The emergence of low-cost sensor architectures for diverse modalities has made it possible to deploy sensor arrays that capture a single event from a large number of vantage points and using multiple modalities. In many scenarios, these sensors acquire very high-dimensional data such as audio signals, images, and video. To cope with such high-dimensional data, we typically rely on low-dimensional models. Manifold models provide a particularly powerful model that captures the structure of high-dimensional data when it is governed by a low-dimensional set of parameters. However, these models do not typically take into account dependencies among multiple sensors. We thus propose a new joint manifold framework for data ensembles that exploits such dependencies. We show that simple algorithms can exploit the joint manifold structure to improve their performance on standard signal processing applications. Additionally, recent results concerning dimensionality reduction for manifolds enable us to formulate a network-scalable data compression scheme that uses random projections of the sensed data. This scheme efficiently fuses the data from all sensors through the addition of such projections, regardless of the data modalities and dimensions.
Joint Manifold Learning and Density Estimation Using Normalizing Flows
University of Tehran（德黑兰大学）
基于流形假设，真实世界的数据通常位于低维流形上，而作为基于可能性的生成模型的归一化流由于其结构约束而无法找到该流形。因此，出现了一个有趣的问题：$\textit{"Can we find sub-manifold(s) of data in normalizing flows and estimate the density of the data on the sub-manifold(s)?"}$。在本文中，我们介绍了两种方法，即每像素惩罚对数似然和分层训练，回答上述问题。我们提出了一种单步方法，用于联合流形学习和密度估计，通过将流归一化为流形和非流形部分而获得的变换空间进行分离。这是通过用于学习数据的子流形的每像素惩罚似然函数完成的。规范化流假设转换后的数据是高斯化的，但这种强加的假设不一定是真的，特别是在高维中。为了解决这个问题，采用分层训练方法来改善子流形上的密度估计。结果验证了所提出的方法的优越性，同时流形学习和密度估计使用归一化流生成的图像质量和似然性。
Joint Representation Learning and Clustering via Gradient-Based Manifold Optimization
Department of Statistics Florida State University Tallahassee, Florida, USA（统计学系佛罗里达州立大学塔拉萨斯, 佛罗里达州, 美国）；Department of Mathematics Florida State University Tallahassee, Florida, USA（数学系佛罗里达州立大学塔拉萨斯, 佛罗里达州, 美国）
A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations
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）
Department of Statistics, Virginia Tech（统计学系，弗吉尼亚理工学院）；Université Côte d’Azur, Inria, CNRS, LJAD, France（蔚蓝海岸大学，法国国家信息与自动化技术研究院，法国国家科学研究中心，LJAD，法国）；Department of Industrial and Systems Engineering, University of Washington（工业与系统工程系，华盛顿大学）
Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets
Department of Electrical and Computer Engineering, Yale University（耶鲁大学电气与计算机工程系）；Department of Biostatistics, Harvard University（哈佛大学生物统计学系）；Program in Applied Mathematics, Yale University（耶鲁大学应用数学项目）；Interdepartmental Program in Computational Biology and Bioinformatics, Yale University（耶鲁大学计算生物学与生物信息学跨学科项目）；Department of Pathology, Yale University School of Medicine（耶鲁大学医学院病理学系）
