We introduce a novel data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space wh…
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48 results for “nonlinear dimensionality reduction”
In this era of data deluge, many signal processing and machine learning tasks are faced with high-dimensional datasets, including images, videos, as well as time series generated from social, commercial and brain network interactions. Their efficient processing calls for dimensionality reduction techniques capable of p…
IKD uses eigen-decomposition for nonlinear dimensionality reduction.
problem Lack of sophisticated and nonlinear dimensionality reduction methods.
method Inverse Kernel Decomposition (IKD) based on eigen-decomposition of sample covariance matrix.
result IKD achieves comparable performance to optimization-based methods with faster running speeds.
We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where ba…
Survey on geometric foundations of data reduction methods.
problem High-dimensional data with intrinsic nonlinear structure.
method Spectral manifold learning methods.
result Derivation and convergence analysis of spectral manifold learning.
WeldNet reduces complex dynamics to simpler, manageable segments.
problem Complex, high-dimensional time-dependent datasets from physical processes are costly to simulate.
method Windowed Encoders for Learning Dynamics, splitting time domain into windows for nonlinear dimension reduction and propagator training.
result WeldNet captures nonlinear latent structures and dynamics, outperforming existing methods.
PGPCA improves PCA for nonlinear data in neuroscience.
problem Nonlinear data distribution in neuroscience.
method Developed PGPCA for nonlinear manifolds, incorporating EM algorithm.
result PGPCA outperforms PPCA in modeling data around nonlinear manifolds.
A new deep neural network tackles nonlinear functional regression with improved dimensionality reduction.
problem Nonlinear functional regression in infinite-dimensional functional data analysis.
method Functional deep neural network with adaptive kernel embedding and projection steps.
result Explicit rates of approximating nonlinear smooth functionals are derived, and the network is shown to be effective in both simulated and real datasets.
A new DDR framework learns low-dimensional data representations using dynamical systems.
problem Learning efficient low-dimensional data representations.
method DDR framework based on nonlinear dynamical systems, using linear combinations of functions and regularization.
result DDR method outperforms other methods on synthetic and real datasets.
Feature Learning for Nonlinear Dimensionality Reduction toward Maximal Extraction of Hidden Patternscs.LG
FEALM learns features for better nonlinear DR of hidden patterns.
problem DR misses important patterns on distorted manifolds.
method FEALM generates optimized projections using an optimization algorithm and neighbor-shape dissimilarity.
result FEALM captures important patterns on hidden manifolds.
While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods are not amenable to the analysis of such manifolds. This is mainly due to the necessary spectral decomposition step, which limits control over the…
A new method for reducing model complexity using neural active manifolds.
problem Uncertainty quantification in computationally expensive models.
method Autoencoders and surrogate models to discover a neural active manifold.
result Neural active manifolds reduce model variance in multifidelity sampling.
A restricted Boltzmann machine (RBM) is a two-layer neural network with shared weights and has been extensively studied for dimensionality reduction, data representation and recommendation systems in the literature. The traditional RBM requires a probabilistic interpretation of the values on both layers and a Markov ch…
Proposes a stratified sampling method for high-dimensional models using neural active manifolds.
problem Uncertainty propagation in computationally expensive models with many inputs.
method Neural active manifolds for nonlinear dimensionality reduction, followed by stratification in the reduced space.
result Effective variance reduction in high-dimensional models using stratified sampling.
MCE reduces embedding instability in nonlinear dimensionality reduction.
problem Embedding instability caused by random initialization.
method Median of multiple embeddings (MCE) based on large deviation theory.
result MCE achieves consistency at an exponential rate and effectively mitigates instability.
New method deflates manifolds to visualize high-dimensional data.
problem Failure of nonlinear dimensionality reduction methods on simple manifolds.
method Iterative deflation of differential operators using single-coordinate estimates.
result Empirically, recovers novel embeddings on real-world and synthetic datasets.
On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reductionstat.ML
GenSDR tackles SDR by leveraging generative models to fully recover lower-dimensional structures.
problem Challenges in identifying low-dimensional sufficient structures in nonlinear SDR.
method Proposes GenSDR, a method that uses modern generative models to fully recover information in the central σ-field.
result Establishes consistency of GenSDR estimator for sample-level data and extends its applicability to non-Euclidean responses.
