-dimensional parameter spaces and highly nonlinear objective landscapes. To effectively leverage neural repr…
Criteo AI Lab（Criteo人工智能实验室）；Amazon（亚马逊）；Meta AI Research（Meta人工智能研究院）
In this paper, we investigate the impact of neural networks (NNs) topology on adversarial robustness. Specifically, we study the graph produced when an input traverses all the layers of a NN, and show that such graphs are different for clean and adversarial inputs. We find that graphs from clean inputs are more centralized around highway edges, whereas those from adversaries are more diffuse, leveraging under-optimized edges. Through experiments on a variety of datasets and architectures, we show that these under-optimized edges are a source of adversarial vulnerability and that they can be us…
Institute of Theoretical Physics（理论物理研究所）；Faculty of Mathematics and Computer Science（数学与计算机科学学院）；Jagiellonian University（雅盖隆大学）
We introduce a flexible setup allowing for a neural network to learn both its size and topology during the course of a standard gradient-based training. The resulting network has the structure of a graph tailored to the particular learning task and dataset. The obtained networks can also be trained from scratch and achieve virtually identical performance. We explore the properties of the network architectures for a number of datasets of varying difficulty observing systematic regularities. The obtained graphs can be therefore understood as encoding nontrivial characteristics of the particular …
Topology design optimization offers tremendous opportunity in design and manufacturing freedoms by designing and producing a part from the ground-up without a meaningful initial design as required by conventional shape design optimization approaches. Ideally, with adequate problem statements, to formulate and solve the topology design problem using a standard topology optimization process, such as SIMP (Simplified Isotropic Material with Penalization) is possible. In reality, an estimated over thousands of design iterations is often required for just a few design variables, the conventional op…
We propose a neural network-based approach to topology optimization that aims to reduce the use of support structures in additive manufacturing. Our approach uses a network architecture that allows the simultaneous determination of an optimized: (1) part segmentation, (2) the topology of each part, and (3) the build direction of each part that collectively minimize the amount of support structure. Through training, the network learns a material density and segment classification in the continuous 3D space. Given a problem domain with prescribed load and displacement boundary conditions, the ne…
TU Kaiserslautern（凯泽斯劳滕工业大学）；MPI-MIS（马克斯·普朗克软件系统研究所）；IST Austria（奥地利科学技术研究所）
Recent works have shown that gradient descent can find a global minimum for over-parameterized neural networks where the widths of all the hidden layers scale polynomially with $N$ ($N$ being the number of training samples). In this paper, we prove that, for deep networks, a single layer of width $N$ following the input layer suffices to ensure a similar guarantee. In particular, all the remaining layers are allowed to have constant widths, and form a pyramidal topology. We show an application of our result to the widely used LeCun's initialization and obtain an over-parameterization requireme…
Topology optimization is computationally demanding that requires the assembly and solution to a finite element problem for each material distribution hypothesis. As a complementary alternative to the traditional physics-based topology optimization, we explore a data-driven approach that can quickly generate accurate solutions. To this end, we propose a deep learning approach based on a 3D encoder-decoder Convolutional Neural Network architecture for accelerating 3D topology optimization and to determine the optimal computational strategy for its deployment. Analysis of iteration-wise progress …
This paper proposes a new topology optimization method that applies a convolutional neural network (CNN), which is one deep learning technique for topology optimization problems. Using this method, we acquire a structure with a little higher performance that could not be obtained by the previous topology optimization method. In particular, in this paper, we solve a topology optimization problem aimed at maximizing stiffness with a mass constraint, which is a common type of topology optimization. In this paper, we first formulate the conventional topology optimization by the solid isotropic mat…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Artificial Intelligence Research Institute（人工智能研究所）；Paris Cité University（巴黎城市大学）；ETH Zurich（苏黎世联邦理工学院）
Connectionist temporal classification (CTC) has matured as an alignment free to sequence transduction and shows competitive for end-to-end speech recognition. In the CTC topology, the blank symbol occupies more than half of the state trellis, which results the spike phenomenon of the non-blank symbols. For classification task, the spikes work quite well, but as to the segmentation task it does not provide boundaries information. In this paper, a novel topology is introduced to combine the temporal classification and segmentation ability in one framework.
