The challenge of describing model drift is an open question in unsupervised learning. It can be difficult to evaluate at what point an unsupervised model has deviated beyond what would be expected from a different sample from the same population. This is particularly true for models without a probabilistic interpretation. One such family of techniques, Topological Data Analysis, and the Mapper algorithm in particular, has found use in a variety of fields, but describing model drift for Mapper graphs is an understudied area as even existing techniques for measuring distances between related constructs like graphs or simplicial complexes fail to account for the fact that Mapper graphs represent a combination of topological, metric, and density information. In this paper, we develop an optimal transport based metric which we call the Network Augmented Wasserstein Distance for evaluating distances between Mapper graphs and demonstrate the value of the metric for model drift analysis by using the metric to transform the model drift problem into an anomaly detection problem over dynamic graphs.
This paper presents a proposal of a faster Wasserstein $k$-means algorithm for histogram data by reducing Wasserstein distance computations and exploiting sparse simplex projection. We shrink data samples, centroids, and the ground cost matrix, which leads to considerable reduction of the computations used to solve optimal transport problems without loss of clustering quality. Furthermore, we dynamically reduced the computational complexity by removing lower-valued data samples and harnessing sparse simplex projection while keeping the degradation of clustering quality lower. We designate this…
A Probabilistic Interpretation of the Ball Mapper Graph
Dioscuri Centre in Topological Data Analysis, Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所拓扑数据分析迪奥斯库里中心）；International Environmental Doctoral School, University of Silesia in Katowice（卡托维兹西里西亚大学国际环境博士生院）
Kyoto University（京都大学）；RIKEN AIP（理化学研究所先进智能项目中心）；Facebook AI Research（脸书人工智能研究院）；SliceX AI（SliceX人工智能）
The Wasserstein barycenter has been widely studied in various fields, including natural language processing, and computer vision. However, it requires a high computational cost to solve the Wasserstein barycenter problem because the computation of the Wasserstein distance requires a quadratic time with respect to the number of supports. By contrast, the Wasserstein distance on a tree, called the tree-Wasserstein distance, can be computed in linear time and allows for the fast comparison of a large number of distributions. In this study, we propose a barycenter under the tree-Wasserstein distan…
Stanford University（斯坦福大学）；University of Utah（犹他大学）；Florida State University（佛罗里达州立大学）
The Gromov-Wasserstein (GW) framework adapts ideas from optimal transport to allow for the comparison of probability distributions defined on different metric spaces. Scalable computation of GW distances and associated matchings on graphs and point clouds have recently been made possible by state-of-the-art algorithms such as S-GWL and MREC. Each of these algorithmic breakthroughs relies on decomposing the underlying spaces into parts and performing matchings on these parts, adding recursion as needed. While very successful in practice, theoretical guarantees on such methods are limited. Inspi…
Measuring similarities between different tasks is critical in a broad spectrum of machine learning problems, including transfer, multi-task, continual, and meta-learning. Most current approaches to measuring task similarities are architecture-dependent: 1) relying on pre-trained models, or 2) training networks on tasks and using forward transfer as a proxy for task similarity. In this paper, we leverage the optimal transport theory and define a novel task embedding for supervised classification that is model-agnostic, training-free, and capable of handling (partially) disjoint label sets. In s…
Fused Partial Gromov-Wasserstein for Structured Objects
Department of Computer Science, Vanderbilt University（范德比大学计算机科学系）；Department of Computer Science, Indiana University–Indianapolis（印第安纳大学印第安纳市计算机科学系）；Department of BioHealth Informatics, Indiana University–Indianapolis（印第安纳大学印第安纳市生物健康信息学系）；Department of Mathematics and Computer Science, Fisk University（菲斯克大学数学与计算机科学系）
We propose a novel method for comparing non-aligned graphs of different sizes, based on the Wasserstein distance between graph signal distributions induced by the respective graph Laplacian matrices. Specifically, we cast a new formulation for the one-to-many graph alignment problem, which aims at matching a node in the smaller graph with one or more nodes in the larger graph. By integrating optimal transport in our graph comparison framework, we generate both a structurally-meaningful graph distance, and a signal transportation plan that models the structure of graph data. The resulting align…
We propose a framework, named Aggregated Wasserstein, for computing a dissimilarity measure or distance between two Hidden Markov Models with state conditional distributions being Gaussian. For such HMMs, the marginal distribution at any time position follows a Gaussian mixture distribution, a fact exploited to softly match, aka register, the states in two HMMs. We refer to such HMMs as Gaussian mixture model-HMM (GMM-HMM). The registration of states is inspired by the intrinsic relationship of optimal transport and the Wasserstein metric between distributions. Specifically, the components of …
Covariance alignment: from maximum likelihood estimation to Gromov-Wasserstein
Center for Data Science, New York University（纽约大学数据科学中心）；Institute for Data, Systems, and Society, Massachusetts Institute of Technology（麻省理工学院数据、系统与社会研究所）；Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems（信息传输问题研究所）
In the machine learning and optimization community, there are two main approaches for the convex risk minimization problem, namely, the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of oracle complexity (required number of stochastic gradient evaluations), both approaches are considered equivalent on average (up to a logarithmic factor). The total complexity depends on the specific problem, however, starting from work \cite{nemirovski2009robust} it was generally accepted that the SA is better than the SAA. % Nevertheless, in case of large-scale problems SA …
Stanford University（斯坦福大学）；Florida State University（佛罗里达州立大学）
We establish a bridge between spectral clustering and Gromov-Wasserstein Learning (GWL), a recent optimal transport-based approach to graph partitioning. This connection both explains and improves upon the state-of-the-art performance of GWL. The Gromov-Wasserstein framework provides probabilistic correspondences between nodes of source and target graphs via a quadratic programming relaxation of the node matching problem. Our results utilize and connect the observations that the GW geometric structure remains valid for any rank-2 tensor, in particular the adjacency, distance, and various kerne…
Wasserstein Parallel Transport for Predicting the Dynamics of Statistical Systems
Department of Statistics and Data Science, Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）；Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Department of Economics, Emory University（埃默里大学经济系）
Decentralized Entropic Optimal Transport for Distributed Distribution Comparison
School of Computer Science and Technology, East China Normal University（华东师范大学计算机科学与技术学院）；Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）
Metric properties of partial and robust Gromov-Wasserstein distances
University of Houston（休斯顿大学）；Colorado School of Mines（科罗拉多矿业学院）；University of Maryland（马里兰大学）；University of Nebraska - Lincoln（内布拉斯加大学林肯分校）；University of Tennessee - Knoxville（田纳西大学诺克斯维尔分校）；Florida State University（佛罗里达州立大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
