On the use of the Gram matrix for multivariate functional principal components analysis

• URL: http://arxiv.org/abs/2306.12949v2
• Date: Thu, 20 Jun 2024 09:31:17 GMT
• Title: On the use of the Gram matrix for multivariate functional principal components analysis
• Authors: Steven Golovkine, Edward Gunning, Andrew J. Simpkin, Norma Bargary,
• Abstract summary: Dimension reduction is crucial in functional data analysis (FDA) Existing approaches for functional principal component analysis usually involve the diagonalization of the covariance operator. We propose to use the inner-product between the curves to estimate the eigenelements of multivariate and multidimensional functional datasets.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Dimension reduction is crucial in functional data analysis (FDA). The key tool to reduce the dimension of the data is functional principal component analysis. Existing approaches for functional principal component analysis usually involve the diagonalization of the covariance operator. With the increasing size and complexity of functional datasets, estimating the covariance operator has become more challenging. Therefore, there is a growing need for efficient methodologies to estimate the eigencomponents. Using the duality of the space of observations and the space of functional features, we propose to use the inner-product between the curves to estimate the eigenelements of multivariate and multidimensional functional datasets. The relationship between the eigenelements of the covariance operator and those of the inner-product matrix is established. We explore the application of these methodologies in several FDA settings and provide general guidance on their usability.

Related papers

• Interpetable Target-Feature Aggregation for Multi-Task Learning based on Bias-Variance Analysis [53.38518232934096]
  Multi-task learning (MTL) is a powerful machine learning paradigm designed to leverage shared knowledge across tasks to improve generalization and performance. We propose an MTL approach at the intersection between task clustering and feature transformation based on a two-phase iterative aggregation of targets and features. In both phases, a key aspect is to preserve the interpretability of the reduced targets and features through the aggregation with the mean, which is motivated by applications to Earth science.
  arXiv  Detail & Related papers  (2024-06-12T08:30:16Z)
• Fast and interpretable Support Vector Classification based on the truncated ANOVA decomposition [0.0]
  Support Vector Machines (SVMs) are an important tool for performing classification on scattered data. We propose solving SVMs in primal form using feature maps based on trigonometric functions or wavelets.
  arXiv  Detail & Related papers  (2024-02-04T10:27:42Z)
• On the estimation of the number of components in multivariate functional principal component analysis [0.0]
  We present extensive simulations to investigate choosing the number of principal components to retain. We show empirically that the conventional approach of using a percentage of variance explained threshold for each univariate functional feature may be unreliable.
  arXiv  Detail & Related papers  (2023-11-08T09:05:42Z)
• Nonlinear Feature Aggregation: Two Algorithms driven by Theory [45.3190496371625]
  Real-world machine learning applications are characterized by a huge number of features, leading to computational and memory issues. We propose a dimensionality reduction algorithm (NonLinCFA) which aggregates non-linear transformations of features with a generic aggregation function. We also test the algorithms on synthetic and real-world datasets, performing regression and classification tasks, showing competitive performances.
  arXiv  Detail & Related papers  (2023-06-19T19:57:33Z)
• A Functional approach for Two Way Dimension Reduction in Time Series [13.767812547998735]
  We propose a non-linear function-on-function approach, which consists of a functional encoder and a functional decoder. Our approach gives a low dimension latent representation by reducing the number of functional features as well as the timepoints at which the functions are observed.
  arXiv  Detail & Related papers  (2023-01-01T06:09:15Z)
• A geometric perspective on functional outlier detection [0.0]
  We develop a conceptualization of functional outlier detection that is more widely applicable and realistic than previously proposed. We show that simple manifold learning methods can be used to reliably infer and visualize the geometric structure of functional data sets. Our experiments on synthetic and real data sets demonstrate that this approach leads to outlier detection performances at least on par with existing functional data-specific methods.
  arXiv  Detail & Related papers  (2021-09-14T17:42:57Z)
• Eigen Analysis of Self-Attention and its Reconstruction from Partial Computation [58.80806716024701]
  We study the global structure of attention scores computed using dot-product based self-attention. We find that most of the variation among attention scores lie in a low-dimensional eigenspace. We propose to compute scores only for a partial subset of token pairs, and use them to estimate scores for the remaining pairs.
  arXiv  Detail & Related papers  (2021-06-16T14:38:42Z)
• Feature Weighted Non-negative Matrix Factorization [92.45013716097753]
  We propose the Feature weighted Non-negative Matrix Factorization (FNMF) in this paper. FNMF learns the weights of features adaptively according to their importances. It can be solved efficiently with the suggested optimization algorithm.
  arXiv  Detail & Related papers  (2021-03-24T21:17:17Z)
• The role of feature space in atomistic learning [62.997667081978825]
  Physically-inspired descriptors play a key role in the application of machine-learning techniques to atomistic simulations. We introduce a framework to compare different sets of descriptors, and different ways of transforming them by means of metrics and kernels. We compare representations built in terms of n-body correlations of the atom density, quantitatively assessing the information loss associated with the use of low-order features.
  arXiv  Detail & Related papers  (2020-09-06T14:12:09Z)
• Supervised Quantile Normalization for Low-rank Matrix Approximation [50.445371939523305]
  We learn the parameters of quantile normalization operators that can operate row-wise on the values of $X$ and/or of its factorization $UV$ to improve the quality of the low-rank representation of $X$ itself. We demonstrate the applicability of these techniques on synthetic and genomics datasets.
  arXiv  Detail & Related papers  (2020-02-08T21:06:02Z)
• Invariant Feature Coding using Tensor Product Representation [75.62232699377877]
  We prove that the group-invariant feature vector contains sufficient discriminative information when learning a linear classifier. A novel feature model that explicitly consider group action is proposed for principal component analysis and k-means clustering.
  arXiv  Detail & Related papers  (2019-06-05T07:15:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.