Abstract reviews recent Lagrangian analysis on immersions into higher dimensions.
problem Analyzing Lagrangians on immersions into higher dimensions.
method Reviews recent progress on Lagrangians on immersions with first and second fundamental forms and their derivatives.
result Recent progress in the analysis of Lagrangians on immersions into higher dimensions.
Extends active subspace analysis to infinite dimensions.
problem Dimension reduction in infinite dimensional functionals.
method Defines an operator for Hilbert space, extends Euclidean properties, proposes Monte Carlo procedure.
result Desirable properties extend to infinite dimensional setting.
Paper compares dimension reduction methods using topological analysis on EEG data.
problem Comparing dimension reduction methods on EEG data.
method Topological data analysis, including persistent homology, Wasserstein distance, and hypothesis tests.
result Different dimension reduction methods show significant qualitative differences across topological homologies.
Painlevé analysis identifies integrable cases of Ricci solitons over specific warped products and line bundles.
problem Finding integrable cases of Ricci solitons over warped products and line bundles.
method Painlevé analysis applied to cohomogeneity one steady Ricci soliton equations for two classes of solitons: warped products and complex line bundles over a Fano Kähler Einstein base.
result Integrable cases identified for specific dimensions and configurations of Ricci solitons.
We carry out a Painlevé analysis of the systems of differential equations corresponding to the steady and the expanding, rotationally symmetric, gradient Ricci solitons on Rn. For the steady case, dimensions of the form n=k2+1 are singled out, with dimensions 2, 5, and 10 being particularly distinguished…
PANDA improves linear discriminant analysis in high dimensions with minimal tuning.
problem Linear discriminant analysis in high-dimensional settings.
method PANDA: a tuning-insensitive method for linear discriminant analysis.
result PANDA achieves optimal convergence rates in estimation error and misclassification rate.
A new method reduces both input and output dimensions for better goal-oriented analysis.
problem Simultaneous reduction of input and output dimensions for more accurate analysis.
method Coupled input-output dimension reduction, optimizing gradient-based bounds.
result Determine most informative sensors and influential parameters efficiently.
Proposes an online method for high-dimensional streaming data.
problem Increasing variable dimensions with sample size in online kernel sliced inverse regression.
method Introduces approximate linear dependence condition and dictionary variable sets to address the problem. Transforms into online generalized eigen-decomposition problem and uses stochastic optimization for updates.
result Achieves close performance to batch processing kernel sliced inverse regression.
Develops robust methods for infinite-dimensional stochastic processes.
problem Measuring covariations in stochastic evolution equations in infinite dimensions.
method Asymptotic theory for jump robust measurement of covariations.
result Identifies scaling limits for realized covariations.
Paper analyzes ensemble Kalman updates for effective dimension and localization.
problem Why small ensemble sizes work well in inverse problems and data assimilation.
method Non-asymptotic analysis of ensemble Kalman updates, focusing on effective dimension and localization.
result Rigorously explains why a small ensemble size is sufficient when prior covariance has moderate effective dimension.
This paper reviews and compares supervised linear dimension-reduction techniques.
problem Lack of information in the response during unsupervised PCA reduces predictive performance.
method Review and comparison of supervised linear dimension-reduction techniques.
result PLS and LSPCA consistently outperform other techniques in simulations.
We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions bel…
Improved learning algorithms with privacy using smoothed analysis.
problem Designing robust and private learning algorithms.
method Smoothed analysis of adversarial and differentially private learning.
result Stronger regret and privacy error guarantees with smoothed adversaries.
Taxicab correspondence analysis visualizes sparse text data sets.
problem Visualization of extremely sparse contingency tables.
method Robust variant of correspondence analysis for sparse data.
result Visualized an 8265-dimensional textual data set.
Study finds all hyperbolic manifolds with specific automorphism group dimensions.
problem Classifying homogeneous Kobayashi-hyperbolic manifolds based on automorphism group size.
method Analyzing holomorphic automorphism groups of dimension n2−3 for manifolds of dimension n≥2. result All connected homogeneous Kobayashi-hyperbolic manifolds of dimension n≥2 with automorphism group of dimension n2−3 are identified. A heuristic framework tests the multi-manifold hypothesis in empirical data.
problem Overestimation of parameters in global linear models.
method Heuristic multiscale framework using spline-interpolated manifolds.
result Validates the multi-manifold hypothesis in empirical data.
