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arxiv: 2509.19250 · v1 · pith:QSRXFINBnew · submitted 2025-09-23 · 📊 stat.ML · cs.LG

Recovering Wasserstein Distance Matrices from Few Measurements

Muhammad Rana , Abiy Tasissa , HanQin Cai , Yakov Gavriyelov , Keaton Hamm This is my paper

Pith reviewed 2026-05-21 21:58 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords Wasserstein distancematrix completionNyström methodmultidimensional scalingmanifold learningdistance matrix recoverymedical image classification
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The pith

Nyström completion recovers Wasserstein distance matrices from O(d log d) columns while preserving MDS stability.

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

The paper develops two ways to estimate full Wasserstein distance matrices from only a small number of entries, since computing every pairwise distance is very expensive. One approach uses standard matrix completion on upper-triangular samples; the other uses Nyström completion that needs only O(d log d) columns, where d is the target embedding dimension. The authors prove that multidimensional scaling embeddings remain stable under the Nyström approximation and show that this method can beat ordinary completion for the same number of distance calculations. On the OrganCMNIST medical-image set, classification accuracy stays essentially unchanged even when only 10 percent of the columns are computed to form the embedding.

Core claim

Nyström completion, by sampling and using only O(d log d) columns of the Wasserstein distance matrix, produces an approximation that keeps multidimensional scaling stable; the resulting low-dimensional embeddings support reliable downstream classification on OrganCMNIST data even when just 10 percent of the columns are available.

What carries the argument

Nyström completion on Wasserstein distance matrices, which reconstructs the full matrix and its MDS embedding from a small set of sampled columns.

If this is right

  • Manifold-learning pipelines that rely on Wasserstein distances can run with far fewer pairwise computations.
  • Nyström sampling outperforms triangular matrix completion when the computational budget is fixed.
  • Classification performance on medical-image benchmarks remains stable under this reduced sampling regime.
  • Stability proofs for MDS extend directly to the Nyström-approximated Wasserstein matrices.

Where Pith is reading between the lines

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

  • The same column-sampling idea could be tested on other optimal-transport distances beyond Wasserstein.
  • If the low-rank structure is common, the method might scale Wasserstein-based learning to much larger data sets.
  • Combining Nyström sampling with faster approximate Wasserstein solvers could reduce cost even further.

Load-bearing premise

The Wasserstein distance matrix admits a sufficiently accurate low-rank or column-space approximation that can be recovered from O(d log d) sampled columns.

What would settle it

An experiment on a dataset where reducing to 10 percent of the columns causes a large increase in MDS stress or a clear drop in classification accuracy.

Figures

Figures reproduced from arXiv: 2509.19250 by Abiy Tasissa, HanQin Cai, Keaton Hamm, Muhammad Rana, Yakov Gavriyelov.

Figure 1
Figure 1. Figure 1: Classification accuracy (mean ± 1 standard deviation of 100 trials) vs. number of columns computed in Algorithm 2 for 2000 OrganCMNIST data points. gap makes it especially important to develop principled methods that reduce the cost of computing W2 distances. In future work, we will explore theoretical conditions under which W2 distance matrices admit approximate low-rank structure or fast spectral decay, … view at source ↗
read the original abstract

This paper proposes two algorithms for estimating square Wasserstein distance matrices from a small number of entries. These matrices are used to compute manifold learning embeddings like multidimensional scaling (MDS) or Isomap, but contrary to Euclidean distance matrices, are extremely costly to compute. We analyze matrix completion from upper triangular samples and Nystr\"{o}m completion in which $\mathcal{O}(d\log(d))$ columns of the distance matrices are computed where $d$ is the desired embedding dimension, prove stability of MDS under Nystr\"{o}m completion, and show that it can outperform matrix completion for a fixed budget of sample distances. Finally, we show that classification of the OrganCMNIST dataset from the MedMNIST benchmark is stable on data embedded from the Nystr\"{o}m estimation of the distance matrix even when only 10\% of the columns are computed.

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 proposes two algorithms for recovering square Wasserstein distance matrices from few entries: one based on matrix completion from upper-triangular samples and another using Nyström completion that computes only O(d log d) columns, where d is the target embedding dimension. It proves stability of multidimensional scaling (MDS) embeddings under the Nyström approach, shows that Nyström can outperform standard matrix completion for a fixed budget of distance computations, and reports that classification accuracy on the OrganCMNIST dataset remains stable when embeddings are obtained from a Nyström estimate using only 10% of the columns.

