Dimensionality reduction techniques are powerful tools for data preprocessing and visualization which typically come with few guarantees concerning the topological correctness of an embedding. The interleaving distance between the persistent homology of Vietoris-Rips filtrations can be used to identify a scale at which topological features such as clusters or holes in an embedding and original data set are in correspondence. We show how optimization seeking to minimize the interleaving distance can be incorporated into dimensionality reduction algorithms, and explicitly demonstrate its use in finding an optimal linear projection. We demonstrate the utility of this framework to data visualization.
Duke University（杜克大学）；Geometric Data Analytics（几何数据分析公司）
We propose a novel approach to dimensionality reduction combining techniques of metric geometry and distributed persistent homology, in the form of a gradient-descent based method called DIPOLE. DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. By fixing an initial embedding method (we use Isomap), DIPOLE can also be viewed as a full dimensionality-reduction pipeline. This framework is based on the strong theoretical and computational properties of distribut…
Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view, and, as an application, we describe an algorithm which takes as input a dataset together with an initial representa…
Finding global optima in high-dimensional optimization problems is extremely challenging since the number of function evaluations required to sufficiently explore the search space increases exponentially with its dimensionality. Furthermore, multimodal cost functions render local gradient-based search techniques ineffective. To overcome these difficulties, we propose to trim uninteresting regions of the search space where global optima are unlikely to be found by means of autoencoders, exploiting the lower intrinsic dimensionality of certain cost functions; optima are then searched over lower-…
Most dimensionality reduction methods employ frequency domain representations obtained from matrix diagonalization and may not be efficient for large datasets with relatively high intrinsic dimensions. To address this challenge, Correlated Clustering and Projection (CCP) offers a novel data domain strategy that does not need to solve any matrix. CCP partitions high-dimensional features into correlated clusters and then projects correlated features in each cluster into a one-dimensional representation based on sample correlations. Residue-Similarity (R-S) scores and indexes, the shape of data i…
In this paper, we investigate Dimensionality reduction (DR) maps in an information retrieval setting from a quantitative topology point of view. In particular, we show that no DR maps can achieve perfect precision and perfect recall simultaneously. Thus a continuous DR map must have imperfect precision. We further prove an upper bound on the precision of Lipschitz continuous DR maps. While precision is a natural measure in an information retrieval setting, it does not measure `how' wrong the retrieved data is. We therefore propose a new measure based on Wasserstein distance that comes with sim…
Recently, Graph Convolutional Networks (GCNs) and their variants have been receiving many research interests for learning graph-related tasks. While the GCNs have been successfully applied to this problem, some caveats inherited from classical deep learning still remain as open research topics in the context of the node classification problem. One such inherited caveat is that GCNs only consider the nodes that are a few propagations away from the labeled nodes to classify them. However, taking only a few propagation steps away nodes into account defeats the purpose of using the graph topologic…
We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning al…
Persistent Homology for High-dimensional Data Based on Spectral Methods
Hertie Institute for AI in Brain Health, University of Tübingen, Germany（图宾根大学希特研究所，德国）；Tübingen AI Center, Germany（图宾根人工智能中心，德国）；IWR, Heidelberg University, Germany（海德堡大学IWR研究所，德国）
Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis
Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain（巴塞罗那大学复杂系统研究所（UBICS）、巴塞罗那大学、西班牙巴塞罗那）；Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD, Delft, Netherlands（代尔夫特理工大学电子工程、数学和计算机科学学院、荷兰代尔夫特）；Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, USA（复杂网络与系统研究中心、信息学、计算与工程学院、印第安纳大学、美国布卢明顿）
Scalar Function Topology Divergence: Comparing Topology of 3D Objects
Skolkovo Institute of Science and Technology（斯克尔科夫科学与技术研究所）；AIRI（AI研究机构）；CNRS, IMJ, Paris Cité University（国家科学研究中心、巴黎城市大学）；AI Foundation and Algorithm Lab（AI基金会与算法实验室）
Dalian University of Technology（大连理工大学）；The American University in Cairo（开罗美国大学）
In this paper, a mechanistic data-driven approach is proposed to accelerate structural topology optimization, employing an in-house developed finite element convolutional neural network (FE-CNN). Our approach can be divided into two stages: offline training, and online optimization. During offline training, a mapping function is built between high and low resolution representations of a given design domain. The mapping is expressed by a FE-CNN, which targets a common objective function value (e.g., structural compliance) across design domains of differing resolutions. During online optimizatio…
