Demystifying UMAP artifacts: An interactive study on diagnosis and steering using 3D probes

Abstract

Uniform Manifold Approximation and Projection (UMAP) has become a ubiquitous tool for high-dimensional data visualization, yet its interpretation is often hindered by the “cartographic fallacy”—a cognitive bias where the embedding layout is assumed to be a faithful map of the data’s intrinsic geometry, leading users to mistake algorithmic side-effects for genuine data properties. These artifacts stem not only from stochastic optimization but also from inherent mathematical assumptions regarding simplicial approximation and metric normalization. In this work, we present an interactive study aimed at diagnosing these mechanisms. We introduce a classification system derived from a suite of 10 synthetic 3D “probe” datasets, categorizing distortions into spatial logic failures, topological loss, and metric distortion. Furthermore, we demonstrate a human-in-the-loop framework that pairs layout steering with parameter tuning to correct optimization traps and reveal topological trade-offs. This approach transforms UMAP, which is in its current form a static black box, into an explorable educational instrument, helping practitioners distinguish between genuine data features and algorithmic artifacts.

Introduction

In the era of high-dimensional data, dimensionality reduction (DR) has become a cornerstone of Exploratory Data Analysis (EDA).^1,2 Among the plethora of techniques available, Uniform Manifold Approximation and Projection (UMAP) ³ has emerged as the de facto standard, largely superseding predecessors like t-SNE ⁴ due to its superior runtime efficiency and its theoretical ability to preserve both local and global structure. From single-cell genomics ⁵ to machine learning model interpretability, ⁶ practitioners rely on UMAP to compress complex, high-dimensional manifolds into intuitive 2D scatterplots, using these visualizations to form hypotheses, identify clusters, and detect outliers.

However, this widespread adoption has fostered a “cartographic fallacy,” a pervasive misinterpretation where the embedding layout is assumed to be a faithful map of the data’s intrinsic geometry.^7,8 This cognitive trap stems from the opaque nature of the algorithm. UMAP is not a direct projection but a complex interplay of Riemannian metric approximation, simplicial complex construction, and non-convex optimization. Consequently, it is prone to generating “specious” artifacts: distortions where the 2D layout misrepresents the high-dimensional reality. ⁹ Global relative positions may be scrambled, continuous manifolds may be torn apart, and density variations may be erased. Crucially, without access to the ground truth, these artifacts are often indistinguishable from genuine data patterns, leading users to draw scientifically invalid conclusions based on algorithmic quirks.

Currently, when users suspect an embedding is flawed, their recourse is limited to a tedious cycle of “blind” parameter tuning: adjusting hyperparameters such as n_neighbors or min_dist and re-running the algorithm in hopes of a better outcome. This trial-and-error approach is inefficient and, more importantly, lacks semantic transparency. Users may observe that the shape changes, but they rarely understand why. The algorithm lacks the semantic intent or domain knowledge that the user possesses; it simply seeks a low-energy state, which often corresponds to a poor local minimum or a mathematically simplified topology.

In this paper, we propose to shift the paradigm from blind tuning to interactive diagnosis and steering. We present an educational instrument targeting (1) novices and students seeking to build intuition about manifold learning “black boxes” and visually grasp why these algorithms are inherently prone to distortions, and (2) practitioners needing a diagnostic sandbox to verify hypotheses by distinguishing genuine data features from algorithmic artifacts. Crucially, our contribution lies not in the discovery of isolated UMAP defects, which are theoretically established, but in the systematization and operationalization of these limitations into a cohesive visual benchmark suite. While prioritizing system design over formal pedagogical validation, we position our system as a concept demonstrator that transforms abstract theoretical knowledge into a practical diagnostic instrument.

To achieve this, our approach utilizes a suite of synthetic 3D datasets as measurable “probes.” Since the ground truth of these probes is known and visually verifiable, they allow us to isolate specific failure modes. We then introduce an interactive workflow that enables users to either diagnose trade-offs by exploring the parameter space or steer the optimization out of local minima by injecting human priors via initialization manipulation.^10,11 An interactive online demo of the instrument is available at https://interactive-umap-diagnostics.vercel.app/.

Our primary contributions are:

• (i) A Classification of Artifacts: Moving beyond a simple catalog of errors, we classify UMAP distortions into three theoretical categories: Inter-manifold spatial logic failures (Type A, driven by optimization entrapment), Intra-manifold topological loss (Type B, driven by simplicial fracture), and Metric distortion (Type C, driven by local normalization).

• (ii) A Benchmark Suite of 3D Probes: We introduce 10 synthetic datasets (D1–D10) specifically designed to stress-test these mechanisms. These probes serve both as a diagnostic testbed for researchers to evaluate algorithm fidelity and as an educational resource for practitioners to intuitively understand the capabilities and limitations of dimensionality reduction algorithms.

• (iii) An Interactive Diagnosis & Steering Framework: We demonstrate a human-in-the-loop framework centered on a coordinated 3D-2D visualization. By juxtaposing the ground truth against the embedding, our system empowers users to visually investigate the probes and validate two strategies: Layout Steering to correct optimization-driven errors, and Parameter Tuning to reveal the intrinsic trade-offs in topology and geometry preservation.