Significance. If the stability result and empirical claims hold, the work offers a practical route to scaling manifold learning methods that rely on Wasserstein distances, which are otherwise prohibitively expensive to compute in full. The combination of a stated proof for MDS stability together with reproducible downstream classification experiments on a public benchmark constitutes a concrete advance for optimal-transport-based dimensionality reduction.

major comments (2)
  1. [Section 4] The stability proof for MDS under Nyström completion (Section 4) and the claim that O(d log d) columns suffice both presuppose that the column space of the Wasserstein distance matrix is captured to sufficient accuracy by the sampled columns. No independent diagnostic—such as the decay rate of singular values of the full distance matrix or the Frobenius reconstruction error on held-out entries—is reported to verify this modeling assumption before the stability argument is invoked.
  2. [Experimental section / Table 2] Table 2 (or the corresponding experimental table) shows classification accuracy on OrganCMNIST remains within a few percent of the full-matrix baseline at 10% columns, yet this downstream metric does not directly test the MDS embedding distortion that the stability theorem bounds. A direct comparison of embedding stress or Procrustes error between the Nyström and full embeddings would be needed to confirm that the observed classifier stability is not merely an artifact of the particular linear head.
minor comments (2)
  1. [Section 3] Notation for the number of sampled columns is introduced as O(d log d) in the abstract but should be stated precisely (with the hidden constants) when the Nyström procedure is defined.
  2. [Section 4] The manuscript would benefit from an explicit statement of the precise assumptions on the Wasserstein kernel (e.g., boundedness or Lipschitz conditions) that are used to obtain the stability guarantee.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below, indicating planned revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Section 4] The stability proof for MDS under Nyström completion (Section 4) and the claim that O(d log d) columns suffice both presuppose that the column space of the Wasserstein distance matrix is captured to sufficient accuracy by the sampled columns. No independent diagnostic—such as the decay rate of singular values of the full distance matrix or the Frobenius reconstruction error on held-out entries—is reported to verify this modeling assumption before the stability argument is invoked.

    Authors: We thank the referee for highlighting this point. The stability result in Section 4 is stated under the modeling assumption that the selected columns provide a sufficiently accurate basis for the column space of the distance matrix. We agree that empirical verification of this assumption would strengthen the presentation. In the revised manuscript we will add (i) plots of the singular-value decay of the full Wasserstein distance matrices for all datasets used in the experiments and (ii) the Frobenius reconstruction error of the Nyström approximation evaluated on a held-out set of entries. These diagnostics will be placed in a new subsection of Section 4 immediately preceding the stability theorem. revision: yes

  2. Referee: [Experimental section / Table 2] Table 2 (or the corresponding experimental table) shows classification accuracy on OrganCMNIST remains within a few percent of the full-matrix baseline at 10% columns, yet this downstream metric does not directly test the MDS embedding distortion that the stability theorem bounds. A direct comparison of embedding stress or Procrustes error between the Nyström and full embeddings would be needed to confirm that the observed classifier stability is not merely an artifact of the particular linear head.

    Authors: We agree that a direct quantification of embedding distortion provides a more immediate test of the stability theorem than the downstream classification task alone. In the revised experimental section we will augment Table 2 (and the associated figures) with two additional metrics: (i) the classical MDS stress value computed on the full versus Nyström embeddings and (ii) the Procrustes distance between the two embeddings, both reported as functions of the column budget. These quantities will be computed on the same OrganCMNIST splits used for classification, allowing readers to relate embedding fidelity directly to the observed classifier stability. revision: yes

Circularity Check

0 steps flagged

No circularity: stability proof and empirical claims rest on independent mathematical analysis and separate dataset evaluation

full rationale

The paper's core claims rest on two algorithms for distance matrix recovery, a stated stability proof for MDS under Nyström sampling of O(d log d) columns, and downstream classification experiments on OrganCMNIST using only 10% columns. No load-bearing step reduces by construction to fitted parameters or self-citations; the low-rank column-space modeling assumption is used for the proof but is not derived from the target results themselves, and the empirical validation uses held-out classification performance rather than reconstruction error on the same samples. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the modeling assumption that Wasserstein distance matrices are amenable to Nyström column sampling and on standard matrix completion theory; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Wasserstein distance matrices admit accurate approximation via Nyström sampling of O(d log d) columns
    This assumption underpins both the proposed Nyström algorithm and the stability result for MDS.

pith-pipeline@v0.9.0 · 5684 in / 1262 out tokens · 29159 ms · 2026-05-21T21:58:38.223970+00:00 · methodology

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