Tokyo Institute of Technology（东京工业大学）；Fujitsu Limited（富士通有限公司）；RIKEN AIP（理化学研究所人工智能研究中心）
We consider the scenario of deep clustering, in which the available prior knowledge is limited. In this scenario, few existing state-of-the-art deep clustering methods can perform well for both non-complex topology and complex topology datasets. To address the problem, we propose a constraint utilizing symmetric InfoNCE, which helps an objective of deep clustering method in the scenario train the model so as to be efficient for not only non-complex topology but also complex topology datasets. Additionally, we provide several theoretical explanations of the reason why the constraint can enhance…
Neural Networks for Generating Better Local Optima in Topology Optimization
Chair of Computational Modeling and Simulation, Technical University of Munich, School of Engineering and Design, Arcisstraße 21, Munich, 80 333, Germany（慕尼黑技术大学计算建模与模拟系）；Department of Civil and Mechanical Engineering, Technical University of Denmark（丹麦技术大学土木与机械工程系）；BMW Group, Knorrstraße 147, 80937 München（宝马集团）；Chair of Data Science in Engineering, Bauhaus-Universität Weimar, Coudraystraße 13 b, 99423, Weimar, Germany（魏玛包豪斯大学工程数据科学系）
Important advances have been made using convolutional neural network (CNN) approaches to solve complicated problems in areas that rely on grid structured data such as image processing and object classification. Recently, research on graph convolutional neural networks (GCNN) has increased dramatically as researchers try to replicate the success of CNN for graph structured data. Unfortunately, traditional CNN methods are not readily transferable to GCNN, given the irregularity and geometric complexity of graphs. The emerging field of GCNN is further complicated by research papers that differ gr…
Exploiting the deep generative model's remarkable ability of learning the data-manifold structure, some recent researches proposed a geometric data interpolation method based on the geodesic curves on the learned data-manifold. However, this interpolation method often gives poor results due to a topological difference between the model and the dataset. The model defines a family of simply-connected manifolds, whereas the dataset generally contains disconnected regions or holes that make them non-simply-connected. To compensate this difference, we propose a novel density regularizer that make t…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Paradigm. Inc.（第四范式有限公司）；Lenovo（联想）
In recent years, Graph Neural Networks (GNNs) have shown superior performance on diverse real-world applications. To improve the model capacity, besides designing aggregation operations, GNN topology design is also very important. In general, there are two mainstream GNN topology design manners. The first one is to stack aggregation operations to obtain the higher-level features but easily got performance drop as the network goes deeper. Secondly, the multiple aggregation operations are utilized in each layer which provides adequate and independent feature extraction stage on local neighbors w…
Solving the optimal power flow (OPF) problem is a fundamental task to ensure the system efficiency and reliability in real-time electricity grid operations. We develop a new topology-informed graph neural network (GNN) approach for predicting the optimal solutions of real-time ac-OPF problem. To incorporate grid topology to the NN model, the proposed GNN-for-OPF framework innovatively exploits the locality property of locational marginal prices and voltage magnitude. Furthermore, we develop a physics-aware (ac-)flow feasibility regularization approach for general OPF learning. The advantages o…
When designing large-scale distributed controllers, the information-sharing constraints between sub-controllers, as defined by a communication topology interconnecting them, are as important as the controller itself. Controllers implemented using dense topologies typically outperform those implemented using sparse topologies, but it is also desirable to minimize the cost of controller deployment. Motivated by the above, we introduce a compact but expressive graph recurrent neural network (GRNN) parameterization of distributed controllers that is well suited for distributed controller and commu…
National Centre for Scientific Research “Demokritos”（德谟克利特国家科学研究中心）
Deep learning models, while effective and versatile, are becoming increasingly complex, often including multiple overlapping networks of arbitrary depths, multiple objectives and non-intuitive training methodologies. This makes it increasingly difficult for researchers and practitioners to design, train and understand them. In this paper we present ANNETT-O, a much-needed, generic and computer-actionable vocabulary for researchers and practitioners to describe their deep learning configurations, training procedures and experiments. The proposed ontology focuses on topological, training and eva…
Komatsu University（小松大学）；University of Electro-Communications（电气通信大学）
Spectral clustering (SC) is one of the most popular clustering methods and often outperforms traditional clustering methods. SC uses the eigenvectors of a Laplacian matrix calculated from a similarity matrix of a dataset. SC has serious drawbacks: the significant increases in the time complexity derived from the computation of eigenvectors and the memory space complexity to store the similarity matrix. To address the issues, I develop a new approximate spectral clustering using the network generated by growing neural gas (GNG), called ASC with GNG in this study. ASC with GNG uses not only refe…
