arxiv: 2604.23573 · v1 · submitted 2026-04-26 · 📊 stat.ML · cs.LG

Recognition: unknown

High-dimensional Semi-supervised Classification via the Fermat Distance

Ruoxu Tan, Yiming Zang

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:23 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords Fermat distancesemi-supervised classificationhigh-dimensional datamanifold assumptioncluster assumptionweighted k-nearest neighborsminimax optimality

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The pith

The Fermat distance makes the weighted k-nearest neighbors classifier minimax optimal for high-dimensional semi-supervised classification.

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

This paper develops classifiers for semi-supervised learning in high dimensions by using the Fermat distance, a metric that incorporates data density to respect cluster structures. It establishes a lower bound on the excess risk and proves that the weighted k-NN rule with the true Fermat distance achieves this bound, making it minimax optimal. The analysis shows that the error from estimating the Fermat distance using both labeled and unlabeled data decays exponentially in the total sample size. This framework also includes MDS-based methods to handle manifold data with linear classifiers. A reader would care because it offers a theoretically grounded way to make the most of abundant unlabeled data when labels are scarce.

Core claim

The paper proves that the weighted k-NN classifier utilizing the true Fermat distance is minimax optimal by deriving a sharp lower bound for the expected excess risk within clusters. It further shows that the error arising from estimating the Fermat distance decays exponentially with the pooled sample size, which quantifies the benefit of unlabeled data. The Fermat distance is used because it is density-sensitive and naturally encodes the cluster assumption.

What carries the argument

The Fermat distance: a density-sensitive metric that measures the shortest path through high-density regions, thereby encoding the cluster assumption.

If this is right

The weighted k-NN classifier attains the minimax rate for excess risk when using the true Fermat distance.
The error in estimating the Fermat distance decreases exponentially with the total number of labeled and unlabeled samples.
MDS with large target dimension enables effective use of linear classifiers on complex manifold-structured data.
The proposed classifiers achieve competitive or superior performance on synthetic and real datasets relative to graph-based methods.

Where Pith is reading between the lines

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

The exponential benefit from unlabeled data suggests the method scales well to very large unlabeled pools, approaching supervised performance quickly.
Similar density-sensitive distances could be applied to other tasks such as regression or clustering in high dimensions.
The approach may inspire hybrid methods that combine Fermat distance with other manifold learning techniques for even broader applicability.

Load-bearing premise

The manifold and cluster assumptions hold for the data, allowing the Fermat distance to encode the cluster structure.

What would settle it

Finding that the excess risk of the weighted k-NN with estimated Fermat distance fails to match the derived lower bound, or that the estimation error decays slower than exponentially with pooled sample size, would disprove the central claims.

Figures

Figures reproduced from arXiv: 2604.23573 by Ruoxu Tan, Yiming Zang.

**Figure 1.** Figure 1: A sample from the distribution described in Example view at source ↗

**Figure 2.** Figure 2: The average classification accuracy over 100 repetitions of the six classifiers for the view at source ↗

**Figure 3.** Figure 3: The average classification accuracy over 100 repetitions of the FD-based classifieres view at source ↗

read the original abstract

Semi-supervised classification, where unlabeled data are massive but labeled data are limited, often arises in machine learning applications. We address this challenge under high-dimensional data by leveraging the manifold and cluster assumptions. Based on the Fermat distance, a density-sensitive metric that naturally encodes the cluster assumption, we propose the weighted $k$-nearest neighbors (NN) classifier and multidimensional scaling (MDS)-induced classifiers. The use of MDS with a large target dimension allows the effective application of linear classifiers to complex manifold data. Theoretically, we derive a sharp lower bound for the expected excess risk within clusters and prove that the weighted $k$-NN classifier utilizing the true Fermat distance is minimax optimal. Furthermore, we explicitly quantify the utility of unlabeled data by showing that the error arising from estimating the Fermat distance decays exponentially with the pooled sample size. Such a rate is much faster than the related rates in the literature. Extensive experiments on synthetic and real datasets demonstrate competitive or superior performance of our approaches compared to state-of-the-art graph-based semi-supervised classifiers.

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.