Related work

Our work sits at the intersection of evaluation methodologies for dimensionality reduction (DR) and interactive techniques for steering these algorithms. While typically treated as separate domains—diagnostics focusing on static metrics and steering on user interaction—we argue that effective analysis of non-linear manifolds requires their integration.

Evaluation and artifacts in dimensionality reduction

The proliferation of non-linear neighbor embedding techniques, particularly t-SNE ⁴ and UMAP, ³ has revolutionized the visualization of high-dimensional data. However, the stochastic nature and non-convex objective functions of these algorithms inherently introduce topological distortions. While newer algorithms like PaCMAP ¹² attempt to better preserve global structure through specialized loss functions (e.g. balancing mid-near pairs), they remain fully automated approaches. When the intrinsic topology is semantically ambiguous, these methods still lack a mechanism for users to rectify misalignments, necessitating rigorous post-hoc evaluation.

Standard evaluation practices rely heavily on quantitative metrics, such as Trustworthiness and Continuity, ¹³ or Shepard diagrams, ¹⁴ which attempt to summarize the preservation of local and global distances into single numerical scores. Despite their prevalence, these metrics are often too abstract to guide practical visual analytics. They quantify that an error exists, but not what the error looks like or why it occurred (e.g. distinguishing between a topological tear and a density cluster).

To mitigate this opacity, visualization researchers have developed diagnostic tools. Classical approaches like CheckViz ¹⁵ used distortion maps to highlight unreliable regions, while recent techniques seek to encode error directly within the embedding. For instance, Bian et al. ¹⁶ proposed projecting local linear subspaces as glyphs utilizing implicit differentiation to reveal deformations. Wattenberg et al. ⁷ famously demonstrated that hyperparameter misconfiguration can induce spurious cluster patterns, yet this “misuse” remains rampant. A recent systematic review by Jeon et al. ⁸ confirms that practitioners frequently misinterpret artifacts as biological or semantic truths, such as the arbitrary distance between clusters in UMAP.

To address the interpretability gap, recent research has pivoted toward visual diagnostics using synthetic ground-truth datasets. Cooley et al. ¹⁷ utilized spherical datasets to demonstrate failures in preserving closed topologies, while others have employed the “Swiss Roll” to test manifold unrolling capabilities. ¹⁸ Sankaran et al. ¹⁹ recently proposed visualizing metric distortion directly on the embedding. However, a gap remains: existing works primarily use these synthetic probes for algorithmic benchmarking in static papers. We lack a systematic, user-facing visual classification that operationalizes these 3D probes (e.g. spheres, knots, rolls) to train users to recognize specific topological failures—such as “tearing” or “semantic flipping”—in their own data.

Interactive dimensionality reduction

Interactive Dimensionality Reduction (iDR) seeks to bridge the cognitive gap between the user’s mental model and the algorithm’s mathematical output. ²⁰ Early systems facilitated exploration primarily through hyperparameter tuning, allowing users to adjust sliders for n_neighbors or perplexity and observe the resulting layout changes. ² While educational, this interaction is indirect; users must guess which parameter combination will resolve a specific local error.

A more direct paradigm is semantic interaction (SI), pioneered by Endert et al., ²¹ where user manipulations in the visual space (e.g. dragging points together) are translated into algorithmic updates. Systems like Dis-Function ²² and Andromeda ²³ update the underlying distance metric (e.g. weighting feature columns) based on user constraints. Similarly, AxSketch ²⁴ allows users to draw axes to define custom linear or generalized linear projections.

However, a critical limitation of most current iDR systems is their reliance on linear projections (PCA, LDA) or Multidimensional Scaling (MDS). While Pezzotti et al. ²⁵ pioneered steering mechanisms for non-linear methods by allowing users to interact with the t-SNE optimization process, inverting user interactions to steer non-linear, non-parametric models like UMAP is significantly more challenging due to the lack of a simple weight matrix to update. Recent systems for large-scale machine learning embeddings, such as WizMap, ²⁶ have achieved impressive scalability through multi-resolution summarization and WebGL-based rendering. However, these approaches primarily facilitate the exploration of a static projection, lacking mechanisms for users to intervene and correct topological distortions through direct human-in-the-loop steering.

Semi-supervised and constrained embeddings

To incorporate user knowledge into non-linear embeddings, recent works have explored semi-supervised and constrained variations of DR algorithms. Semi-supervised UMAP ²⁷ utilizes class labels to guide the graph construction, effectively pulling points with the same label closer together during optimization. Similarly, Xia et al. proposed Contrastive Dimensionality Reduction (CDR), ²⁸ which allows users to define “must-link” and “cannot-link” constraints that modify the loss function to enforce cluster separation.