Bayesian nonparametric PCA infers the number of significant components.
problem Selecting the number of significant components in PCA is challenging.
method Introduces a Bayesian nonparametric approach using a Stiefel manifold prior and Indian buffet process for uncertainty modeling.
result Proposes a new estimator of the subspace dimension and a refined statistical significance test.
NCC is inefficient in higher dimensions, NCDA improves performance.
problem Inefficiency of NCC in higher dimensions.
method Combining NCC with LDA to create NCDA.
result NCDA outperforms NCC and competes with LDA and QDA.
Proposes a linear dimension reduction method for high-dimensional classification.
problem High-dimensional classification with unequal covariance matrices.
method Simultaneous variable selection and linear dimension reduction followed by quadratic discriminant analysis.
result The method doesn't require estimating precision matrices and scales linearly with the number of measurements.
This paper explores the impact of metric choice on Fréchet regression.
problem Choosing the right metric for Fréchet regression in complex data.
method Review and extensive numerical studies of existing dimension reduction methods.
result Different metrics significantly affect the estimation of central and central mean space.
This research simplifies PCA model selection using MDL principle.
problem Choosing the right number of principal components in PCA.
method Reduces NML problems to lower-dimension problems and bounds PCA NML.
result Bound the NML of PCA by terms of the NML of linear regression.
Proposes a method for evaluating multiple dimensions of organizational effectiveness using DEA.
problem Evaluating multiple dimensions of organizational effectiveness in large data sets.
method Introduces two regularized DEA models (SBM and GP-SBM) to estimate both dimension-specific and aggregate efficiency scores.
result Demonstrates improved efficiency and validity compared to conventional methods.
A new method uses Gram matrix for efficient multivariate functional principal components.
problem Efficiently estimating eigencomponents of multidimensional functional datasets.
method Proposes using inner-product matrix to estimate eigenelements of multivariate and multidimensional functional datasets.
result Established relationship between eigenelements of covariance operator and inner-product matrix.
PSMM method optimizes matrix sufficient dimension reduction.
problem Feature matrices with row- and column-wise interpretations require efficient dimension reduction.
method PSMM method converts matrix problem into classification problems using rank-1 normal matrix.
result PSMM outperforms existing methods and provides strong interpretability.
New algebraic framework for Jacobi manifolds connects geometric mechanics and dimensional analysis.
problem Lack of clear algebraic interpretation for Jacobi manifolds.
method Developed a dimensioned algebra approach to capture algebraic counterparts of Jacobi manifolds.
result Poly-Jacobi manifolds provide a new connection between geometric mechanics and dimensional analysis.
The paper corrects biases in estimating intrinsic dimension and differential entropy.
problem Systematic bias in estimating intrinsic dimension and differential entropy.
method A bias-corrected estimator for both measures is proposed, highlighting shared steps and useful consequences.
result Simultaneous estimation of differential entropy and intrinsic dimension provides complementary perspectives on underlying manifolds.
Wasserstein archetypal analysis finds optimal data summaries using Wasserstein metric.
problem Finding optimal data summaries using Wasserstein metric.
method Alternative formulation of archetypal analysis based on Wasserstein metric, with regularization and gradient-based computational approach.
result Existence and consistency of solutions for the regularized problem.
Constructs Gabor frames for curved manifolds to detect boundaries.
problem Signal analysis on curved manifolds with boundaries.
method Higher-dimensional Gabor frames for local linearizations.
result Detection of higher-dimensional boundaries in curved manifolds.
Note on failure of Martingale Wasserstein Inequality in higher dimensions.
problem Analyzing failure of Martingale Wasserstein Inequality in higher dimensions.
method Checking failure in dimension d≥2 and proving a stronger inequality in all dimensions.
result A stronger Maximal Martingale Wasserstein Inequality holds in all dimensions.
New algorithm balances spatial data approximation and prediction accuracy.
problem Lack of methods considering spatial correlation and downstream modeling in dimension reduction.
method Formalizes approximation and modeling utility as metrics, proposes a balanced algorithm.
result Optimal trade-off between approximation accuracy and downstream modeling utility.
