A Sparse Grid Based Generative Topographic Mapping for the Dimensionality Reduction of High-Dimensional Data

Griebel, Michael; Hullmann, Alexander

doi:10.1007/978-3-319-09063-4_5

Cited by 4 publications

(2 citation statements)

References 13 publications

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“…Consequently, it is not suited for a drastic initial dimensionality reduction (also called hard dimensionality reduction [18]) of high-dimensional data to a moderate amount of latent space dimensions, e.g., L ≈ 6. While there are modifications of the original GTM with a semilinear approach [3] or sparse grids [12] which allow us to treat somewhat higher latent space dimensions, their practical use is still limited.…”

Section: Introductionmentioning

confidence: 99%

Dimensionality Reduction of High-Dimensional Data with a NonLinear Principal Component Aligned Generative Topographic Mapping

Griebel¹,

Hullmann²

2014

SIAM J. Sci. Comput.

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Most high-dimensional real-life data exhibit some dependencies such that data points do not populate the whole data space but lie approximately on a lower-dimensional manifold. A major problem in many data mining applications is the detection of such a manifold and the expression of the given data in terms of a moderate number of latent variables. We present a method which is derived from the generative topographic mapping (GTM) and can be seen as a nonlinear generalization of the principal component analysis (PCA). It can detect certain nonlinearities in the data but does not suffer from the curse of dimensionality with respect to the latent space dimension as the original GTM and thus allows for higher embedding dimensions. We provide experiments that show that our approach leads to an improved data reconstruction compared to the purely linear PCA and that it can furthermore be used for classification.

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Section: Introductionmentioning

confidence: 99%

Dimensionality Reduction of High-Dimensional Data with a NonLinear Principal Component Aligned Generative Topographic Mapping

Griebel¹,

Hullmann²

2014

SIAM J. Sci. Comput.

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“…[25,32] ) relating the idea to our approach. Since, the current work focuses on generating hierarchical basis functions, it is also closely related to works such as [5,28] that implement the idea of sparse grids for data analysis and learning tasks.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical regularization networks for sparsification based learning on noisy datasets

Shekhar¹,

Patra²

2020

Preprint

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We propose a hierarchical learning strategy aimed at generating sparse representations and associated models for large noisy datasets. The hierarchy follows from approximation spaces identified at successively finer scales. For promoting model generalization at each scale, we also introduce a novel, projection based penalty operator across multiple dimension, using permutation operators for incorporating proximity and ordering information. The paper presents a detailed analysis of approximation properties in the reconstruction Reproducing Kernel Hilbert Spaces (RKHS) with emphasis on optimality and consistency of predictions and behavior of error functionals associated with the produced sparse representations. Results show the performance of the approach as a data reduction and modeling strategy on both synthetic (univariate and multivariate) and real datasets (time series). The sparse model for the test datasets, generated by the presented approach, is also shown to efficiently reconstruct the underlying process and preserve generalizability.

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Hierarchical approximations for data reduction and learning at multiple scales

Shekhar¹,

Patra²

2020

Foundations of Data Science

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A Sparse Grid Based Generative Topographic Mapping for the Dimensionality Reduction of High-Dimensional Data

Cited by 4 publications

References 13 publications

Dimensionality Reduction of High-Dimensional Data with a NonLinear Principal Component Aligned Generative Topographic Mapping

Dimensionality Reduction of High-Dimensional Data with a NonLinear Principal Component Aligned Generative Topographic Mapping

Hierarchical regularization networks for sparsification based learning on noisy datasets

Hierarchical approximations for data reduction and learning at multiple scales

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