Visualizing high-dimensional data is an important routine for understanding biomedical data and interpreting deep learning models. Neighbor embedding methods, such as t-SNE, UMAP, and LargeVis, among others, are a family of popular visualization methods which reduce high-dimensional data to two dimensions. However, recent studies suggest that these methods often produce visual artifacts, potentially leading to incorrect scientific conclusions. Recognizing that the current limitation stems from a lack of data-independent notions of embedding maps, we introduce a novel conceptual and computational framework, LOO-map, that learns the embedding maps based on a classical statistical idea known as the leave-one-out. LOO-map extends the embedding over a discrete set of input points to the entire input space, enabling a systematic assessment of map continuity, and thus the reliability of the visualizations. We find for many neighbor embedding methods, their embedding maps can be intrinsically discontinuous. The discontinuity induces two types of observed map distortion: ``overconfidence-inducing discontinuity," which exaggerates cluster separation, and ``fracture-inducing discontinuity," which creates spurious local structures. Building upon LOO-map, we propose two diagnostic point-wise scores -- perturbation score and singularity score -- to address these limitations. These scores can help identify unreliable embedding points, detect out-of-distribution data, and guide hyperparameter selection. Our approach is flexible and works as a wrapper around many neighbor embedding algorithms. We test our methods across multiple real-world datasets from computer vision and single-cell omics to demonstrate their effectiveness in enhancing the interpretability and accuracy of visualizations.

It is often used to justify the effectiveness of machine learning algorithms in high-dimensional settings, since the curse of dimensionality can be circumvented if the data concentrates on a lowdimensional manifold. It is, however, evident that several low-dimensional (and hence, visualisable) datasets do not satisfy the Manifold Hypothesis. Instead, such data can have singularities -- points at which the local geometry does not resemble n-dimensional Euclidean space for any n. Prime examples of singular loci of datasets include branching points in neurons and cosmic filaments. Furthermore, standard image datasets (such as MNIST and CIFAR-10) are known to have non-constant intrinsic dimension [17], whereas a connected manifold must possess the same intrinsic dimension throughout. Whenever such non-manifold behaviour within datasets is of interest, it becomes natural to wonder whether it can be accurately and automatically identified. Particularly in large, high-dimensional datasets where visual inspection is impossible, we seek tools to identify and locate singularities within datasets. Our focus here is on unsupervised singularity detection, where one has recourse neither to a plethora of training data, nor the opportunity to regenerate samples along an unknown probability measure.