Proposes a method to reveal nonlinearities in tensor data.
problem Capturing nonlinear relationships in high-dimensional tensor data.
method Linear tensor projection method to maximize prediction accuracy.
result Effective in revealing nonlinear relationships in tensor data.
NKI integrates obfuscated datasets using nonlinear kernels for improved data collaboration.
problem Privacy-preserving data collaboration with reduced reconstruction risk.
method Formulates linear kernel integration, kernelizes it, and introduces graph regularization and centering constraints.
result NKI improves classification accuracy over existing linear integration methods under nonlinear dimensionality reduction.
DMT enhances deep neural networks to better preserve data structures.
problem Preserving geometric, topological, and distributional structures of data in NLDR.
method Deep manifold transformation (DMT) using cross-layer LGP constraints.
result DMT networks outperform existing NLDR methods in preserving data structures.
New neural network method simplifies high-dimensional data.
problem Scalability issues in nonlinear sufficient dimension reduction.
method Stochastic neural network with adaptive gradient algorithm.
result Proposed method outperforms existing methods on large-scale data.
Ensemble learning has had many successes in supervised learning, but it has been rare in unsupervised learning and dimensionality reduction. This study explores dimensionality reduction ensembles, using principal component analysis and manifold learning techniques to capture linear, nonlinear, local, and global feature…
Randomized ICA and LDA Dimensionality Reduction Methods for Hyperspectral Image Classificationstat.ML
Dimensionality reduction is an important step in processing the hyperspectral images (HSI) to overcome the curse of dimensionality problem. Linear dimensionality reduction methods such as Independent component analysis (ICA) and Linear discriminant analysis (LDA) are commonly employed to reduce the dimensionality of HS…
This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead of linear features. DRR identifies the nonlinear features through multivariate r…
Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques over sparse and noisy data is not guaranteed. In fact, the embed…
Sensor data analysis plays a key role in health assessment of critical equipment. Such data are multivariate and exhibit nonlinear relationships. This paper describes how one can exploit nonlinear dimension reduction techniques, such as the t-distributed stochastic neighbor embedding (t-SNE) and kernel principal compon…
New DR method uses Gromov-Wasserstein distance for high-dimensional data.
problem Analyzing relationships between high-dimensional objects.
method Optimal transportation theory and Gromov-Wasserstein distance.
result Robust and efficient solution for complex high-dimensional datasets.
Wassmap reduces image complexity while preserving key features.
problem Global nonlinear dimensionality reduction in imaging.
method Wassmap uses Wasserstein space and pairwise distances to create isometric embeddings.
result Wassmap can recover parameters of image manifolds like translations and dilations.
New methods for functional data analysis improve manifold methods for continuous data.
problem Challenges in evaluating embeddings for functional data.
method Transfer manifold methods from tabular and image data to functional data, define a theoretical framework, and propose nuanced evaluation strategies.
result Manifold methods can be successfully applied to functional data, but careful evaluation is needed.
This paper surveys various methods for dimensionality reduction and nearest neighbor search.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving essential information.
method Linear and nonlinear random projections, including sparse random projections, random Fourier Features, and Random Kitchen Sinks.
result Various methods for dimensionality reduction and nearest neighbor search are explained and compared.
Quantum dynamics algorithm learns manifold from data.
problem Learning manifolds from high-dimensional datasets.
method Simulation of quantum dynamics on a graph embedding of data.
result Algorithm reveals connections between data sampling and quantization.
In this paper we introduce and analyze the learning scenario of \emph{coupled nonlinear dimensionality reduction}, which combines two major steps of machine learning pipeline: projection onto a manifold and subsequent supervised learning. First, we present new generalization bounds for this scenario and, second, we int…
Novel autoencoder method approximates Koopman operator in low dimensions.
problem Challenges in approximating finite Koopman operators using data-driven methods.
method Mori-Zwanzig autoencoder (MZ-AE) for robust Koopman operator approximation.
result Improved predictive capability and robust long-term statistical performance.