While effective for classification tasks, these methods typically rely on hard constraints in the form of categorical labels or strict logic. They assume the user wants to enforce class structure, rather than correct spatial topology. For instance, a user might want to manually “unroll” a manifold that UMAP has incorrectly folded, without necessarily assigning discrete labels to the points. Our approach addresses this gap by treating initialization as a soft constraint. Instead of altering the loss function or requiring class labels, we allow users to spatially position “anchor points” that define the initial state of the embedding. This leverages Kobak and Linderman’s finding^10,11 that initialization dictates global structure, using it as a feature rather than a bug to permit “human-in-the-loop” steering of the non-convex landscape via spatial intuition.

classification of UMAP distortions via 3D synthetic probes

To systematically characterize the failure modes of UMAP, we require a controlled environment where the intrinsic topological ground truth is explicitly known and visually verifiable. Diagnosing projection artifacts in real-world high-dimensional data is inherently challenging because the “true” shape of the manifold is invisible to the human analyst.

Therefore, we adopt a strategy analogous to using “model organisms” in biology: we constructed a comprehensive suite of 10 synthetic 3D “probe” datasets designed to serve as proxies for high-dimensional manifolds. We restrict our probes to ${\mathbb{R}}^3$ for two critical reasons:

• Visual Verifiability: 3D is the maximum dimensionality that allows for a direct, side-by-side comparison with the generated 2D embedding, enabling us to unambiguously confirm artifacts.

• Minimal Counterexamples: These datasets represent fundamental topological primitives. By demonstrating that UMAP generates distortions even in these minimal complexity settings, we establish a baseline for potential failures in higher dimensions.

Based on observations across this suite, we classify UMAP distortions into three categories: Spatial Logic Misalignment, Topological Structure Loss, and Metric Distortion.

Design rationale: Mathematical primitives versus algorithmic constraints

To ensure our classification provides a systematic diagnosis, we categorize the distortions based on the specific scope of the geometric failure (inter-manifold vs intra-manifold) and the algorithmic assumptions (local metric normalization).

Inter-manifold logic (governing type A)

This category concerns the spatial arrangement of distinct, disconnected manifolds (clusters). In topological terms, this relates to the placement of connected components, quantified by the zeroth Betti number ${\beta }_0>1$ . The core failure mode here is optimization entrapment: UMAP’s non-convex objective function prioritizes local attraction, often settling into local minima that satisfy local connectivity but violate global semantic logic.

• Relative Orientation & Symmetry Loss: UMAP imposes no constraints on long-range relative positioning. As a result, clusters mispositioned in initialization tend to remain trapped during optimization. This makes the embedding prone to local minima where the global symmetry is lost (e.g. Antipodal-Clusters).

• Spatial Configuration Failure: UMAP tends to favor simple, compact cluster separations. This leads to two opposing artifacts: spurious separation for nested structures, where the algorithm un-nests the geometry into a side-by-side layout (e.g. Enclosed-Blob); or spurious entanglement for interleaved structures, where the algorithm fails to cleanly disentangle proximal manifolds, leaving them mixed or bridged (e.g. Two-Moons).

Intra-manifold continuity (governing type B)

This category concerns the structural integrity within a single continuous manifold. The theoretical challenge is constructing a simplicial complex that faithfully approximates the manifold’s topology without tearing or collapsing.

• Simplicial Fracture (Continuity): Approximating a continuous curve or surface with a discrete $k$ -NN graph requires sufficient sampling relative to curvature. If $k$ is insufficient, the simplicial complex fails to cover the manifold continuously, causing the graph to fracture and producing artificial “tearing” artifacts (e.g., Swiss-Roll, Uniform-Strip, and S-Curve).

• Topological Obstruction (Homology): Embedding manifolds with non-trivial homology (loops ${\beta }_1\ne 0$ , voids ${\beta }_2\ne 0$ ) into 2D faces, mathematical obstructions, as formalized in computational topology. ²⁹ To minimize embedding cross-entropy, the optimizer tends to collapse holes or flatten voids unless hyperparameters are tuned to penalize these topological violations (e.g. Torus-Surface).

Algorithmic mechanisms (governing type C)

Type C probes target the side-effects of UMAP’s Riemannian metric approximation. The algorithm computes a local distance scaler ${\sigma }_i$ for each point such that $\sum _j\exp (-d_{\mathrm{ij}}/{\sigma }_i)={\log }_{2}k$ . This creates a locally adaptive metric but introduces specific artifacts:

• Density Blindness: By normalizing the sum of weights, UMAP assumes data is uniformly distributed on the manifold. ³ This implies that variations in absolute density are mathematically removed (e.g. Density-Contrast).

• Non-isometry: Since distances are measured in terms of “number of neighbors” rather than Euclidean units, inter-cluster distances in the low-dimensional embedding do not scale linearly with the original high-dimensional distances (e.g. Distance-Contrast).

• Gradient Intolerance: In datasets with density gradients, the local normalization can create a disconnection artifact. The adaptive metric expands the volume of sparse regions (large ${\sigma }_i$ ) while compressing dense regions (small ${\sigma }_i$ ). During optimization, this disparity can cause sparse connectors to be statistically overwhelmed by the attractive forces of dense clusters or treated as disconnected noise (e.g. Connected-Blobs).

