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CS-313 · computer science · mathematics · Algorithms and data structures · Computational geometry · Computational statistics · Graph theory and combinatorics

Density and Geometry Recovery from Nearest-Neighbor Graphs

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A directed $k$-nearest-neighbor graph links each sample to its $k$ closest distinct samples. An unweighted adjacency matrix records only adjacency, without distances. Similarities consist of translation, orthogonal transformation, and common rescaling.

Statement recorded in the supplied source.
Let $X_1,\ldots,X_n$ be i.i.d. samples from a density $p$ on $\mathbb R^d$, and let $A$ be the adjacency matrix of their unweighted $k$-nearest-neighbor graph. Can $p(X_i)$ be estimated up to constant factors from $A$ alone, and can the sample locations be approximately reconstructed up to translation, rotation, and rescaling when $n$ is large? [S2]
Scope qualification.
The supplied question does not specify directed versus symmetrized adjacency, density regularity, scaling of $k$, or reconstruction-error norm.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Recover sample densities and geometric positions using only nearest-neighbor adjacency data.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30002105 and source code OWR-11793-003 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-313-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-313

Density and Geometry Recovery from Nearest-Neighbor Graphs

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A directed $k$-nearest-neighbor graph links each sample to its $k$ closest distinct samples. An unweighted adjacency matrix records only adjacency, without distances. Similarities consist of translation, orthogonal transformation, and common rescaling.

Statement recorded in the supplied source.
Let $X_1,\ldots,X_n$ be i.i.d. samples from a density $p$ on $\mathbb R^d$, and let $A$ be the adjacency matrix of their unweighted $k$-nearest-neighbor graph. Can $p(X_i)$ be estimated up to constant factors from $A$ alone, and can the sample locations be approximately reconstructed up to translation, rotation, and rescaling when $n$ is large? [S2]
Scope qualification.
The supplied question does not specify directed versus symmetrized adjacency, density regularity, scaling of $k$, or reconstruction-error norm.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Recover sample densities and geometric positions using only nearest-neighbor adjacency data.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30002105 and source code OWR-11793-003 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

Scale invariance and a finite geometric ambiguity

The primary report uses an undirected unweighted neighbor graph; the preserved catalogue also explains the directed convention. The following obstructions apply to either convention and leave that distinction explicit. Later primary work studies density recovery and synchronization-based localization under additional assumptions. We test whether the graph alone can supply an absolute scale and whether every finite neighbor pattern determines a geometry.
Proposition: Let $p$ be a density on $\mathbb R^d$ and $a>0$. The coupled samples $Y_i=aX_i$ have density $p_a(y)=a^{-d}p(y/a)$ and exactly the same nearest-neighbor adjacency as the $X_i$. Consequently no graph-only estimator can estimate absolute density values with one fixed multiplicative factor $C\ge1$, uniformly over all rescalings, with success probability greater than $1/2$ at a fixed sample index where $p(X_i)>0$ almost surely.
Proof: Change of variables gives the stated density. Every interpoint distance is multiplied by $a$, preserving its order and any consistent tie convention. Couple the estimator's random seed as well, so its output $T_i$ is identical on the two graphs. The permitted intervals for successful estimation in the two models are \[ [p(X_i)/C,Cp(X_i)]\quad\hbox{and}\quad [a^{-d}p(X_i)/C,Ca^{-d}p(X_i)]. \] Choose $a^d>C^2$. These intervals are disjoint on every coupled sample with positive density. The two success indicators therefore sum to at most one, and the same is true of their probabilities. They cannot both exceed $1/2$. ∎
This excludes only a uniform absolute-scale guarantee. Density ratios or recovery up to an unknown common multiplicative normalization do not have this obstruction, and "up to constant factors" might intend such a normalization. Thus the argument does not refute the intended density-recovery results. For a separate finite example, the labeled configurations $(0,1,10,11)$ and $(0,1,100,101)$ on the line have identical directed one-nearest-neighbor graphs: the first two points choose each other, as do the last two. Their undirected versions are also identical. They are not related by a labeled similarity $x\mapsto ax+b$ with $a\ne0$: the first two coordinates force $b=0,a=1$, after which the third coordinates disagree. The invariant ratio of the third-to-first distance to the second-to-first distance is 10 versus 100.

Verification and remaining gaps

All four nearest-neighbor choices and the two distance ratios were checked exactly. A scale example with $d=1,C=2,a=5$ yields disjoint density intervals for a positive unit reference density. The attempted reconstruction route from a single finite adjacency pattern therefore needs more assumptions. The finite example is not an asymptotic impossibility theorem for regular connected supports and a suitable growing $k$. No estimator, error rate, or general reconstruction theorem for that regime is obtained. The later papers were checked at their stated contribution level, not fully reproved.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2012/31. Relevant passages read: 2012/31 pp. 1914--1915, undirected neighbor-graph density and position questions.

https://proceedings.neurips.cc/paper/2013/hash/eae27d77ca20db309e056e3d2dcd7d69-Abstract.html. Read primary 2013 abstract describing density estimation from local link counts and integration along graph paths; full assumptions and proof not independently verified here.

https://arxiv.org/abs/1504.00722. Read primary abstract describing synchronization-based directed-neighbor localization up to similarities and density-estimation experiments; not a verified universal reconstruction theorem.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch019/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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