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Dimensionality Reduction (DR) methods are widely used to visualize high-dimensional data. One key task in DR-based analysis is discovering neighborhoods, which relies on analyzing the fine-grained local structure of a projection. However, DR is an inherently lossy process; no technique can perfectly preserve the high-dimensional relationships, and projections therefore contain visual artifacts. In this paper, we highlight a typically overlooked source of visual artifacts: ambiguous instances. These are instances that are highly similar to multiple mutually dissimilar neighborhoods in the high-dimensional space. Standard DR methods cannot faithfully project such instances, since each data instance is mapped to a single point in the visual space. As a result, such an instance is placed in only one of its neighborhoods (or in none at all), so only part of its neighborhood structure is represented. We call this distortion partial neighborhood embedding. In this paper, we introduce a graph-based approach that identifies ambiguous instances and replicates them as multiple points in the projection, placing each copy within its respective neighborhood. We use UMAP for our results, but our approach also generalizes to other local graph-based DR techniques. We show that our approach reveals previously hidden neighborhood memberships in projections and reduces partial neighborhood embedding across multiple examples, and is further supported by quantitative analyses.

As data becomes increasingly central across engineering and scientific disciplines, effective visualization is essential for interpreting complex, high-dimensional structures. Dimensionality reduction techniques project high-dimensional data into lower dimensions while aiming to preserve structural properties such as pairwise distances and local neighborhoods. In this paper, we focus on improving homological preservation, that is, the retention of topological features such as connected components and loops, which is critical for maintaining global shape and continuity. We first introduce AdaMapper, a Mapper-based algorithm that leverages persistence diagrams to guide both skeleton construction and landmark selection. AdaMapper incorporates an adaptive refinement strategy that automatically increases cover resolution in regions exhibiting topological loops. We then propose AdaHIsomap, which extends landmark Isomap by incorporating homology-informed landmark selection and augmenting it with random anchor points to better balance distance and homology preservation. We evaluate both methods on a diverse set of datasets, including high-dimensional point clouds, scientific simulations, networks, and image data, and benchmark their performance against state-of-the-art approaches.

We present a new nonlinear dimensionality reduction method, MAPLE, that enhances UMAP by improving manifold modeling. MAPLE employs a self-supervised learning approach to more efficiently encode low-dimensional manifold geometry. Central to this approach are maximum manifold capacity representations (MMCRs), which help untangle complex manifolds by compressing variances among locally similar data points while amplifying variance among dissimilar data points. This design is particularly effective for high-dimensional data with substantial intra-cluster variance and curved manifold structures, such as biological or image data. Our qualitative and quantitative evaluations demonstrate that MAPLE can produce clearer visual cluster separations and finer subcluster resolution than UMAP while maintaining tractable computational cost.

Nonlinear dimensionality reduction (NDR) techniques are widely used to visualize high-dimensional data. However, they often lack explainability, making it challenging for analysts to relate patterns in projections to original high-dimensional features. Existing interactive methods typically separate user interactions from the feature space, treating them primarily as post-hoc explanations rather than integrating them into the exploration process. This separation limits insight generation by restricting users' understanding of how features dynamically influence projections. To address this limitation, we propose a bidirectional interaction method that directly bridges the feature space and the projections. By allowing users to adjust feature weights, our approach enables intuitive exploration of how different features shape the embedding. We also define visual semantics to quantify projection changes, enabling structured pattern discovery through automated query-based interaction. To ensure responsiveness despite the computational complexity of NDR, we employ a neural network to approximate the projection process, enhancing scalability while maintaining accuracy. We evaluated our approach through quantitative analysis, assessing accuracy and scalability. A user study with a comprehensive visual interface and case studies demonstrated its effectiveness in supporting hypothesis generation and exploratory tasks with real-world data. The results confirmed that our approach supports diverse analytical scenarios and enhances users' ability to explore and interpret high-dimensional data through interactive exploration grounded in the feature space.

This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation. First, we provide a novel theoretical analysis of its associated loss and show that a zero loss indeed induces identical persistence pairs (in high and low dimensions) for the 0-dimensional persistent homology (PH0) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PHd for d>=1. Based on this observation, we introduce a novel generalization of TopoAE to 1-dimensional persistent homology (PH), called TopoAE++, for the accurate generation of cycle-aware planar embeddings, addressing the above failure case. This generalization is based on the notion of cascade distortion, a new penalty term favoring an isometric embedding of the 2-chains filling persistent 1-cycles, hence resulting in more faithful geometrical reconstructions of the 1-cycles in the plane. We further introduce a novel, fast algorithm for the exact computation of PH for Rips filtrations in the plane, yielding improved runtimes over previously documented topology-aware methods. Our method also achieves a better balance between the topological accuracy, as measured by the Wasserstein distance, and the visual preservation of the cycles in low dimensions. Our C++ implementation is available at https://github.com/MClemot/TopologicalAutoencodersPlusPlus.

Due to the intrinsic complexity of high-dimensional (HD) data, dimensionality reduction (DR) techniques cannot preserve all the structural characteristics of the original data. Therefore, DR techniques focus on preserving either local neighborhood structures (local techniques) or global structures such as pairwise distances between points (global techniques). However, both approaches can mislead analysts to erroneous conclusions about the overall arrangement of manifolds in HD data. For example, local techniques may exaggerate the compactness of individual manifolds, while global techniques may fail to separate clusters that are well-separated in the original space. In this research, we provide a deeper insight into Uniform Manifold Approximation with Two-phase Optimization (UMATO), a DR technique that addresses this problem by effectively capturing local and global structures. UMATO achieves this by dividing the optimization process of UMAP into two phases. In the first phase, it constructs a skeletal layout using representative points, and in the second phase, it projects the remaining points while preserving the regional characteristics. Quantitative experiments validate that UMATO outperforms widely used DR techniques, including UMAP, in terms of global structure preservation, with a slight loss in local structure. We also confirm that UMATO outperforms baseline techniques in terms of scalability and stability against initialization and subsampling, making it more effective for reliable HD data analysis. Finally, we present a case study and a qualitative demonstration that highlight UMATO's effectiveness in generating faithful projections, enhancing the overall reliability of visual analytics using DR.

We present a comprehensive pipeline, integrated with a visual analytics system called GapMiner, capable of exploring and exploiting untapped opportunities within the empty regions of high-dimensional datasets. Our approach utilizes a novel Empty-Space Search Algorithm (ESA) to identify the center points of these uncharted voids, which represent reservoirs for potentially valuable new configurations. Initially, this process is guided by user interactions through GapMiner, which visualizes Empty-Space Configurations (ESCs) within the context of the dataset and allows domain experts to explore and refine ESCs for subsequent validation in domain experiments or simulations. These activities iteratively enhance the dataset and contribute to training a connected deep neural network (DNN). As training progresses, the DNN gradually assumes the role of identifying and validating high-potential ESCs, reducing the need for direct user involvement. Once the DNN achieves sufficient accuracy, it autonomously guides the exploration of optimal configurations by predicting performance and refining configurations through a combination of gradient ascent and improved empty-space searches. Domain experts were actively involved throughout the system's development. Our findings demonstrate that this methodology consistently generates superior novel configurations compared to conventional randomization-based approaches. We illustrate its effectiveness in multiple case studies with diverse objectives.