Develops an efficient method for real-time data analysis and visualization.
problem Challenges of analyzing high-dimensional data.
method Incremental non-linear manifold approximation using GMRA framework.
result Accurately represents non-linear manifolds with small initial samples.
New method constructs Floer homologies without hard analysis.
problem Constructing homological invariants in infinite dimensions.
method Soft, finite-dimensional tools to avoid hard analysis.
result Demonstrates construction of Floer homologies without PDE.
CorrCA identifies reliable dimensions in multivariate data across repetitions.
problem Finding consistent dimensions in multivariate data across trials, subjects, or raters.
method Maximizes the ratio of between-repetition to within-repetition covariance.
result CorrCA leads to repeat-reliability maximization and is equivalent to Linear Discriminant Analysis for zero-mean signals.
A new method for real-time CCA on streaming data.
problem Finding correlated features in online data streams.
method Sliding Window Informative Canonical Correlation Analysis (SWICCA) using streaming PCA.
result SWICCA provides real-time CCA components in high dimensions with theoretical guarantees.
Estimates multiple means in high dimensions using convex combinations.
problem Estimating multiple multi-dimensional means from samples.
method Convex combinations of empirical means with data-dependent weights.
result Our methods asymptotically approach oracle (minimax) improvement.
Data-driven method solves multiscale elliptic PDEs with random coefficients.
problem Solving multiscale elliptic PDEs with random coefficients.
method Data-driven approach based on intrinsic dimension reduction.
result Efficient solution of multiscale elliptic PDEs with random coefficients.
Paper develops a novel approach for unsupervised dimension selection.
problem Tackles the combinatorial problem of identifying top-k dimensions in high-dimensional data.
method Develops a novel approach based on graph signal analysis to measure feature influence.
result Demonstrates the superiority of the proposed approach over existing techniques in capturing crucial characteristics of high-dimensional spaces using only a small subset of features.
New statistic κ-profile helps monitor weather, soundscapes, and dynamical systems.
problem Monitoring intrinsic dimensionality of large data sets.
method Optimization problem to find κ-profile, which is the norm of the shortest projected secant. result The κ-profile provides a useful statistic for understanding and monitoring large data sets. Proved dynamical Alekseevskii conjecture in 5D.
problem Proving the Alekseevskii conjecture in 5D.
method Detailed analysis of homogeneous Ricci flows.
result Proved the dynamical Alekseevskii conjecture in 5D.
We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimens…
The purpose of sufficient dimension reduction (SDR) is to find the low-dimensional subspace of input features that is sufficient for predicting output values. In this paper, we propose a novel distribution-free SDR method called sufficient component analysis (SCA), which is computationally more efficient than existing …
Modern techniques simplify complex high-dimensional data.
problem Complex, high-dimensional data.
method Unsupervised dimension reduction techniques.
result Simplified representation of high-dimensional data.
This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower-dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimension…
A review of contrastive dimension reduction methods for treatment vs control studies.
problem Traditional dimension reduction techniques fail to isolate treatment-specific signals.
method Systematic overview and taxonomy of CDR methods.
result Unified framework for CDR methods and applications.
SignSGD analysis quantifies its effects in high dimensions.
problem Understanding signSGD's effects in high-dimensional settings.
method High-dimensional analysis of signSGD, deriving SDE and ODE for risk.
result Quantification of signSGD's effects: effective learning rate, noise compression, diagonal preconditioning, gradient noise reshaping.
Estimates intrinsic dimensionality from minimal neighbor distances.
problem Analyzing high-dimensional datasets with complex manifolds.
method Minimal neighborhood information approach to estimate intrinsic dimensionality.
result The method provides consistent measures of intrinsic dimensionality.
This paper evaluates fractal dimension and persistent homology for neural network generalization.
problem Bounding and predicting the generalization gap of neural networks.
method Empirical evaluation of fractal dimension and persistent homology as generalization measures.
result Fractal dimension and persistent homology fail to predict generalization of models trained from poor initializations.
Analyzes the Levi form on CR manifolds of any dimension.
problem Understanding the Levi form on CR manifolds of varying dimensions and codimensions.
method Analytical and geometrical study of the Levi form.
result Comprehensive insights into the Levi form on CR manifolds.