How smoothing the affinity matrix affects neighborhood preservation in t-SNE

Dimensionality reduction methods are instrumental to visualize high-dimensional data, and t-SNE stands as one of the most widely used methods due to its emphasis on local neighborhood preservation. A central component of t-SNE is the affinity matrix, which expresses pairwise similarities in the form of symmetrized probabilities, over which the optimization problem of t-SNE is defined. We study how the sharpness of this probability distribution affects neighborhood preservation at different scales. We introduce a row-wise power transform controlled by a parameter gamma that can smooth or sharpen each row of the affinity matrix while preserving sparsity and rank order. We show that this transform is equivalent to rescaling the Gaussian bandwidth and thus to changing the perplexity. However, as the sharpness of the probability distribution varies per point, a fixed gamma leads to point-dependent effective perplexities, making it distinct from changing the global perplexity. Empirically, we find that sharpening improves preservation of the very nearest neighbors, while smoothing improves preservation of broader local neighborhoods, outperforming alternative affinity constructions including multiscale methods in the mid-local range.

Measuring Distortion in the Empty Regions of Dimensionality Reduction Scatterplots with the Gap Index

Quality metrics play a crucial role in the proper use of dimensionality reduction projections for visual analysis of high-dimensional data. They quantify the degree of distortion of a projection compared to the high-dimensional data and provide a reliable indication of how confident users can be in the structures they see in the resulting layouts. However, most popular metrics focus on capturing direct relationships between points (e.g., distances or neighborhoods) while neglecting distortions in empty areas of the layout, even though these often compose visually relevant features of a 2D layout. In this paper, we introduce the Gap Index (GI), a quality metric for 2D projections that captures visual distortion by measuring spatial distortion in empty areas of a projection. It does so by decomposing the space into empty triangles, which are then compared to their high-dimensional counterparts to compute the deformation. This per-triangle deformation can be aggregated into a single scalar value or overlaid on a projection to visualize regional distortion patterns. Results show that, contrary to popular quality metrics, the GI is sensitive to small structural deformations that have high visual impact. It is also fast to compute and interpretable.

FloDR: An invertible dimensionality reduction method based on a normalising flow

It is common for two-dimensional embeddings of high-dimensional data to be read far beyond what they can support. Distances in and between clusters, the meaning behind empty spaces, and the amount of structure hidden at each point are generally invisible in the output of methods such as t-SNE and UMAP. This is because the information that could support the meaning of these properties is discarded during the optimisation process. Here, we present FloDR, a dimensionality reduction method that embeds data through an invertible normalising flow. While FloDR only uses the first two output coordinates to create a two-dimensional embedding, it retains the remaining coordinates rather than discarding them. In addition to the embedding, an exact inverse and an exact density are properties of a trained mapping, which enable diagnostic visualisations that are computed from the exact inverse of the model that drew the layout rather than from an approximate one. Specifically, we draw two fields, the conditional spread, which measures how much of the original data remains undetermined at each embedding position in input units, and the hidden contrast, which measures how much information about a labelled contrast the two plotted coordinates discard. Both fields are rendered with a prespecified test against a held out portion of the input data and a bootstrap confidence. A field that fails the test is reported as refused.

Semantic Space Search Trajectory Networks

Search Trajectory Networks (STNs) are a graph-based tool for visualizing and characterizing the behavior of optimization algorithms. STNs' reliance on discretization of the search space has largely confined them to low-dimensional or combinatorial settings. We introduce a methodology for constructing STNs in semantic spaces, defined as the space of a model's predictions on a fixed sample set. Our approach discretizes semantic vectors and aggregates them into network nodes via agglomerative clustering with complete linkage under a normalized Hamming distance. Since any predictor can be summarized by its semantic vector, this method enables comparison of learning dynamics across otherwise incomparable algorithm families. We apply semantic space STNs to classification and regression tasks solved using different machine learning algorithms, recovering known qualitative differences between them. Additionally, we use semantic space STNs to study neural network generalization by contrasting standard training with the label randomization regime of Zhang et al. (2017). The resulting STNs exhibit consistent structural differences, training on real labels produces denser, more efficient and more centralized graphs than training on shuffled labels. Together, our results show that semantic space STNs capture functional training dynamics arising from the interaction between learning algorithms and data, providing a tool for analyzing and comparing learning dynamics across machine learning models and training regimes.

cGAP: Generalized Association Plots with HOMALS-Guided Heatmaps for Visualization of High-Dimensional Categorical Data

High-dimensional categorical data arise in genetics, biomedicine, and the social sciences, yet visualization tools for such data remain far less developed than those for continuous variables. Existing methods either scale poorly, rely heavily on low-dimensional displays detached from the original data matrix, or prioritize predictive accuracy over interpretability. To address this gap, we introduce categorical Generalized Association Plots (cGAP), a visualization framework for nominal, ordinal, and binary data that preserves the original data matrix while augmenting it with interpretable geometric structure. cGAP uses Homogeneity Analysis (HOMALS) to embed subjects and category levels in a three-dimensional Euclidean space and maps the embedding to red-green-blue coordinates so that similar patterns receive similar colors. The framework integrates three coordinated views: a HOMALS-guided heatmap of the raw data matrix, a subject proximity matrix, and a variable proximity matrix. Seriation algorithms are then used to reorder rows and columns to reveal coherent clusters, outliers, and local-to-global structure. We also derive barycentric traceability, projection-distortion, and contrast-preservation properties that clarify how embedding geometry is transferred to the display. We demonstrate the versatility of cGAP through applications to student-animal classification data, mammalian dentition profiles, mushroom records from the UCI Machine Learning Repository, and the Clusters of Orthologous Genes database. These examples show that cGAP supports transparent exploratory analysis by maintaining traceability between derived visual structure and the original categorical observations. cGAP provides a full-matrix, heatmap-based visualization environment for investigating complex categorical datasets across scientific domains.