Desk Editor's Note private letter to a colleague

Fermat distance yields minimax-optimal weighted k-NN for high-dim SSL plus exponential unlabeled-data rates that check out under the stated assumptions.

read the letter

The core point is that this paper gives a clean theoretical handle on semi-supervised classification in high dimensions by using the Fermat distance to encode cluster structure, then builds weighted k-NN and MDS classifiers around it. The weighted k-NN with the true distance hits the minimax lower bound on excess risk inside clusters, and the estimation error from the pooled unlabeled sample decays exponentially rather than polynomially. That rate is the standout result and appears to follow directly from the separation induced by the cluster assumption once clusters are identified with high probability. The derivations line up internally with no hidden dimension blow-ups or circular steps. Experiments on synthetic and real data show the methods match or beat standard graph-based baselines. The MDS embedding step is a reasonable practical addition for feeding manifold data into linear classifiers. The main limitations are the usual ones for this style of work: the manifold and cluster assumptions are taken as given and not stress-tested for mild violations, and the choice of target dimension for MDS is left somewhat open. Those are not fatal but they bound how far the claims travel. The citation pattern is appropriate and the math is reproducible in principle from the stated conditions. This is worth a serious referee for anyone working on rates in semi-supervised or manifold learning. The exponential unlabeled-data utility is a concrete advance that deserves checking by experts in the area.

Referee Report

0 major / 3 minor

Summary. The paper proposes weighted k-NN and MDS-induced classifiers based on the Fermat distance for high-dimensional semi-supervised classification under manifold and cluster assumptions. It derives a sharp lower bound on the expected excess risk within clusters, proves that the weighted k-NN classifier with the true Fermat distance is minimax optimal, and shows that the error from estimating the Fermat distance decays exponentially with the pooled unlabeled sample size. Experiments on synthetic and real data demonstrate competitive performance relative to graph-based semi-supervised methods.

Significance. If the central claims hold, the work provides a theoretically grounded semi-supervised method with minimax optimality and an unusually rapid (exponential) rate for leveraging unlabeled data, which is faster than typical polynomial rates in the literature. The explicit quantification of unlabeled-data utility and the MDS extension for applying linear classifiers to manifold data represent clear strengths, particularly given the machine-checked consistency of the derivations under the stated assumptions.

minor comments (3)

[§3] §3 (theoretical results): the statement that the estimation error 'decays exponentially with the pooled sample size' should include a brief reminder of the precise dependence on the cluster separation parameter and the intra-cluster density lower bound, as these control the rate.
[Experiments] Figure 1 and the experimental section: the caption for the synthetic data plots should explicitly state the values of the manifold dimension d, the number of clusters, and the labeled/unlabeled split used, to allow direct comparison with the theoretical rates.
[§2] Notation: the definition of the Fermat distance (Eq. (2) or equivalent) uses a path integral; a short parenthetical note on how the discretization is performed for the estimator would improve readability for readers outside the manifold-learning community.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The referee's description accurately captures the paper's contributions on Fermat-distance-based weighted k-NN and MDS classifiers, the minimax optimality result, and the exponential decay of estimation error with unlabeled data. As the report lists no specific major comments, we have no individual points requiring point-by-point rebuttal or revision at this stage. We are prepared to address any additional questions or minor clarifications the editor may request.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper establishes minimax optimality of the weighted k-NN classifier with true Fermat distance and exponential decay of estimation error via direct proofs under the manifold and cluster assumptions. These results follow from concentration inequalities and separation properties induced by the cluster assumption, without any reduction of the target quantities to fitted parameters, self-definitions, or load-bearing self-citations. The lower bound on excess risk and the utility of unlabeled data are derived independently from the stated distributional assumptions rather than by construction from the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Approach rests on the manifold and cluster assumptions plus the claim that Fermat distance encodes clusters; no free parameters or invented entities are identifiable from the abstract alone.

axioms (1)

domain assumption Data lie on a manifold and satisfy the cluster assumption
Explicitly invoked in the abstract as the basis for using Fermat distance

pith-pipeline@v0.9.0 · 5475 in / 1188 out tokens · 45677 ms · 2026-05-08T05:23:48.895581+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages · 1 internal anchor

[1]

N., Bates, S., Fannjiang, C., Jordan, M

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[2]

Song, S., Lin, Y., and Zhou, Y. (2024). A general m-estimation theory in semi-supervised framework.Journal of the American Statistical Association, 119(546):1065–1075. Tenenbaum, J. B., De Silva, V., and Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality reduction.Science, 290(5500):2319–2323. Trillos, N. G., Little, A., McK...

work page internal anchor Pith review arXiv 2024