The Isomap Algorithm and Topological Stability

By M. Balasubramanian, Eric L. Schwartz. 2002

818 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition

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Citations in the corpus

A Comprehensive Survey of Graph Embedding: Problems, Techniques, and Applications (2017, methodology): “For example, Isomap [78] incorporates the geodesic distance in A.”
Riemannian Manifold Learning (2008, methodology): “An appropriately chosen neighborhood size is the key to the success of these algorithms [36].”
Graph Learning: A Survey (2021, background): “and the subsequent studies [67], [71]–[73] tackle this issue.”
Stochastic k-Neighborhood Selection for Supervised and Unsupervised Learning (2013, methodology): “While this approach to dimensionality reduction is taken by popular methods such as Locally Linear Embedding (LLE) (Roweis & Saul, 2000) and Isomap (Balasubramanian et al., 2002), the SNE method differs from these methods because it uses a fundamentally smooth objective that is based on matching dis”
Non-linear Dimensionality Reduction by Locally Linear Isomaps (2004, methodology): “There is a serious problem in the Isomap algorithm which is referred to as Short Circuiting [7].”
A Review of Feature Selection and Feature Extraction Methods Applied on Microarray Data (2015, methodology): “Many variants of Isomap have been used; for example, Balasubramanian and Schwartz proposed a tree connected version which differs in the way the neighbourhood graph is constructed [72].”
A General Exponential Framework for Dimensionality Reduction (2014, background): “The algorithmic performance is often sensitive to the parameter k and the performance may vary each time k changes [24], [25].”
Exploiting manifold geometry in hyperspectral imagery (2005, methodology): “to be more vulnerable to noise (see response of Tenenbaum, de Silva, and Langford in [8]).”
Feature dimensionality reduction: a review (2022, background): “ISOMAP believes that when the data set has an embedded manifold structure [92, 93], the corresponding description of the observation space data set in the low-dimensional structure can be obtained according to the distance map.”
Large-scale manifold learning (2008, background): “Since Isomap needs to compute shortest paths in the neighborhood graph, the presence of bad edges can cause leakage or ‘short-circuits’[2].”
Interpolation-based reduced-order modelling for steady transonic flows via manifold learning (2014, background): “The optimal choice of these parameters is problem dependent and the success of Isomap depends largely on the selected values (Balasubramanian et al. 2002).”
Dimensionality Reduction: A Comparative Review (2008, background): “An important weakness of the Isomap algorithm is its topological instability [7].”
View-invariant human action recognition via robust locally adaptive multi-view learning (2015, methodology): “These methods can also be divided into two groups: linear methods, such as principal component analysis (PCA) (Wold et al., 1987), latent Dirichlet allocation (LDA) (Balakrishnama and Ganapathiraju, 1998), and non-negative matrix factorization (Lee and Seung, 1999), and nonlinear methods, such as te”
Charting a Manifold (2002, background): “It has been noted that trusted-set methods are vulnerable to noise because they consider the subset of point-to-point relationships that has the lowest signal-to-noise ratio; small changes to the trusted set can induce large changes in the set of constraints on the embedding, making solutions unstab”
Conceptual and empirical comparison of dimensionality reduction algorithms (PCA, KPCA, LDA, MDS, SVD, LLE, ISOMAP, LE, ICA, t-SNE) (2021, background): “, A global geometric framework for nonlinear dimensionality reduction, Science, 290, 5500, 2319-2323 (2000) [45] Balasubramanian, M.”