Consider a set of $n$ data points in the Euclidean space $\mathbb{R}^d$. This set is called dataset in machine learning and data science. Manifold hypothesis states that the dataset lies on a low-dimensional submanifold with high probability. All dimensionality reduction and manifold learning methods have the assumption of manifold hypothesis. In this paper, we show that the dataset lies on an embedded hypersurface submanifold which is locally $(d-1)$-dimensional. Hence, we show that the manifold hypothesis holds at least for the embedding dimensionality $d-1$. Using an induction in a pyramid …
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Artificial Intelligence Research Institute (AIRI)（人工智能研究院（AIRI））；Huawei Noah’s Ark lab（华为诺亚方舟实验室）；CNRS（法国国家科学研究中心）；Université Paris Cité（巴黎西岱大学）
Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology.…
Cover Learning for Large-Scale Topology Representation
Centre de Recherches Mathématiques et Institut des sciences mathématiques, Laboratoire de combinatoire et d'informatique mathématique de l'Université du Québec à Montréal, Université de Sherbrooke, Canada（数学研究中心和数学科学研究所，魁北克大学蒙特利尔分校组合与数学信息学实验室， Sherbrooke 大学，加拿大）；Queen Mary University of London, United Kingdom（伦敦女王学院，英国）；Max Planck Institute of Molecular Cell Biology and Genetics（分子细胞生物学与遗传学马克斯·普朗克研究所）；Centre for Systems Biology, Dresden, Germany（系统生物学中心，德累斯顿，德国）；Faculty of Mathematics, Technische Universität Dresden, Germany（数学系，德累斯顿技术大学，德国）；Mathematical Institute, University of Oxford, United Kingdom（数学研究所，牛津大学，英国）
Sparse cubical complexes for efficient topology-preservation in image data
Weill Cornell Medicine（威尔康奈尔医学院）；Technical University of Munich（慕尼黑工业大学）；Imperial College London（伦敦帝国理工学院）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Cornell Tech（康奈尔科技校区）；Munich Data Science Institute（慕尼黑数据科学研究所）
Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Max Planck Institute for Human Cognitive and Brain Sciences（马克斯·普朗克人类认知与脑科学研究所）；Max Planck School of Cognition（马克斯·普朗克认知学校）；ScaDS.AI (Center for Scalable Data Analytics and AI)（ScaDS.AI（可扩展数据与人工智能中心））；Santa Fe Institute for the Sciences of Complexity（复杂性科学圣菲研究所）
SPHENIC: Topology-Aware Multi-View Clustering for Spatial Transcriptomics
College of Medicine and Biological Information Engineering, Northeastern University（医学与生物信息工程学院，东北大学）；Guangzhou Institutes of Biomedicine and Health, Chinese Academy of Sciences（广州生物医学与健康研究院，中国科学院）；College of Computer Science and Technology, National University of Defense Technology（计算机科学与技术学院，国防科技大学）；Intelligent Game and Decision Lab, Academy of Military Sciences（智能游戏与决策实验室，军事科学院）；University of Chinese Academy of Sciences（中国科学院大学）
Topology optimization (TO) is a popular and powerful computational approach for designing novel structures, materials, and devices. Two computational challenges have limited the applicability of TO to a variety of industrial applications. First, a TO problem often involves a large number of design variables to guarantee sufficient expressive power. Second, many TO problems require a large number of expensive physical model simulations, and those simulations cannot be parallelized. To address these issues, we propose a general scalable deep-learning (DL) based TO framework, referred to as SDL-T…
PHINN-EEG: Topological Time-Series Analysis of Dream-State EEG -- Dynamic Betti Curves for Dream Content Classification and Topology-Conditioned Neural Signal Synthesis
MIT（麻省理工学院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Complex biological systems have been successfully modeled by biochemical and genetic interaction networks, typically gathered from high-throughput (HTP) data. These networks can be used to infer functional relationships between genes or proteins. Using the intuition that the topological role of a gene in a network relates to its biological function, local or diffusion based "guilt-by-association" and graph-theoretic methods have had success in inferring gene functions. Here we seek to improve function prediction by integrating diffusion-based methods with a novel dimensionality reduction techn…
Classification of Epileptic iEEG using Topological Machine Learning
Department of Mechanical Engineering, Michigan State University（密歇根州立大学机械工程系）；Faculty of Medicine and Health Technology, Tampere University（坦佩雷大学医学与健康技术学院）；A.I.Virtanen Insttute of Molecular Sciences, University of Eastern Finland（东芬兰大学A.I.Virtanen分子科学研究所）；Department of Computational Mathematics, Science and Engineering, Michigan State University（密歇根州立大学计算数学、科学与工程系）
This paper considers improved forecasting in possibly nonlinear dynamic settings, with high-dimension predictors ("big data" environments). To overcome the curse of dimensionality and manage data and model complexity, we examine shrinkage estimation of a back-propagation algorithm of a deep neural net with skip-layer connections. We expressly include both linear and nonlinear components. This is a high-dimensional learning approach including both sparsity L1 and smoothness L2 penalties, allowing high-dimensionality and nonlinearity to be accommodated in one step. This approach selects signific…
University of Waterloo（滑铁卢大学）；Waterloo Artificial Intelligence Institute（滑铁卢人工智能研究所）；DarwinAI Corp.（DarwinAI公司）
The quantitative analysis of information structure through a deep neural network (DNN) can unveil new insights into the theoretical performance of DNN architectures. Two very promising avenues of research towards quantitative information structure analysis are: 1) layer similarity (LS) strategies focused on the inter-layer feature similarity, and 2) intrinsic dimensionality (ID) strategies focused on layer-wise data dimensionality using pairwise information. Inspired by both LS and ID strategies for quantitative information structure analysis, we introduce two novel complimentary methods for i…
