Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport. We study the asymptotic properties of POTD and show that in the cases when the class labels contain no error, POTD estimates the SDR subspace exclusively. Empirical studies show POTD outperforms most of the state-of-the-art linear dimension reduction methods.

Dimension reduction methods are a fundamental class of techniques in data analysis, which aim to find a lower-dimensional representation of higher-dimensional data while preserving as much of the original information as possible. These methods are extensively used in practice, including in exploratory data analyses to visualize data--arguably, one of the first and most vital steps in any data analysis (Ray et al., 2021). Notably, in genomics, dimension reduction methods are ubiquitously applied to visualize high-dimensional single-cell RNA sequencing data in two dimensions (Becht et al., 2019). Beyond visualization, dimension reduction methods are also frequently employed to mitigate the curse of dimensionality (Bellman, 1957), engineer new features to improve downstream tasks like prediction (e.g., Massy, 1965), and enable scientific discovery in unsupervised learning settings (Chang et al., 2025). For example, many researchers have used dimension reduction in conjunction with clustering to discover new cell types and cell states (Wu et al., 2021), new cancer subtypes (Northcott et al., 2017), and other substantively-meaningful structure in a variety of domains (Bergen et al., 2019; Traven et al., 2017). Given the widespread use and need for dimension reduction methods, numerous dimension reduction techniques have been developed. Popular techniques include but are not limited to principal component analysis (PCA) (Pearson, 1901; Hotelling, 1933), multidimensional scaling (MDS) (Torgerson, 1952; Kruskal, 1964a), Isomap (Tenenbaum et al., 2000), locally linear embedding (LLE) (Roweis and Saul, 2000), t-distributed stochastic neighbor embedding (t-SNE) (van der 1

We develop the first polynomial-time algorithms that return a near-optimal solution with high probability.

With the rise of IoT-based botnet attacks, researchers have explored various learning models for detection, including traditional machine learning, deep learning, and hybrid approaches. A key advancement involves deploying attention mechanisms to capture long-term dependencies among features, significantly improving detection accuracy. However, most models treat attack instances independently, overlooking inter-instance relationships. Graph Neural Networks (GNNs) address this limitation by learning an embedding space via iterative message passing where similar instances are placed closer based on node features and relationships, enhancing classification performance. To further improve detection, attention mechanisms have been embedded within GNNs, leveraging both long-range dependencies and inter-instance connections. However, transforming the high dimensional IoT attack datasets into a graph structured dataset poses challenges, such as large graph structures leading computational overhead. To mitigate this, this paper proposes a framework that first reduces dimensionality of the NetFlow-based IoT attack dataset before transforming it into a graph dataset. We evaluate three dimension reduction techniques--Variational Autoencoder (VAE-encoder), classical autoencoder (AE-encoder), and Principal Component Analysis (PCA)--and compare their effects on a Graph Attention neural network (GAT) model for botnet attack detection

In this paper, we introduce a simple yet effective reward dimension reduction method to tackle the scalability challenges of multi-objective reinforcement learning algorithms. While most existing approaches focus on optimizing two to four objectives, their abilities to scale to environments with more objectives remain uncertain. Our method uses a dimension reduction approach to enhance learning efficiency and policy performance in multi-objective settings. While most traditional dimension reduction methods are designed for static datasets, our approach is tailored for online learning and preserves Pareto-optimality after transformation. We propose a new training and evaluation framework for reward dimension reduction in multi-objective reinforcement learning and demonstrate the superiority of our method in environments including one with sixteen objectives, significantly outperforming existing online dimension reduction methods.

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.

Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport.