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arxiv: 2605.23540 · v1 · pith:3DLXHY6Rnew · submitted 2026-05-22 · 💻 cs.LG

When One Point Is Not Enough: Addressing Ambiguous Instances in Dimensionality Reduction by Splitting

Diede P.M. van der Hoorn , Alessio Arleo , Fernando V. Paulovich This is my paper

Pith reviewed 2026-05-25 05:01 UTC · model grok-4.3

classification 💻 cs.LG
keywords dimensionality reductionambiguous instancespartial neighborhood embeddinggraph-based methodsUMAPvisualization artifactsneighborhood structure
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The pith

Ambiguous data points similar to multiple neighborhoods can be replicated as separate points in dimensionality reduction to show their full structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies a source of distortion in dimensionality reduction where points that are highly similar to several dissimilar groups in high dimensions get mapped to only one location in the projection. This leaves part of their neighborhood relationships invisible. The proposed fix detects these ambiguous instances through graph analysis of similarities and creates multiple copies of each point, positioning every copy inside one of its original neighborhoods. Demonstrated on UMAP but intended to apply to other local graph-based methods, the approach is said to expose previously hidden memberships and lessen the effect across examples, backed by both visual and quantitative checks.

Core claim

Ambiguous instances, defined as points highly similar to multiple mutually dissimilar neighborhoods in high-dimensional space, cause partial neighborhood embedding when projected to a single point; replicating each such instance as multiple points in the low-dimensional space and placing each copy in its respective neighborhood reveals the full set of neighborhood memberships.

What carries the argument

Graph-based detection of ambiguous instances via high-dimensional similarity relations, followed by replication of each instance into multiple projected points.

If this is right

  • Projections display previously invisible neighborhood memberships for the replicated points.
  • Partial neighborhood embedding is reduced across multiple tested examples.
  • The method extends to other local graph-based dimensionality reduction techniques beyond UMAP.
  • Quantitative measures confirm the reduction in embedding distortion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Interactive visualization systems could allow users to toggle between the multiple placements of a split point.
  • The replication step might be combined with existing quality metrics to automatically flag remaining projection artifacts.
  • Downstream tasks such as clustering performed on the augmented projection could recover groupings that standard single-point embeddings miss.

Load-bearing premise

High-dimensional similarity relations supply a reliable ground truth for identifying which neighborhoods an instance belongs to, without the detection process itself being compromised by projection loss.

What would settle it

A test set of known ambiguous instances where the method's splits fail to match independent neighborhood labels or where the added points increase rather than decrease measured partial embedding error.

Figures

Figures reproduced from arXiv: 2605.23540 by Alessio Arleo, Diede P.M. van der Hoorn, Fernando V. Paulovich.

Figure 1
Figure 1. Figure 1: We project the latent space of a CNN classifier trained on the Street View House Numbers dataset to study misclassifications. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An example of the ambiguous data instance [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic overview of our approach. Given a graph [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Vertex a is identified as LAP at r = 1 and at r = 2 because the neighborhoods are disjoint if a was removed. At r = 3, vertex h connects the previously disjoint neighborhoods. and edges of points b,c,g and e. Removing a creates two separate con￾nected components; {b,c,d} and {e, f,g}. At r = 3, vertex h connects the two previously separate components, thus a ∈/ LAP3. Note, however, that at r = 1, only dire… view at source ↗
Figure 5
Figure 5. Figure 5: Three scenarios for vertex splitting, showing the resulting graph [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Projection of the MNIST testset, first reduced to [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Two projections of the same text dataset containing information on papers, the standard projection suffers from partial neighborhood [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Projection of a mice cell landscape, colored by cell-type as in [ [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Ratios of G or G¯ embeddings vs high dimensional neighborhood preservation. the alignment between the embedding and the true high-dimensional neighborhoods; we call it ρHD. We used the datasets from the previous examples with the same pa￾rameters described before. We created five embeddings for each graph G and G¯, and set k to match the number of neighbors used to construct the graph. We use spectral ini… view at source ↗
Figure 11
Figure 11. Figure 11: Distribution of trustworthiness and continuity for the examples [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
read the original abstract

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, and 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper identifies ambiguous instances—points in high-dimensional space that are highly similar to multiple mutually dissimilar neighborhoods—as a source of visual artifacts in dimensionality reduction (DR). It proposes a graph-based method to detect such instances entirely in the original high-dimensional similarity graph, replicate each as multiple points, and embed the copies into their respective neighborhoods during projection (demonstrated with UMAP but claimed to generalize to other local graph-based DR methods). The central claims are that this reveals previously hidden neighborhood memberships and reduces partial neighborhood embedding, supported by examples and quantitative analyses.

Significance. If the claims hold, the work addresses an under-recognized source of distortion in neighborhood discovery from DR projections by operating the detection step upstream of any lossy embedding. The high-dimensional graph construction insulates identification from projection artifacts, avoiding the circularity concern raised in the stress-test note. Strengths include the parameter-free framing of the core procedure and the attempt at quantitative support beyond visual examples.

major comments (2)
  1. [§3] §3 (Method): The definition of 'ambiguous instances' and the procedure for identifying 'mutually dissimilar neighborhoods' via the high-dimensional similarity graph are not accompanied by explicit thresholds, similarity measures, or pseudocode; without these, the quantitative analyses in §4 cannot be reproduced or verified as load-bearing evidence for reduced partial neighborhood embedding.
  2. [§4] §4 (Quantitative analyses): No error bars, baseline comparisons, or controls for post-hoc parameter choices in the splitting decision are reported; the claim that the approach 'reduces partial neighborhood embedding across multiple examples' therefore rests on unverifiable metrics.
minor comments (2)
  1. [Abstract] The abstract states the method 'generalizes to other local graph-based DR techniques' but provides no explicit statement of the required interface (e.g., neighborhood graph input) in the main text.
  2. [Figures] Figure captions should explicitly state whether the shown projections use the original UMAP or the modified splitting procedure for direct visual comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential significance of addressing ambiguous instances upstream of the embedding step. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [§3] §3 (Method): The definition of 'ambiguous instances' and the procedure for identifying 'mutually dissimilar neighborhoods' via the high-dimensional similarity graph are not accompanied by explicit thresholds, similarity measures, or pseudocode; without these, the quantitative analyses in §4 cannot be reproduced or verified as load-bearing evidence for reduced partial neighborhood embedding.

    Authors: The procedure is deliberately parameter-free and operates exclusively on the input similarity graph (using the same similarity measure and neighborhood construction already required by the downstream DR method such as UMAP). No additional thresholds are introduced. To improve reproducibility and allow direct verification of the §4 metrics, we will insert explicit pseudocode for the detection and replication steps into the revised §3. revision: yes

  2. Referee: [§4] §4 (Quantitative analyses): No error bars, baseline comparisons, or controls for post-hoc parameter choices in the splitting decision are reported; the claim that the approach 'reduces partial neighborhood embedding across multiple examples' therefore rests on unverifiable metrics.

    Authors: Because the splitting decision is deterministic and parameter-free once the high-dimensional graph is fixed, post-hoc parameter controls are not applicable. We agree, however, that the current quantitative section would be strengthened by baseline comparisons (standard DR without splitting) and by explicit reporting of the metrics used to quantify partial neighborhood embedding. We will add these elements and clarify any sources of variation in the revised §4. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method operates entirely in high-dimensional space prior to projection

full rationale

The paper's core procedure constructs a high-dimensional similarity graph to detect ambiguous instances (points similar to multiple dissimilar neighborhoods) and replicates them before any DR is applied. Identification and splitting decisions occur in the original space, insulated from projection artifacts. No equations or steps reduce by construction to fitted parameters, self-definitions, or self-citation chains. The derivation is self-contained against external benchmarks with no load-bearing self-referential elements.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate specific free parameters or axioms; the method implicitly assumes high-dimensional similarities define true neighborhoods and requires at least one threshold or criterion for identifying ambiguity.

pith-pipeline@v0.9.0 · 5768 in / 1029 out tokens · 27208 ms · 2026-05-25T05:01:01.271345+00:00 · methodology

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