Type A: Spatial logic & semantic misalignment

Type A distortions represent failures in inter-manifold relations, which are shown in Figure 1. These occur when the UMAP optimizer, driven by local connectivity and global repulsion, settles into a layout that minimizes overlap but contradicts the high-dimensional semantic relationship between distinct clusters.

Figure 1.

Type A Distortions: Spatial Logic & Semantic Misalignment. We identify three manifestations corresponding to failures in inter-manifold positioning driven by optimization entrapment. (D1) Antipodal-Clusters: Exhibits Symmetry Loss, where semantically opposing clusters collapse into adjacency because the optimizer lacks global constraints to enforce relative orientation. (D2) Enclosed-Blob: Shows False Separation, where a nested “parent-child” hierarchy is flattened into a side-by-side “sibling” layout to satisfy local graph disjointness. (D3) Two-Moons: Demonstrates Spurious Entanglement, where the algorithm fails to cleanly disentangle proximal, interleaved manifolds due to local minima in the optimization landscape.

Symmetry loss (Antipodal-Clusters)

Our primary probe consists of a central cluster surrounded by pairs of opposing clusters. UMAP collapses this global metric structure, placing opposing clusters adjacent to each other. This confirms that without explicit constraints, the optimization landscape lacks the gradient information required to enforce relative orientation, creating a spurious semantic proximity.

False separation of nested structure (Enclosed-Blob)

This probe tests the embedding of a dense core enclosed by a spherical shell. UMAP resolves the graph disjointness by placing the “unrolled” shell side-by-side with the core. This destroys the semantic containment hierarchy, effectively converting a vertical “parent-child” (inside-outside) relationship into a horizontal “sibling” (adjacent) relationship, solely to satisfy the optimizer’s preference for compact, non-overlapping separation.

Spurious entanglement (Two-Moons)

Here, we evaluate the algorithm’s ability to disentangle two proximal, interleaved manifolds. Despite the manifolds being topologically distinct, their spatial proximity in the ambient space often traps the optimizer in a local minimum. The projection frequently exhibits spurious entanglement, where the two moons fail to separate cleanly or are bridged by points that should belong to distinct geodesics.

Type B: Topological & global structure loss

Type B distortions represent failures in intra-manifold continuity (Figure 2). These artifacts occur when the algorithm’s simplicial approximation fails to cover the entire manifold, or when the dimension reduction process collapses high-dimensional topological features (e.g., holes) to minimize embedding energy.

Figure 2.

Type B Distortions: Topological & Global Structure Loss. Failures arise when the simplicial approximation cannot maintain intra-manifold continuity or when high-order topology is sacrificed to minimize embedding energy. (D4-D5) Swiss-Roll & Uniform-Strip: Exhibit Simplicial Fracture, where continuous manifolds artificially tear into disjoint segments due to high curvature or sampling sensitivity relative to the $k$ -NN graph construction. (D6) Torus-Surface: Shows Homological Collapse, where the central topological hole (Genus-1 feature) is obliterated to flatten the embedding. (D7) S-Curve: Demonstrates Extrinsic Geometry Erasure, where crucial 3D curvature information is lost during the manifold unrolling process.

Simplicial fracture (Swiss-Roll & Uniform-Strip)

This probe category tests the algorithm’s ability to maintain the connectedness ( ${\beta }_0=1$ ) of a single continuous component.

• Swiss-Roll: Represents a classic manifold learning challenge involving high curvature. UMAP frequently fractures the continuous roll into disjoint segments. This occurs because the $k$ -NN graph fails to traverse the high-curvature regions smoothly, treating the single surface as multiple disconnected patches.

• Uniform-Strip: Unlike “false structure discovery” in random noise, this dataset contains a valid, uniform manifold. The artifact here is artificial tearing due to sampling sensitivity. Even with uniform density, stochastic sampling variations can cause the local connectivity graph to break, splitting a continuous strip into two or more spurious clusters.

Homological collapse (Torus-Surface)

This probe targets the preservation of non-trivial topology, specifically the central hole (Genus-1, ${\beta }_1=1$ ). Projecting a torus into 2D creates a topological conflict. UMAP typically resolves this by collapsing the hole entirely, reducing the ring to a dense, featureless disc. This demonstrates the algorithm’s tendency to sacrifice high-order topological features (voids) to preserve local neighbor distances.

Extrinsic geometry erasure (S-Curve)

While similar to the Swiss-Roll, the S-Curve probe highlights the loss of extrinsic geometry. UMAP successfully “unrolls” the S-shape into a flat sheet (restoring intrinsic topology), but in doing so, it completely erases the 3D curvature information. While this is mathematically justifiable for manifold learning, it can be misleading in visual analytics, as the user loses the context of the data’s original geometric complexity (e.g., a folding trajectory in state space).

Type C: Metric & density distortion

Unlike previous types, Type C distortions (Figure 3) are direct consequences of UMAP’s local connectivity normalization mechanism.

Figure 3.