Theoretical Connection between Locally Linear Embedding, Factor Analysis, and Probabilistic PCAstat.ML
This work connects LLE, factor analysis, and probabilistic PCA through a stochastic perspective.
problem Exploring the theoretical connection between LLE, factor analysis, and probabilistic PCA.
method Solving the stochastic linear reconstruction of LLE using expectation maximization.
result LLE, factor analysis, and probabilistic PCA are shown to be connected through a stochastic perspective.
Spectral Overlap and a Comparison of Parameter-Free, Dimensionality Reduction Quality Metricsstat.ML
Nonlinear dimensionality reduction methods are a popular tool for data scientists and researchers to visualize complex, high dimensional data. However, while these methods continue to improve and grow in number, it is often difficult to evaluate the quality of a visualization due to a variety of factors such as lack of…
We demonstrate that almost all non-parametric dimensionality reduction methods can be expressed by a simple procedure: regularized loss minimization plus singular value truncation. By distinguishing the role of the loss and regularizer in such a process, we recover a factored perspective that reveals some gaps in the c…
Inverting Nonlinear Dimensionality Reduction with Scale-Free Radial Basis Function Interpolationmath.NA
Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map ever…
Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to detect the existence of a low-dimensional manifold of a given dimension that lies …
Grassmannian diffusion maps based dimension reduction and classification for high-dimensional datacs.LG
GDMaps reduces high-dimensional data to lower dimensions for better classification.
problem High-dimensional data classification and representation.
method Grassmannian Diffusion Maps technique for nonlinear dimensionality reduction.
result GDMaps effectively identifies intrinsic subspace structures in high-dimensional data.
Principal component analysis (PCA) is widely used for feature extraction and dimensionality reduction, with documented merits in diverse tasks involving high-dimensional data. Standard PCA copes with one dataset at a time, but it is challenged when it comes to analyzing multiple datasets jointly. In certain data scienc…
LNPE enhances local connections in embeddings using extended neighbor propagation.
problem Improving local connections and interactions in nonlinear dimensionality reduction.
method Inspired by GCN, LNPE extends 1-hop neighbors to n-hop neighbors in LLE.
result LNPE produces more faithful and robust embeddings with better topological and geometrical properties.
Machine learning methods are used to discover complex nonlinear relationships in biological and medical data. However, sophisticated learning models are computationally unfeasible for data with millions of features. Here we introduce the first feature selection method for nonlinear learning problems that can scale up t…
New method for nonlinear SDR of complex non-Euclidean data.
problem Nonlinear SDR for complex non-Euclidean random objects.
method Fréchet Cumulative Covariance (FCCov) and neural networks.
result Robust and unbiased nonlinear SDR for complex data.
Deep neural networks are vulnerable to adversarial examples, i.e., carefully-perturbed inputs aimed to mislead classification. This work proposes a detection method based on combining non-linear dimensionality reduction and density estimation techniques. Our empirical findings show that the proposed approach is able to…
In our work, we propose a novel formulation for supervised dimensionality reduction based on a nonlinear dependency criterion called Statistical Distance Correlation, Szekely et. al. (2007). We propose an objective which is free of distributional assumptions on regression variables and regression model assumptions. Our…
GD-VAEs learn dynamics from observations using geometric and topological information.
problem Learning parsimonious representations of nonlinear dynamics from observations.
method Develops data-driven methods incorporating geometric and topological information using Variational Autoencoders (VAEs).
result GD-VAEs provide methods for learning reduced dimensional representations of nonlinear dynamics.
Develops a new method for nonlinear dimension reduction using random features.
problem Statistical challenges in generalizing Gaussian process-based latent variable models to non-Gaussian data.
method Random feature latent variable models (RFLVMs) that approximate nonlinear relationships with linear functions of random features.
result RFLVMs produce comparable results to state-of-the-art methods on various data types.
t-SNE fails to reveal clusters even in well-clusterable data.
problem t-SNE's failure to show clusters in well-clusterable data.
method Empirical evidence and theoretical analysis of t-SNE's limitations.
result t-SNE may hide clustering patterns even with strong signals in the data.