Multidimensional Scaling

Also known as MDS

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11 papers1 paper in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Sep 30, 2026

Gromov-Wasserstein Distillation for Inductive Multi-View Embedding

Gromov-Wasserstein multidimensional scaling (GW-MDS) learns low-dimensional representations from relational data but remains transductive, providing no explicit mapping for unseen samples. We introduce an inductive framework based on barycentric distillation. A GW-MDS teacher learns a latent support and an optimal transport plan from the training data, and barycentric projection converts the resulting coupling into sample-aligned targets. A neural student then learns an explicit out-of-sample mapping, avoiding additional relational-matrix construction and GW optimization at inference. We formulate the approach for single-view data and extend it to Mean-GWMDS and Multi-GWMDS teachers through consensus and selected-projection targets learned by a multi-view student with view-specific encoders. We also investigate a direct neural baseline trained solely with a GW objective. Experiments on synthetic and real-world data using Euclidean, geodesic, and cosine relations show that the distilled models preserve the teacher geometry on unseen samples and consistently outperform direct neural GW training in sample-indexed relational preservation. These results establish barycentric projection as an effective bridge between transductive GW embeddings and inductive neural mappings.

Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Instituto Federal do Ceará (IFCE) Canindé, Ceará, Brazil · Federal University of Ceará (UFC) Fortaleza, Ceará, Brazil · Sigma Nova Paris, France

Representation LearningMulti-View LearningManifold LearningRelational Knowledge DistillationMultidimensional Scaling
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Sep 5, 2026

Robust conditional dimension reduction for dissimilarity data

Conditional dimension reduction (cDR) learns low-dimensional latent coordinates while accounting for observed covariates that represent known sources of variation in the data. Conditional Multidimensional Scaling (cMDS) is a cDR technique that works directly with dissimilarity data. Its standard squared-stress formulation, however, is sensitive to contaminated dissimilarity, since outliers can dominate the objective and distort the learned configuration. We proposed Robust Conditional Multidimensional Scaling (rcMDS) by replacing the squared-stress criterion with a Fair M-estimation objective. We developed a reweighted conditional SMACOF algorithm to optimize this objective. The proposed algorithm admits computationally tractable updates, and its stabilized objective values decrease monotonically and converge to a finite limit. Experiments on synthetic and real data show that the pro

Xiao Ling, Anh Bui

Department of Mathematics, Auburn University at Montgomery Montgomery, AL 36117, USA · Department of Mathematics and Statistics, Virginia Commonwealth University Richmond, VA 23284, USA

Multidimensional ScalingDimensionality ReductionRobust Estimation
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Jun 29, 2026

Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS

This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.

Flor Martinez-Sermeno, Arturo Jaramillo, Johan Van Horebeek

Multidimensional ScalingDimensionality ReductionHeavy-Tailed DistributionsSliced Wasserstein DistanceWasserstein Distance
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May 24, 2026

Optimizing Multidimensional Scaling in Gini Metric Spaces

The Gini Multidimensional Scaling (Gini MDS) framework extends the Euclidean multidimensional scaling. We introduce a Gini pseudo-distance based on values and their ranks that depends on a fine-tunable hyperparameter. This pseudo-distance allows flexible exploration of latent configurations, enabling embeddings that best match observed dissimilarities. The Gini MDS is shown to be robust to noise and outliers, making it well-suited for real-world applications. We provide experiments on 16 UCI datasets with outliers and on MNIST images with noise to show that the Gini MDS outperforms the Euclidean MDS on noisy data. Finally, a tensor-based implementation in PyTorch provides GPU acceleration and efficient computation compared to the standard MDS of the sklearn library.

Cassandra Mussard, Stéphane Mussard

CESBIO, Univ. Toulouse Toulouse, France · CHROME, Univ. Nˆımes UM6P, AIRESS Nˆımes, France

Multidimensional ScalingDimensionality Reduction
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May 1, 2026

Bridging Graph Drawing and Dimensionality Reduction with Stochastic Stress Optimization

Both Dimensionality Reduction (DR) and Graph Drawing (GD) aim to visualize abstract, non-linear structures, yet rely on different optimization paradigms. This contrast is evident in Multidimensional Scaling (MDS), which typically depends on the SMACOF algorithm despite graph drawing results showing that simpler stochastic optimization schemes can be more effective for the same objective. We bridge these domains by adapting Stochastic Gradient Descent (SGD) techniques from graph drawing to vector data embedding. We present a scikit-learn compatible estimator that minimizes global stress through local pairwise updates, improving upon the existing implementation. Experiments on standard high-dimensional benchmarks show that our stochastic solver converges substantially faster than SMACOF while achieving comparable or lower stress.

Daniel Hangan, Stephen Kobourov, Jacob Miller

Technical University of Munich

Scientific VisualizationData VisualizationMultidimensional ScalingDimensionality ReductionStochastic Gradient Descent
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Apr 26, 2026

Gromov-Wasserstein Methods for Multi-View Relational Embedding and Clustering

Learning low-dimensional representations from multi-view relational data is challenging when underlying geometries differ across views. We propose Bary-GWMDS, a Gromov-Wasserstein-based method that operates directly on distance matrices to learn a consensus embedding preserving shared relational structure. By leveraging intrinsic distances, the approach naturally handles nonlinear distortions across views. We also introduce Mean-GWMDS-C, a clustering-oriented formulation that averages distance matrices and learns reduced-support representations via a consensus Gromov-Wasserstein transport. Experiments on synthetic and real-world datasets show that the proposed framework yields stable and geometrically meaningful embeddings.

Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Canindé, Ceará, Brazil · Federal University of Ceara, Fortaleza, Ceará, Brazil · Sigma Nova, Paris, France

Representation LearningMultidimensional ScalingMulti-View LearningMulti-View ClusteringDimensionality ReductionOptimal Transport
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Apr 26, 2026

High-dimensional Semi-supervised Classification via the Fermat Distance

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 kk-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 kk-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.

Ruoxu Tan, Yiming Zang

School of Mathematical Sciences and School of Economics and Management, Tongji University · Department of Sciences, North China University of Technology

Multidimensional Scalingk-Nearest NeighborsSemi-Supervised LearningStatistical Learning Theory
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Apr 6, 2026

On the Geometry of Positional Encodings in Transformers

Neural language models process sequences of words, but the mathematical operations inside them are insensitive to the order in which words appear. Positional encodings are the component added to remedy this. Despite their importance, positional encodings have been designed largely by trial and error, without a mathematical theory of what they ought to do. This paper develops such a theory. Four results are established. First, any Transformer without a positional signal cannot solve any task sensitive to word order (Necessity Theorem). Second, training assigns distinct vector representations to distinct sequence positions at every global minimiser, under mild and verifiable conditions (Positional Separation Theorem). Third, the best achievable approximation to an information-optimal encoding is constructed via classical multidimensional scaling (MDS) on the Hellinger distance between positional distributions; the quality of any encoding is measured by a single number, the stress (Proposition 5, Algorithm 1). Fourth, the optimal encoding has effective rank r = rank(B) <= n-1 and can be represented with r(n+d) parameters instead of nd (minimal parametrisation result). Appendix A develops a proof of the Monotonicity Conjecture within the Neural Tangent Kernel (NTK) regime for masked language modelling (MLM) losses, sequence classification losses, and general losses satisfying a positional sufficiency condition, through five lemmas. Experiments on SST-2 and IMDB with BERT-base confirm the theoretical predictions and reveal that Attention with Linear Biases (ALiBi) achieves much lower stress than the sinusoidal encoding and Rotary Position Embedding (RoPE), consistent with a rank-1 interpretation of the MDS encoding under approximate shift-equivariance.

Giansalvo Cirrincione

Laboratoire LTI, Université de Picardie Jules Verne Chemin du Thil, 80025 Amiens, France

Multidimensional ScalingPositional EncodingRepresentation Geometry in Language Models
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Apr 3, 2026

Structure-Preserving Multi-View Embedding Using Gromov-Wasserstein Optimal Transport

Multi-view data analysis seeks to integrate multiple representations of the same samples in order to recover a coherent low-dimensional structure. Classical approaches often rely on feature concatenation or explicit alignment assumptions, which become restrictive under heterogeneous geometries or nonlinear distortions. In this work, we propose two geometry-aware multi-view embedding strategies grounded in Gromov-Wasserstein (GW) optimal transport. The first, termed Mean-GWMDS, aggregates view-specific relational information by averaging distance matrices and applying GW-based multidimensional scaling to obtain a representative embedding. The second strategy, referred to as Multi-GWMDS, adopts a selection-based paradigm in which multiple geometry-consistent candidate embeddings are generated via GW-based alignment and a representative embedding is selected. Experiments on synthetic manifolds and real-world datasets show that the proposed methods effectively preserve intrinsic relational structure across views. These results highlight GW-based approaches as a flexible and principled framework for multi-view representation learning.

Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Canindé, Ceará, Brazil · Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, Ceará, Brazil · Sigma Nova, Paris, France

Multidimensional ScalingMulti-View LearningDimensionality ReductionMulti-View GeometryOptimal Transport
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Mar 17, 2026

Manifold-Matching Autoencoders

We study a simple unsupervised regularization scheme for autoencoders called Manifold-Matching (MMAE): we align the pairwise distances in the latent space to those of the input data space by minimizing mean squared error. Because alignment occurs on pairwise distances rather than coordinates, it can also be extended to a lower-dimensional representation of the data, adding flexibility to the method. We find that this regularization outperforms similar methods on metrics based on preservation of nearest-neighbor distances and persistent homology-based measures. We also observe that MMAE provides a scalable approximation of Multi-Dimensional Scaling (MDS).

Laurent Cheret, Vincent Létourneau, Isar Nejadgholi +3

Department of Computer Science, University of Ottawa, Ottawa, Canada · MILA, Montréal, Canada · National Research Council of Canada, Ottawa, Canada

Multidimensional ScalingAutoencodersLatent-Space RegularizationGeometric Representation LearningDimensionality ReductionLatent Space Alignment+1
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May 10, 2025

Out-of-Sample Embedding with Proximity Data: Projection versus Restricted Reconstruction

The problem of using proximity (similarity or dissimilarity) data for the purpose of "adding a point to a vector diagram" was first studied by J.C. Gower in 1968. Since then, a number of methods -- mostly kernel methods -- have been proposed for solving what has come to be called the problem of out-of-sample embedding. We survey the various kernel methods that we have encountered and show that each can be derived from one or the other of two competing strategies: projection or restricted reconstruction. Projection can be analogized to a well-known formula for adding a point to a principal component analysis. Restricted reconstruction poses a different challenge: how to best approximate redoing the entire multivariate analysis while holding fixed the vector diagram that was previously obtained. This strategy results in a nonlinear optimization problem that can be simplified to a unidimensional search. Various circumstances may warrant either projection or restricted reconstruction.

Michael W. Trosset, Kaiyi Tan, Minh Tang +1

Department of Statistics, Indiana University. · Department of Statistics, North Carolina State University. · Department of Applied Mathematics & Statistics, Johns Hopkins University.

Multidimensional ScalingKernel MethodsDimensionality Reduction
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