Type C Distortions: Metric & Density Errors. Direct consequences of the algorithm’s local normalization mechanism ( ${\sigma }_i$ ). (D8) Density-Contrast: Shows Density Blindness, where clusters with different densities appear uniform. (D9) Distance-Contrast: Demonstrates Non-isometry, where long-range Euclidean distances are compressed. (D10) Connected-Blobs: Exhibits Gradient Intolerance, where a sparse bridge connecting dense clusters is artificially severed due to the disparity in local metric scaling.

Density blindness (Density-Contrast)

To ensure a uniform topological representation, UMAP expands sparse regions and compresses dense ones. Consequently, the ultra-dense cluster and the sparse cluster in our probe are rendered with visually identical densities in 2D. This confirms that standard UMAP embeddings are mathematically designed to erase absolute density information. Researchers requiring density fidelity must resort to specialized variants like DensMAP, ³⁰ which explicitly incorporates a local radius term to preserve these variations.

Non-isometry (Distance-Contrast)

Because UMAP constructs the graph based on local neighborhoods, it lacks a global metric ruler. Inter-cluster distances in the embedding reflect the graph traversal distance rather than the ambient Euclidean distance. As a result, the embedding compresses the empty space between distant clusters. In our probe, clusters that are extremely distant in 3D appear equidistant to nearby clusters in 2D. This demonstrates that inter-cluster distances in the embedding reflect topological separability rather than the true magnitude of separation.

Gradient intolerance (Connected-Blobs)

This probe highlights the side-effect of local normalization on heterogeneous structures. The “bridge” connects two dense clusters via a sparse gradient. To satisfy the uniform distribution assumption, the algorithm assigns large ${\sigma }_i$ values to the sparse bridge points (expanding them) and small ${\sigma }_i$ values to the cluster points (compressing them). This metric disparity weakens the edge weights between the bridge and the clusters. Consequently, the optimizer often “snaps” the bridge, creating an artificial separation where a continuous density gradient actually exists.

The interactive instrument for diagnosis and steering

Having established the classification of distortions, we now present the interactive diagnostic instrument developed to operationalize these theoretical insights. Rather than serving as a general-purpose production tool for massive datasets, this system acts as a “Concept Demonstrator” and an educational platform. It integrates a coordinated dual-view interface with a human-in-the-loop steering mechanism, allowing ML practitioners to diagnose algorithmic outputs against ground truth visually, explore topological trade-offs and rectify optimization entrapment.

Coordinated dual-view diagnosis

To enable effective diagnosis, the system employs a Coordinated Dual-View Design that juxtaposes the intrinsic reality against the algorithmic projection (Figure 4):

• Ground Truth & Embedding Views: The left panel (Figure 4(b)) renders the rotatable 3D probe to reveal the true topological relationships. The right panel (Figure 4(c)) displays the real-time UMAP result.

• Bi-directional Linking: Operations such as hovering, clicking, or box-selecting in either view immediately highlight the corresponding points in both views, while reducing the opacity of unselected points. This allows users to definitively verify, for example, whether a 2D visual gap corresponds to a genuine separation in 3D or an artificial tear.

Figure 4.

Overview of the Interactive Diagnosis and Steering Interface. (a) The Integrated Control Sidebar includes three functional modules: (1) dataset selection, (2) interaction controls (hyperparameter sliders and Manual Mode toggles), and (3) the comparison panel featuring history thumbnails and the temporal morphing slider. (b) The Ground Truth View (3D) serves as the rotatable visual reference for topological verification. (c) The Embedding View (2D) supports interactive probing and direct layout steering. The Interactive system is available at: https://interactive-umap-diagnostics.vercel.app/.

Parametric exploration & iterative comparison

While the dual-view interface reveals static artifacts, understanding the stability of these structures requires examining how they evolve under different hyperparameters. To track structural evolution across re-runs, our system implements a Temporal Morphing Mechanism (the Compare Panel in 4A):

History thumbnails

Upon each re-optimization, the system caches the previous embedding state ( $Y_{t-1}$ ) and generates a snapshot in the sidebar.

Interpolated trajectory tracking

A progress slider visualizes the structural shift between the previous ( $Y_{t-1}$ ) and current ( $Y_t$ ) layouts via linear interpolation:

$$Y_{\mathrm{view}}\left(\alpha \right)=(1-\alpha )·Y_{t-1}+\alpha ·Y_t,\alpha \in [0,1]$$

(1)

Dragging the slider animates the transition, allowing users to track point trajectories and distinguish stable structural trends from sudden collapses.

Layout steering via spatial injection

For distortions that cannot be resolved by parameter tuning alone (e.g. Type A failures where the layout is trapped in a semantic local minimum), our system offers a direct Steering mechanism.

Direct manipulation

We provide a “Manual Mode” that transforms the embedding view into an editable canvas. Users can click to select a labeled cluster or box-select fragmented islands. Once a target subset is defined, users can manually drag the misaligned clusters or scattered points in the 2D view to their semantically correct relative positions, thereby injecting explicit spatial constraints into the layout.

Algorithmic “Warm Start”

Crucially, these manual adjustments do not serve as the final visual output but as a spatial prior for the algorithm. We employ the user-modified coordinates $Y_{\mathrm{user}}$ as the initialization state for a re-optimization run:

$$Y_{\mathrm{final}}=\mathrm{UMAP}(X,\mathrm{init}=Y_{\mathrm{user}})$$

(2)

This “Warm Start” strategy effectively transports the optimization solver out of a semantically invalid local minimum. It forces the algorithm to respect the user’s imposed spatial logic (global structure) while still optimizing the local neighborhood relationships defined by the high-dimensional graph.

Case studies: Diagnosis and interactive correction

To validate the efficacy of our interactive instrument, we conducted a series of case studies using the probe datasets. These cases are categorized into two primary workflow scenarios: (1) Optimization Steering, where users inject human priors to overcome algorithmic local minima; and (2) Parameter Tuning, where users explore the parameter space to reveal multi-scale topological structures. Figures 5 and 6 illustrate the diagnosis and correction process for these cases.

Figure 5.

Visual Validation of Scenario I (Optimization Steering). Top Row (Case 1): The default UMAP layout fails to capture the antipodal symmetry of the dataset (a). The user manually drags the clusters to opposite sides to enforce a spatial constraint (b), resulting in a mathematically stable and semantically correct layout after re-optimization (c). Bottom Row (Case 2): The algorithm fragments the continuous Two-Moons manifold (d). The user collects the scattered points and drags them toward their parent cluster (e). This “warm start” allows the optimizer to recover the correct topological continuity (f).

Figure 6.

Visual Validation of Scenario II (Parameter Tuning). Left (Case 3): At default settings, the nested blobs appear as separate clusters (a). Increasing n_neighbors reveals the global containment relationship (b). Middle (Case 4): The central hole of the torus is initially obscured (c). Increasing both neighborhood size and minimum distance forces the layout to respect the genus-1 topology, recovering the loop structure (d). Right (Case 5): A low-density bridge is visible at default settings (e). However, a “stress test” with increased neighborhood size causes the bridge to snap (f), revealing that the connection, while topologically present, is density-fragile and sensitive to global optimization pressures.

Scenario I: Integrating human knowledge for layout steering

The non-convex objective function of UMAP can often trap the embedding in local minima, resulting in layouts that preserve local neighborhoods but violate global structural logic. In this scenario, our system allows users to manually adjust the positions of clusters. Crucially, this interaction does not simply “draw” the final result; instead, it provides a spatial sketch that serves as an informative initialization. The final embedding is computed by the UMAP optimizer, ensuring that the local manifold structure remains mathematically valid while respecting user-imposed global constraints.

Case 1: Restoring symmetry in antipodal-clusters (D1)

Diagnosis

In the default 2D embedding of the Antipodal-Clusters dataset (Figure 5(a)), clusters C1 and C2 are projected onto the same side of the central cluster O, falsely suggesting proximity. By rotating the 3D Ground Truth view, the user confirms that these clusters are in fact antipodal (located on opposite sides) relative to the center. This diagnosis reveals that the optimization has settled into a local minimum where the global symmetry is lost due to poor initialization.

Interactive correction

To enforce the correct spatial logic, the user enables Manual Mode to select and drag the misplaced cluster C2 to the opposite side of the canvas (Figure 5(b)). This action imposes a rough “opposition” constraint serving as a spatial prior.

Outcome

Upon re-executing UMAP with this user-defined layout as initialization, the algorithm converges to a stable solution where the global symmetry is restored while local cluster structures remain intact (Figure 5(c)). This case demonstrates that a semantically valid structural configuration exists within the optimization landscape, but the algorithm requires a “warm start” via human intervention to find it.

Case 2: Disentangling the Two-Moons (D3)

Diagnosis

The Two-Moons dataset represents two continuous, interleaved manifolds. Due to the spatial proximity of the two crescents, the default 2D embedding exhibits a severe tearing and entanglement artifact: the continuous structures are fragmented, with subsets of points incorrectly adhering to the opposing manifold (Figure 5(d)). By cross-referencing with the 3D Ground Truth view, the user confirms that these fragments are actually part of continuous curves, identifying the 2D layout as a topological failure.

Interactive correction

To resolve this, the user employs Manual Mode to spatially disentangle the manifolds. By lasso-selecting the fragmented points and dragging them back toward their parent cluster (Figure 5(e)), the user provides a strong spatial prior for separation and continuity.

Outcome

Re-running the optimization with these corrected positions as a warm start successfully repairs the manifold continuity (Figure 5(f)). This demonstrates the system’s capability to steer the algorithm out of complex local minima (where topology is compromised) without requiring exhaustive hyperparameter grid searches.

Scenario II: Recovering global topology via parameter tuning

While Scenario I addresses optimization failures, Scenario II focuses on the intrinsic trade-offs of the UMAP algorithm. In this scenario, our interactive system acts as a “probe,” allowing users to dynamically explore the parameter space (specifically n_neighbors and min_dist). By observing the morphological evolution of the embedding alongside the 3D ground-truth reference, users can effectively diagnose the true topology of the data.