As a network-based functional approximator, we have proposed a "Lagrangian Density Space-Time Deep Neural Networks" (LDDNN) topology. It is qualified for unsupervised training and learning to predict the dynamics of underlying physical science governed phenomena. The prototypical network respects the fundamental conservation laws of nature through the succinctly described Lagrangian and Hamiltonian density of the system by a given data-set of generalized nonlinear partial differential equations. The objective is to parameterize the Lagrangian density over a neural network and directly learn fr…
Nanjing University（南京大学）；Alibaba Groups（阿里巴巴集团）；University of Texas at Dallas（德克萨斯大学达拉斯分校）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Graph neural networks (GNNs) have been widely used in many real applications, and recent studies have revealed their vulnerabilities against topology attacks. To address this issue, existing efforts have mainly been dedicated to improving the robustness of GNNs, while little attention has been paid to the detection of such attacks. In this work, we study the victim node detection problem under topology attacks against GNNs. Our approach is built upon the key observation rooted in the intrinsic message passing nature of GNNs. That is, the neighborhood of a victim node tends to have two competin…
In this empirical paper, we investigate how learning agents can be arranged in more efficient communication topologies for improved learning. This is an important problem because a common technique to improve speed and robustness of learning in deep reinforcement learning and many other machine learning algorithms is to run multiple learning agents in parallel. The standard communication architecture typically involves all agents intermittently communicating with each other (fully connected topology) or with a centralized server (star topology). Unfortunately, optimizing the topology of commun…
Shanghai Jiao Tong University（上海交通大学）；Shanghai AI Laboratory（上海人工智能实验室）
We study a new paradigm of knowledge transfer that aims at encoding graph topological information into graph neural networks (GNNs) by distilling knowledge from a teacher GNN model trained on a complete graph to a student GNN model operating on a smaller or sparser graph. To this end, we revisit the connection between thermodynamics and the behavior of GNN, based on which we propose Neural Heat Kernel (NHK) to encapsulate the geometric property of the underlying manifold concerning the architecture of GNNs. A fundamental and principled solution is derived by aligning NHKs on teacher and studen…
Multiscale topology optimization (M-TO) entails generating an optimal global topology, and an optimal set of microstructures at a smaller scale, for a physics-constrained problem. With the advent of additive manufacturing, M-TO has gained significant prominence. However, generating optimal microstructures at various locations can be computationally very expensive. As an alternate, graded multiscale topology optimization (GM-TO) has been proposed where one or more pre-selected and graded (parameterized) microstructural topologies are used to fill the domain optimally. This leads to a significan…
Northeastern University（东北大学）；Kellogg School of Management（凯洛格管理学院）；Northwestern University（西北大学）；Dana Farber Cancer Institute（达纳-法伯癌症研究所）；Brigham and Women’s Hospital（布莱根妇女医院）；Harvard Medical School（哈佛医学院）；Central European University（中欧大学）；Khoury College of Computer Sciences（胡里计算机科学学院）
To deepen our understanding of graph neural networks, we investigate the representation power of Graph Convolutional Networks (GCN) through the looking glass of graph moments, a key property of graph topology encoding path of various lengths. We find that GCNs are rather restrictive in learning graph moments. Without careful design, GCNs can fail miserably even with multiple layers and nonlinear activation functions. We analyze theoretically the expressiveness of GCNs, concluding a modular GCN design, using different propagation rules with residual connections could significantly improve the p…
Expressivity plays a fundamental role in evaluating deep neural networks, and it is closely related to understanding the limit of performance improvement. In this paper, we propose a three-pipeline training framework based on critical expressivity, including global model contraction, weight evolution, and link's weight rewiring. Specifically, we propose a pyramidal-like skeleton to overcome the saddle points that affect information transfer. Then we analyze the reason for the modularity (clustering) phenomenon in network topology and use it to rewire potential erroneous weighted links. We cond…
Iowa State University（爱荷华州立大学）；Ansys Corporation（Ansys公司）
Topology optimization has emerged as a popular approach to refine a component's design and increase its performance. However, current state-of-the-art topology optimization frameworks are compute-intensive, mainly due to multiple finite element analysis iterations required to evaluate the component's performance during the optimization process. Recently, machine learning (ML)-based topology optimization methods have been explored by researchers to alleviate this issue. However, previous ML approaches have mainly been demonstrated on simple two-dimensional applications with low-resolution geome…
Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and substantiate the geometric and topological view of the learning process of neural networks. Our attention is focused on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of the data manifold on different layers. We also propose a method for assessing the generalizing ability of neural networks based on top…
