Multidimensional scaling: I. Theory and method

By W. Torgerson. 1952

Subfields: Statistics and Probability, Algorithms & Complexity

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Citations in the corpus

Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, methodology): “Early approaches toward this problem, including Principal Component Analysis (PCA) (Pearson, 1901) and Multidimensional Scaling (MDS) (Torgerson, 1952), mostly fall into the global category.”
A practical tutorial on autoencoders for nonlinear feature fusion: Taxonomy, models, software and guidelines (2018, methodology): “Multidimensional Scaling (MDS) [101] is a well known technique and a foundation for other algorithms.”
Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization (2020, methodology): “Prominent examples of global methods include PCA (Pearson, 1901) and MDS (Torgerson, 1952).”
Visualizing Large-scale and High-dimensional Data (2016, methodology): “…to extremely low-dimensional spaces (e.g., 2D), the linear methods such as the Principle Component Analysis [15] and the multidimensional scaling [25] usually do not work as eﬀectively as the non-linear methods as most high-dimensional data usually lies on or near low-dimensional non-linear…”
Interaction of genotype, environment and processing in the chemical composition expression and sensorial quality of Arabica coffee (2016, background): “In addition, it allows for the rearranging of the variable distribution in order to detect the smallest significant dimensions to explain their similarities or dissimilarities (Torgerson, 1952).”
Icasso: software for investigating the reliability of ICA estimates by clustering and visualization (2003, background): “However, for |rij | the same transformation is not guaranteed to give a distance matrix that would exactly originate from Euclidean distances between points in Rk.”
Spectral embedding of graphs (2003, methodology): “The classical MDS method was proposed by Torgenson [19] and Gower [20].”
Trace optimization and eigenproblems in dimension reduction methods (2011, background): “Indeed, under the assumption that the data is centered at zero, it can be shown that [47] where In the matrix form, it is As a result of the above equality, in order to ﬁnd a d -dimensional projection which preserves inter-distances as best possible, we need to ﬁnd a d × n matrix Y whose Grammian Y…”
Random forest-based similarity measures for multi-modal classification of Alzheimer's disease (2013, unknown): “MDS was applied to each similarity matrix, and a goodness-of-fit value of 90% was used to determine an appropriate dimensionality for the resulting embeddings.”
Information Retrieval Perspective to Nonlinear Dimensionality Reduction for Data Visualization (2010, methodology): “PCA is very closely related to linear multidimensional scaling (linear MDS, also called classical scaling; Torgerson, 1952; Gower, 1966), which tries to find lowdimensional coordinates preserving squared distances.”
A New Benchmark Dataset for Handwritten Character Recognition (2009, methodology): “In Figure 3, we present the results of the experiments with Principal Components Analysis [7] (which is identical to classical scaling [19]), Isomap [18], Locally Linear Embedding (LLE) [12], and t-Distributed Stochastic Neighbor Embedding (t-SNE) [20].”
Quality assessment of dimensionality reduction: Rank-based criteria (2009, background): “…assessment of dimensionality reduction: Rank-based criteria John A. Lee ,1,2, Michel Verleysen 3 Machine Learning Group, Université catholique de Louvain, Place du Levant, 3, B-1348 Louvain-la-Neuve, Belgium a r t i c l e i n f o Available online 10 January…”
A Hebbian/Anti-Hebbian Neural Network for Linear Subspace Learning: A Derivation from Multidimensional Scaling of Streaming Data (2015, methodology): “…to principal subspace tracking. x 1 x n . . . 2 Derivation of online algorithms from the CMDS cost function CMDS represents high-dimensional input data in a lower-dimensional output space while preserving pairwise similarities between samples 2 (Young and Householder, 1938; Torgerson, 1952).”
Data Visualization With Multidimensional Scaling (2008, background): “The oldest version of MDS, called classical scaling, is due to Torgerson (1952).”