Case 3: Resolving spatial logic in enclosed-blob (D2)

Diagnosis

In the 3D view, the Enclosed-Blob dataset features a dense Gaussian cluster surrounded by a spherical shell. With a default low n_neighbors (e.g. 15), the 2D projection depicts two completely separate, adjacent clusters (Figure 6(a)). This layout fails to preserve the topological containment, incorrectly suggesting that the two manifolds are independent.

Interactive correction

Using the parameter tuner, the user increases n_neighbors to expand the manifold approximation graph. As the parameter increases, the layout transforms: the outer cluster curves and extends, eventually enveloping the inner one (Figure 6(b)).

Outcome

While this high-parameter view introduces visual overlap, it correctly reconstructs the global nesting structure. This case provides a critical educational insight: UMAP parameters function as a multi-scale lens: low values emphasize local disjointedness, while high values reveal global topological relationships.

Case 4: Rescuing closed surface in torus-surface (D6)

Diagnosis

The Torus-Surface dataset is topologically characterized by a central hole (genus-1). However, standard UMAP settings often collapse the torus into an amorphous, featureless cloud in 2D (Figure 6(c)). The essential topology (a closed loop with a hole) is obscured because the algorithm, optimizing for local neighborhood preservation, sacrifices the global void to minimize embedding space.

Interactive correction

Guided by the 3D Ground Truth view, the user hypothesizes that retaining the central hole requires prioritizing global loop closure over local density. Consequently, the user significantly increases n_neighbors (to $>350$ ) and min_dist (to $>0.6$ ).

Outcome

The expanded neighborhood forces the algorithm to recognize the manifold’s loop structure, while the increased minimum distance prevents the embedding from crowding the center. As a result, the central hole re-emerges in the 2D layout (Figure 6(d)). This case demonstrates that preserving non-trivial topology often requires hyperparameter configurations that favor global structure.

Case 5: Assessing sensitivity to sparse structures in connected-blobs (D10)

Diagnosis

In the 3D view, the Connected-Blobs dataset exhibits a subtle feature: a low-density bridge connecting two dense clusters. While the default UMAP projection renders this bridge (Figure 6(e)), the visual reliability of such sparse structures is often ambiguous. Users must determine if it is a robust topological feature or a transient artifact of stochastic initialization.

Interactive exploration

To verify the structural integrity of this connection, the user performs a “stress test” by interactively increasing n_neighbors. This action forces the algorithm to prioritize broader global structures over local nuances.

Outcome

Counter-intuitively, as the neighborhood size expands, the bridge does not strengthen; instead, it is “absorbed” into the adjacent dense clusters, causing the projection to snap into two separated blobs (Figure 6(f)). This phenomenon reveals that the connection is topologically valid but density-fragile. The interaction serves as a critical warning: aggressive global optimization can sometimes obscure subtle density contrasts, treating sparse but valid connectors as noise.

Summary and generalization

The presented cases illustrate that UMAP artifacts are multifaceted, stemming from initialization traps (Cases 1 & 2), scale mismatches (Cases 3 & 4), or density heterogeneity (Case 5). Beyond these highlighted cases, we observed similar repair capabilities in continuous manifolds. For the Uniform-Strip (D5) and S-Curve (D7), parameter tuning sometimes resolves fragmentation and structural distortion. The Swiss-Roll (D4) presents a more nuanced case: while interactive steering or parameter tuning can repair tears to restore a continuous band, the result remains an “unfolded” flat strip. This highlights a fundamental distinction: while our tool allows users to enforce topological correctness, the loss of extrinsic geometry (e.g., the 3D curl) is an intrinsic characteristic of the algorithm that users must acknowledge. Collectively, these studies demonstrate that our system transforms dimensionality reduction from a static “black-box” output into a dynamic, explorable diagnosis.

Discussion

Our study investigates the intersection of non-linear dimensionality reduction and human-in-the-loop interaction. By utilizing synthetic 3D probes, we have exposed the fragility of UMAP, ranging from global structure instability (Type A) to density homogenization (Type C). Here, we discuss the broader implications of these findings for visual analytics and explainable AI (XAI).

Initialization as a channel for human intent

A core technical insight reinforced by our instrument is that the initialization state of a non-convex optimization algorithm is not merely a transient starting point, but a potent channel for structural injection. Standard implementations often treat initialization as a “cold start” (random or spectral). Our steering experiments (Scenario I) confirm that this lack of semantic guidance is a primary cause of spatial logic failures.

By allowing users to manipulate the initialization via direct manipulation, we bridge the “semantic gap” between the data’s intrinsic geometry and the user’s domain knowledge. Unlike abstract hyperparameters, which indirectly alter the graph topology, our spatial injection mechanism provides explicit geometric constraints. As demonstrated in Case 1 (Antipodal-Clusters), the user does not need to understand the cross-entropy loss function, but simply provides a structural “sketch.” The success of this “warm start” strategy highlights the plasticity of the manifold embedding, proving that the algorithm can support multiple valid minima, and human priors are often the best way to disambiguate them.

