We introduce \textbf{N}on-\textbf{Euc}lidean-\textbf{MDS} (Neuc-MDS), which extends Multidimensional Scaling (MDS) to generate outputs that can be non-Euclidean and non-metric. The main idea is to generalize the inner product to other symmetric bilinear forms to utilize the negative eigenvalues of dissimiliarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods.

In this paper, we aim to learn alow-dimensional Euclidean representation from aset of constraints of the form "itemj is closer to itemithan itemk".

Multidimensional scaling visualizes dissimilarities among objects and reduces data dimensionality. While many methods address symmetric proximity data, asymmetric and especially three-way proximity data (capturing relationships across multiple occasions) remain underexplored. Recent developments, such as the h-plot, enable the analysis of asymmetric and non-reflexive relationships by embedding dissimilarities in a Euclidean space, allowing further techniques like archetypoid analysis to identify representative extreme profiles. However, no existing methods extract archetypal profiles from three-way asymmetric proximity data. This work extends the h-plot methodology to three-way proximity data under both symmetric and asymmetric, conditional and unconditional frameworks. The proposed approach offers several advantages: intuitive interpretability through a unified Euclidean representation; an explicit, eigenvector-based analytical solution free from local minima; scale invariance under linear transformations; computational efficiency for large matrices; and a straightforward goodness-of-fit evaluation. Furthermore, it enables the identification of archetypal profiles and clustering structures for three-way asymmetric proximities. Its performance is compared with existing models for multidimensional scaling and clustering, and illustrated through a financial application. All data and code are provided to facilitate reproducibility.

This paper presents evomap, a Python package for dynamic mapping. Mapping methods are widely used across disciplines to visualize relationships among objects as spatial representations, or maps. However, most existing statistical software supports only static mapping, which captures objects' relationships at a single point in time and lacks tools to analyze how these relationships evolve. evomap fills this gap by implementing the dynamic mapping framework EvoMap, originally proposed by Matthe, Ringel, and Skiera (2023), which adapts traditional static mapping methods for dynamic analyses. The package supports multiple mapping techniques, including variants of Multidimensional Scaling (MDS), Sammon Mapping, and t-distributed Stochastic Neighbor Embedding (t-SNE). It also includes utilities for data preprocessing, exploration, and result evaluation, offering a comprehensive toolkit for dynamic mapping applications. This paper outlines the foundations of static and dynamic mapping, describes the architecture and functionality of evomap, and illustrates its application through an extensive usage example.

We introduce \textbf{N}on-\textbf{Euc}lidean-\textbf{MDS} (Neuc-MDS), which extends Multidimensional Scaling (MDS) to generate outputs that can be non-Euclidean and non-metric. The main idea is to generalize the inner product to other symmetric bilinear forms to utilize the negative eigenvalues of dissimiliarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods.

The full-dimensional (metric, Euclidean, least squares) multidimensional scaling stress loss function is combined with a quadratic external penalty function term. The trajectory of minimizers of stress for increasing values of the penalty parameter is then used to find (tentative) global minima for low-dimensional multidimensional scaling. This is illustrated with several one-dimensional and two-dimensional examples.

Metric Multidimensional scaling (MDS) is a classical method for generating meaningful (non-linear) low-dimensional embeddings of high-dimensional data. MDS has a long history in the statistics, machine learning, and graph drawing communities. In particular, the Kamada-Kawai force-directed graph drawing method is equivalent to MDS and is one of the most popular ways in practice to embed graphs into low dimensions. Despite its ubiquity, our theoretical understanding of MDS remains limited as its objective function is highly non-convex. In this paper, we prove that minimizing the Kamada-Kawai objective is NP-hard and give a provable approximation algorithm for optimizing it, which in particular is a PTAS on low-diameter graphs.

Multidimensional scaling of gene sequence data has long played a vital role in analysing gene sequence data to identify clusters and patterns. However the computation complexities and memory requirements of state-of-the-art dimensional scaling algorithms make it infeasible to scale to large datasets. In this paper we present an autoencoder-based dimensional reduction model which can easily scale to datasets containing millions of gene sequences, while attaining results comparable to state-of-the-art MDS algorithms with minimal resource requirements. The model also supports out-of-sample data points with a 99.5%+ accuracy based on our experiments. The proposed model is evaluated against DAMDS with a real world fungi gene sequence dataset. The presented results showcase the effectiveness of the autoencoder-based dimension reduction model and its advantages.

Multidimensional Scaling (MDS) is one of the first fundamental manifold learning methods. It can be categorized into several methods, i.e., classical MDS, kernel classical MDS, metric MDS, and non-metric MDS. Sammon mapping and Isomap can be considered as special cases of metric MDS and kernel classical MDS, respectively. In this tutorial and survey paper, we review the theory of MDS, Sammon mapping, and Isomap in detail. We explain all the mentioned categories of MDS. Then, Sammon mapping, Isomap, and kernel Isomap are explained. Out-of-sample embedding for MDS and Isomap using eigenfunctions and kernel mapping are introduced. Then, Nystrom approximation and its use in landmark MDS and landmark Isomap are introduced for big data embedding. We also provide some simulations for illustrating the embedding by these methods.

Isotonic regression is a free-form linear model that can be fit to predict sequences of observations. However, there are two major differences between isotonic regression and a similar model like weighted least squares. An isotonic function must not be non-decreasing. This is because an isotonic function is a monotonic function, meaning a function that preserves or reverses a given order. Isotonic regressors use an interesting concept called " Order Theory."