Visualizing High-Dimensional Graph Embeddings via Informed Multi-View Projections

While UMAP is widely used for exploring high-dimensional data, typical workflows focus on its lower-dimensional embedding, largely overlooking the rich k-nearest-neighbor (kNN) graph that UMAP constructs internally. This graph encodes the data manifold in its original high-dimensional space, before the distortion that UMAP's 2D projection introduces. We demonstrate the untapped potential of this internal representation, showing how standard graph algorithms applied to this graph enhance data sensemaking: (1) PageRank identifies representative data points, (2) k-core decomposition reveals dense core regions versus sparse periphery, and (3) clustering coefficient detects tight-knit neighborhoods with highly-similar data points. Through quantitative and qualitative evaluation on MNIST and Fashion MNIST, we show that these graph-based analyses are not only practical but also competitive with or complementary to purpose-built methods (e.g., k-medoids for exemplar selection, HDBSCAN for density-based clustering).

A graph layout is normally a table of $N$ free coordinates. We optimise a function with a fixed number of parameters instead. This gives a drawing a sample complexity and an extensible domain. Force-directed algorithms remain the standard tools for graph drawing. The most accurate among them minimise stress in the Kamada-Kawai formulation by directly optimising the node coordinates, at a full objective cost of $O(N^2)$ in time and space. Here, we propose Fling (Field Layout via Implicit Neural Geometry), a small neural network mapping the distances of each node to a set of landmarks, positioning it in the plane by training on the layout energy. The full spring system then becomes tractable without its distance matrix, as rest lengths follow from a landmark bound in constant time per pair while a second network learns the majorisation sums from exact anchor rows, at $O(|\mathcal{A}|N)$ per step for $|\mathcal{A}|\ll N$ anchors. Unlike neural drawers that read the graph by message passing, we represent the drawing as a function of node features. An unseen node costs one forward pass, where sparse and low-rank majorisation remain transductive. As the unknowns are weights rather than coordinates, the energy only requires a small fraction of the nodes, and a field fitted that way outperforms PivotMDS, landmark MDS, and a kernel ridge trained on the same energy and features, when the task is fitting the energy of a graph from a sample of its nodes. In addition, the same parameterisation enables a stochastic pivot stress variant, an aesthetics-optimised variant carrying a neighbour-embedding energy with node-edge clearance and crossing terms on the same field, and conditioning on the weight between two energies gives a whole layout family from one run.

Both Dimensionality Reduction (DR) and Graph Drawing (GD) aim to visualize abstract, non-linear structures, yet rely on different optimization paradigms. This contrast is evident in Multidimensional Scaling (MDS), which typically depends on the SMACOF algorithm despite graph drawing results showing that simpler stochastic optimization schemes can be more effective for the same objective. We bridge these domains by adapting Stochastic Gradient Descent (SGD) techniques from graph drawing to vector data embedding. We present a scikit-learn compatible estimator that minimizes global stress through local pairwise updates, improving upon the existing implementation. Experiments on standard high-dimensional benchmarks show that our stochastic solver converges substantially faster than SMACOF while achieving comparable or lower stress.