Synthetic probes as explainable AI instruments

While applying dimensionality reduction to high-dimensional data (e.g., scRNA-seq) is standard practice, it suffers from the “ground truth problem”: we cannot verify distortions without knowing the true manifold shape. This is where our approach of using 3D synthetic probes becomes critical as an educational tool in the context of XAI.

Our dual-view instrument forces a confrontation between the verifiable reality (3D Ground Truth) and the algorithmic artifact. This setup shifts the user’s mental model: UMAP is no longer viewed as a deterministic “truth generator” but as a sensitive, stochastic instrument requiring calibration. By enabling users to physically correct a “torn” manifold or stress-test density preservation (Case 5), the system fulfills a core goal of XAI: using interactive visualization not just to display outputs, but to interrogate the behavior and reliability of the underlying algorithm.

Design reflection and usage analysis

To contextualize the instrument’s utility, we discuss its design trade-offs with respect to usability and learning curves.

Accessibility versus flexibility

While the “Steering” interaction empowers users to correct artifacts (Scenario I), it relies on a prior knowledge of the ground truth. For novices, the direct manipulation is intuitive; however, understanding the mathematical implications of “injecting spatial priors” requires instructional guidance.

Cognitive load

The Dual-View interface effectively links 3D topology to 2D embeddings, yet managing simultaneous interactions (steering in 2D while observing 3D) may impose a cognitive load. We anticipate that the “Diagnostic” workflow (Scenario II) is easier for practitioners familiar with hyperparameter tuning, whereas the “Steering” workflow serves as a more powerful, albeit complex, learning tool for understanding optimization landscapes.

Limitations and future work

While our proof-of-concept demonstrates the educational and diagnostic value of interactive steering, several limitations outline the path for future research.

The 3D proxy constraint

Our classification relies on 3D synthetic probes where ground truth is visually verifiable. In real-world scenarios (e.g. 20,000-gene transcriptomics), no such visual reference exists. While 3D manifolds cannot fully capture the complexity of high-dimensional phenomena (e.g., the curse of dimensionality), we argue that success in ${\mathbb{R}}^3$ is a necessary condition for validity. If an algorithm fails to preserve topology even in ${\mathbb{R}}^3$ , it cannot be trusted in ${\mathbb{R}}^N$ . Future work could integrate Topological Data Analysis (TDA) metrics, such as persistent homology, to provide guidance when visual ground truth is absent.

Scalability of interaction

Our current prototype relies on dragging individual points or clusters. For datasets with millions of points, this pixel-level manipulation is impractical. Consequently, the current manual steering is well-suited for diagnostic purposes on data subsets rather than production-level layout adjustment for massive datasets. Future iterations should explore “control point” steering, where users manipulate a sparse set of representative landmarks, and the system propagates these constraints to the full dataset.

Persistence of Type C distortions

While our mechanism effectively resolves topological tearing (Type B) and spatial misalignment (Type A), it is less effective against metric distortions (Type C). Issues like density homogenization are deeply rooted in UMAP’s local connectivity normalization ( $\rho$ ), which is fundamental to its mathematical definition. This highlights a hard theoretical boundary: interactive steering can correct the layout, but it cannot easily alter the underlying metric assumptions of the algorithm.

Longitudinal educational evaluation

The current case study validates the technical feasibility and diagnostic potential of our instrument. Future work will focus on quantifying its pedagogical efficacy. User studies with data science students to measure learning outcomes (e.g., pre- and post-tests on UMAP mechanisms) and evaluate how interactive steering alters their mental models of dimensionality reduction would be a promising next step.

Conclusion

Uniform Manifold Approximation and Projection (UMAP) has become a de facto standard for high-dimensional visualization, yet its stochastic nature often generates misleading artifacts that are difficult to diagnose and correct. In this work, we presented an interactive educational instrument aimed at demystifying these failures. By utilizing synthetic 3D probes as verifiable ground truth, our system serves as a transparent laboratory for understanding the mechanisms of non-linear dimensionality reduction.

Our contributions are threefold. First, we established a classification of UMAP distortions derived from 10 synthetic 3D probes, which serve as verifiable “unit tests” for detecting topological pathologies (Type A/B) and metric inconsistencies (Type C). Second, we demonstrated the diagnostic power of Dual-View Linking and Temporal Morphing, revealing how parameter tuning functions as a multi-scale lens to distinguish stable features from artifacts. Third, we validated Steering via Spatial Injection, showing that injecting human priors as a “warm start” effectively guides the optimization out of semantically invalid local minima where standard tuning fails.

Ultimately, this work advocates for a paradigm shift in visual analytics: moving from the passive observation of static embeddings to an active, exploratory interrogation of the algorithm itself. By empowering practitioners to diagnose artifacts and inject domain knowledge, we take a significant step toward making black-box dimensionality reduction more transparent, trustworthy, and accessible.