An official website of the United States government
Abstract Optimal transport (OT) methods seek a transformation map (or plan) between two probability measures, such that the transformation has the minimum transportation cost. Such a minimum transport cost, with a certain power transform, is called the Wasserstein distance. Recently, OT methods have drawn great attention in statistics, machine learning, and computer science, especially in deep generative neural networks. Despite its broad applications, the estimation of high‐dimensional Wasserstein distances is a well‐known challenging problem owing to the curse‐of‐dimensionality. There are some cutting‐edge projection‐based techniques that tackle high‐dimensional OT problems. Three major approaches of such techniques are introduced, respectively, the slicing approach, the iterative projection approach, and the projection robust OT approach. Open challenges are discussed at the end of the review. This article is categorized under:Statistical and Graphical Methods of Data Analysis > Dimension ReductionStatistical Learning and Exploratory Methods of the Data Sciences > Manifold Learning
more »
« less
Manifold Learning: What, How, and Why
Manifold learning (ML), also known as nonlinear dimension reduction, is a set of methods to find the low-dimensional structure of data. Dimension reduction for large, high-dimensional data is not merely a way to reduce the data; the new representations and descriptors obtained by ML reveal the geometric shape of high-dimensional point clouds and allow one to visualize, denoise, and interpret them. This review presents the underlying principles of ML, its representative methods, and their statistical foundations, all from a practicing statistician's perspective. It describes the trade-offs and what theory tells us about the parameter and algorithmic choices we make in order to obtain reliable conclusions. Expected final online publication date for the Annual Review of Statistics and Its Application, Volume 11 is March 2024. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
more »
« less
- Award ID(s):
- 2015272
- PAR ID:
- 10489495
- Editor(s):
- Reid, Nancy
- Publisher / Repository:
- Annual Reviews
- Date Published:
- Journal Name:
- Annual Review of Statistics and Its Application
- Volume:
- 11
- Issue:
- 1
- ISSN:
- 2326-8298
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
High-dimensional multimodal data arises in many scientific fields. The integration of multimodal data becomes challenging when there is no known correspondence between the samples and the features of different datasets. To tackle this challenge, we introduce AVIDA, a framework for simultaneously performing data alignment and dimension reduction. In the numerical experiments, Gromov-Wasserstein optimal transport and t-distributed stochastic neighbor embedding are used as the alignment and dimension reduction modules respectively. We show that AVIDA correctly aligns high-dimensional datasets without common features with four synthesized datasets and two real multimodal single-cell datasets. Compared to several existing methods, we demonstrate that AVIDA better preserves structures of individual datasets, especially distinct local structures in the joint low-dimensional visualization, while achieving comparable alignment performance. Such a property is important in multimodal single-cell data analysis as some biological processes are uniquely captured by one of the datasets. In general applications, other methods can be used for the alignment and dimension reduction modules.more » « less
-
We consider the semi-supervised dimension reduction problem: given a high dimensional dataset with a small number of labeled data and huge number of unlabeled data, the goal is to find the low-dimensional embedding that yields good classification results. Most of the previous algorithms for this task are linkage-based algorithms. They try to enforce the must-link and cannot-link constraints in dimension reduction, leading to a nearest neighbor classifier in low dimensional space. In this paper, we propose a new hyperplane-based semi-supervised dimension reduction method---the main objective is to learn the low-dimensional features that can both approximate the original data and form a good separating hyperplane. We formulate this as a non-convex optimization problem and propose an efficient algorithm to solve it. The algorithm can scale to problems with millions of features and can easily incorporate non-negative constraints in order to learn interpretable non-negative features. Experiments on real world datasets demonstrate that our hyperplane-based dimension reduction method outperforms state-of-art linkage-based methods when very few labels are available.more » « less
-
Imaging data-based prognostic models focus on using an asset’s degradation images to predict its time to failure (TTF). Most image-based prognostic models have two common limitations. First, they require degradation images to be complete (i.e., images are observed continuously and regularly over time). Second, they usually employ an unsupervised dimension reduction method to extract low-dimensional features and then use the features for TTF prediction. Because unsupervised dimension reduction is conducted on the degradation images without the involvement of TTFs, there is no guarantee that the extracted features are effective for failure time prediction. To address these challenges, this article develops a supervised tensor dimension reduction-based prognostic model. The model first proposes a supervised dimension reduction method for tensor data. It uses historical TTFs to guide the detection of a tensor subspace to extract low-dimensional features from high-dimensional incomplete degradation imaging data. Next, the extracted features are used to construct a prognostic model based on (log)-location-scale regression. An optimization algorithm for parameter estimation is proposed, and analytical solutions are discussed. Simulated data and a real-world data set are used to validate the performance of the proposed model. History: Bianca Maria Colosimo served as the senior editor for this article Funding: This work was supported by National Science Foundation [2229245]. Data Ethics & Reproducibility Note: The code capsule is available on Code Ocean at https://github.com/czhou9/Code-and-Data-for-IJDS and in the e-Companion to this article (available at https://doi.org/10.1287/ijds.2022.x022 ).more » « less
-
Dimension reduction (DR) algorithms have proven to be extremely useful for gaining insight into large-scale high-dimensional datasets, particularly finding clusters in transcriptomic data. The initial phase of these DR methods often involves converting the original high-dimensional data into a graph. In this graph, each edge represents the similarity or dissimilarity between pairs of data points. However, this graph is frequently suboptimal due to unreliable high-dimensional distances and the limited information extracted from the high-dimensional data. This problem is exacerbated as the dataset size increases. If we reduce the size of the dataset by selecting points for a specific sections of the embeddings, the clusters observed through DR are more separable since the extracted subgraphs are more reliable. In this paper, we introduce LocalMAP, a new dimensionality reduction algorithm that dynamically and locally adjusts the graph to address this challenge. By dynamically extracting subgraphs and updating the graph on-the-fly, LocalMAP is capable of identifying and separating real clusters within the data that other DR methods may overlook or combine. We demonstrate the benefits of LocalMAP through a case study on biological datasets, highlighting its utility in helping users more accurately identify clusters for real-world problems.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1146/annurev-statistics-040522-115238
