Unsupervised Functional Data Analysis via Nonlinear Dimension Reduction
- URL: http://arxiv.org/abs/2012.11987v1
- Date: Tue, 22 Dec 2020 13:19:32 GMT
- Title: Unsupervised Functional Data Analysis via Nonlinear Dimension Reduction
- Authors: Moritz Herrmann and Fabian Scheipl
- Abstract summary: We describe and investigate the challenges for nonlinear dimension reduction posed by the functional data setting.
We show that manifold methods can be used successfully in this setting.
We propose a nuanced approach to make trustworthy decisions for or against competing nonconforming embeddings more objectively.
- Score: 0.0
- License: http://creativecommons.org/licenses/by-sa/4.0/
- Abstract: In recent years, manifold methods have moved into focus as tools for
dimension reduction. Assuming that the high-dimensional data actually lie on or
close to a low-dimensional nonlinear manifold, these methods have shown
convincing results in several settings. This manifold assumption is often
reasonable for functional data, i.e., data representing continuously observed
functions, as well. However, the performance of manifold methods recently
proposed for tabular or image data has not been systematically assessed in the
case of functional data yet. Moreover, it is unclear how to evaluate the
quality of learned embeddings that do not yield invertible mappings, since the
reconstruction error cannot be used as a performance measure for such
representations. In this work, we describe and investigate the specific
challenges for nonlinear dimension reduction posed by the functional data
setting. The contributions of the paper are three-fold: First of all, we define
a theoretical framework which allows to systematically assess specific
challenges that arise in the functional data context, transfer several
nonlinear dimension reduction methods for tabular and image data to functional
data, and show that manifold methods can be used successfully in this setting.
Secondly, we subject performance assessment and tuning strategies to a thorough
and systematic evaluation based on several different functional data settings
and point out some previously undescribed weaknesses and pitfalls which can
jeopardize reliable judgment of embedding quality. Thirdly, we propose a
nuanced approach to make trustworthy decisions for or against competing
nonconforming embeddings more objectively.
Related papers
- Selecting Features by their Resilience to the Curse of Dimensionality [0.0]
Real-world datasets are often of high dimension and effected by the curse of dimensionality.
Here we step in with a novel method that identifies the features that allow to discriminate data subsets of different sizes.
Our experiments show that our method is competitive and commonly outperforms established feature selection methods.
arXiv Detail & Related papers (2023-04-05T14:26:23Z) - Interpretable Linear Dimensionality Reduction based on Bias-Variance
Analysis [45.3190496371625]
We propose a principled dimensionality reduction approach that maintains the interpretability of the resulting features.
In this way, all features are considered, the dimensionality is reduced and the interpretability is preserved.
arXiv Detail & Related papers (2023-03-26T14:30:38Z) - An evaluation framework for dimensionality reduction through sectional
curvature [59.40521061783166]
In this work, we aim to introduce the first highly non-supervised dimensionality reduction performance metric.
To test its feasibility, this metric has been used to evaluate the performance of the most commonly used dimension reduction algorithms.
A new parameterized problem instance generator has been constructed in the form of a function generator.
arXiv Detail & Related papers (2023-03-17T11:59:33Z) - DimenFix: A novel meta-dimensionality reduction method for feature
preservation [64.0476282000118]
We propose a novel meta-method, DimenFix, which can be operated upon any base dimensionality reduction method that involves a gradient-descent-like process.
By allowing users to define the importance of different features, which is considered in dimensionality reduction, DimenFix creates new possibilities to visualize and understand a given dataset.
arXiv Detail & Related papers (2022-11-30T05:35:22Z) - Dimensionality Reduction using Elastic Measures [0.6445605125467572]
We present a method for incorporating elastic metrics into the t-distributed Neighbor Embedding (t-SNE) and Uniform Approximation and Projection (UMAP)
We demonstrate improved performance on three benchmark data sets on shape identification and classification tasks.
arXiv Detail & Related papers (2022-09-07T21:09:38Z) - Learning from few examples with nonlinear feature maps [68.8204255655161]
We explore the phenomenon and reveal key relationships between dimensionality of AI model's feature space, non-degeneracy of data distributions, and the model's generalisation capabilities.
The main thrust of our present analysis is on the influence of nonlinear feature transformations mapping original data into higher- and possibly infinite-dimensional spaces on the resulting model's generalisation capabilities.
arXiv Detail & Related papers (2022-03-31T10:36:50Z) - On Modality Bias Recognition and Reduction [70.69194431713825]
We study the modality bias problem in the context of multi-modal classification.
We propose a plug-and-play loss function method, whereby the feature space for each label is adaptively learned.
Our method yields remarkable performance improvements compared with the baselines.
arXiv Detail & Related papers (2022-02-25T13:47:09Z) - Efficient Multidimensional Functional Data Analysis Using Marginal
Product Basis Systems [2.4554686192257424]
We propose a framework for learning continuous representations from a sample of multidimensional functional data.
We show that the resulting estimation problem can be solved efficiently by the tensor decomposition.
We conclude with a real data application in neuroimaging.
arXiv Detail & Related papers (2021-07-30T16:02:15Z) - Hard-label Manifolds: Unexpected Advantages of Query Efficiency for
Finding On-manifold Adversarial Examples [67.23103682776049]
Recent zeroth order hard-label attacks on image classification models have shown comparable performance to their first-order, gradient-level alternatives.
It was recently shown in the gradient-level setting that regular adversarial examples leave the data manifold, while their on-manifold counterparts are in fact generalization errors.
We propose an information-theoretic argument based on a noisy manifold distance oracle, which leaks manifold information through the adversary's gradient estimate.
arXiv Detail & Related papers (2021-03-04T20:53:06Z) - Nonlinear Dimensionality Reduction for Data Visualization: An
Unsupervised Fuzzy Rule-based Approach [5.5612170847190665]
We propose an unsupervised fuzzy rule-based dimensionality reduction method primarily for data visualization.
We use a first-order Takagi-Sugeno type model and generate rule antecedents using clusters in the input data.
We apply the proposed method on three synthetic and three real-world data sets and visually compare the results with four other standard data visualization methods.
arXiv Detail & Related papers (2020-04-08T10:33:06Z)
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.