This paper identifies knot projections with reductivity two.
problem Determining knot projections with a specific reductivity level.
method Examined four types of reductivity (Seifert type splice, non-Seifert type splice, recursively, simultaneously) and their combinations.
result Identified all knot projections with reductivity two for the four definitions.
Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.
problem Existence of compact Clifford-Klein forms in homogeneous spaces.
method Extend Kobayashi's method to non-reductive subgroups and compare Cartan projections and non-compact dimensions.
result Examples of homogeneous spaces without compact Clifford-Klein forms.
Abstract reviews Kähler geometry of complex projective spaces using reduction and unfolding.
problem Describing the Kähler geometry of complex projective spaces.
method Reduction and unfolding procedures associated with a momentum map.
result Describes Kähler geometry of complex projective spaces.
The paper compares PCA and PP for scRNA sequencing data.
problem Limitations of PCA in scRNA sequencing data.
method Applied PCA and PP (using negative Shannon's entropy) on scRNA sequencing data.
result PP outperforms PCA in scRNA sequencing data.
CCP clusters correlated features and projects them to 1D for efficient dimensionality reduction.
problem Efficiency in handling large datasets with high intrinsic dimensions.
method CCP partitions features into correlated clusters and projects them to 1D based on sample correlations.
result CCP achieves efficient dimensionality reduction without matrix diagonalization.
Classifies star products on Lie algebroid duals and extends to projectable quantizations.
problem Classifying and extending quantizations on Lie algebroid duals.
method Classification through second Lie algebroid cohomology, extension to projectable quantizations.
result Quantization commutes with reduction in the considered setting.
Projective geometry simplifies Sasaki-Einstein structures and their compactification.
problem Understanding and compactifying Sasaki-Einstein structures.
method Projective differential geometry and holonomy reductions.
result Characterization of Sasaki-Einstein structures and their compactification.
A new framework for dimension reduction using ensemble of random projections.
problem High-dimensional regression problems with limited data.
method Aggregating an ensemble of carefully chosen random projections, retaining based on empirical performance, and selecting singular vectors.
result The proposed method stabilizes error as the number of projection groups increases.
Data-driven model reduction captures non-Markovian dynamics using Koopman and Mori-Zwanzig formalisms.
problem Modeling complex, non-Markovian dynamics efficiently and understanding their underlying mechanisms.
method Formulates data-driven model reduction within Koopman and Mori-Zwanzig formalisms, deriving NARMAX models from dynamical systems.
result Shows how data-driven methods can represent non-Markovian dynamics using Koopman and Mori-Zwanzig formalisms.
This paper compares and analyzes random projections and column sub-sampling for dimension reduction in regression.
problem Computational efficiency in dimension reduction for large datasets.
method Analysis of random projections and column sub-sampling methods for regression.
result Random projections and column sub-sampling can achieve similar prediction error to Principal Components Regression (PCR) but with less computational cost.
Orthogonal projections improve learning accuracy in clinical image segmentation and music classification.
problem Improving accuracy in learning tasks with high-dimensional data.
method Investigation and application of orthogonal projections to balance variance and pairwise distances in dimension reduction. Extension to deep learning with augmented target loss functions.
result Augmented target loss functions increase accuracy in clinical image segmentation and music classification.
We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of red…
Paper proposes PPMM for fast estimation of large-scale OTM.
problem Estimation of large-scale optimal transport maps (OTM) is challenging due to the curse of dimensionality.
method Combines projection pursuit regression and sufficient dimension reduction to adaptively select projection directions.
result PPMM consistently estimates the most informative projection direction and weakly converges to the target OTM.
TTRP method preserves distances in high-dimensional data with reduced storage and speed.
problem Preserving distances in high-dimensional datasets efficiently and accurately.
method Tensor train random projection (TTRP) using TT-ranks of one.
result TTRP is an expected isometric projection with bounded variance.
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.
Paper proposes a method to reduce sensor drift in electronic noses.
problem Sensor drift in electronic noses.
method Discriminative subspace projection approach.
result The method minimizes within-class variance and maximizes between-class variance using label information.
LS-RPCA reduces dimensionality of large datasets better than random projections.
problem Reducing dimensionality of very large datasets for better classification performance.
method Developed LS-RPCA, an extension of RPCA for large datasets, to compare with random projections.
result LS-RPCA significantly improves classification performance over random projections.
We investigate reductions of the two-dimensional Dirac equation imposed by the requirement of the existence of a differential operator Dn of order n mapping its eigenfunctions to adjoint eigenfunctions. For first order operators these reductions (and multi-component analogs thereof) lead to the Lame equations desc…
Paper introduces S-SSE for stable sparse subspace embedding.
problem Inefficient sparse random projection matrices with uneven non-zero distribution.
method Uses uniform sampling without replacement to create a stable sparse subspace embedded matrix (S-SSE).
result S-SSE maintains Euclidean distance better after dimension reduction.
UMAP simplifies data visualization while preserving global structure.
problem Data visualization and dimension reduction challenges
method UMAP combines geometric and topological principles for efficient data embedding
result UMAP outperforms t-SNE in run time and global structure preservation
CDP reduces point cloud dimensions by preserving detour-induced local non-convexity.
problem Preserving local non-convexity in point cloud dimensionality reduction.
method CDP builds a k-NN graph, identifies admissible pairs, aggregates normalized directions, and uses top-k eigenvectors for projection.
result CDP provides verifiable guarantees on post-projection distortion and direction energy.
Survey of Laplacian-based methods for data dimensionality reduction and embedding.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving important features and structures.
method Laplacian-based methods including spectral clustering, Laplacian eigenmap, locality preserving projection, graph embedding, and diffusion map.
result Comprehensive overview of various optimization variants and applications of Laplacian-based techniques.
Adaptive framework improves nonparametric dimensionality reduction.
problem Optimal hyper-parameter tuning for nonparametric dimensionality reduction.
method Adaptive framework using intrinsic dimension estimator and optimal local neighbourhood sizes.
result Significant improvements in various learning tasks through better low-dimensional visualizations.
Improves cancer classification accuracy using Random Projection combined with other methods.
problem Improving classification accuracy of Random Projection for cancer classification.
method Combining Random Projection with Principle Component Analysis, Linear Discriminant Analysis, and Feature Selection.
result FS followed by RP yields a 14.77% increase in classification accuracy on BC-TCGA dataset.
3-Sasaki structures linked to projective geometry.
problem Understanding 3-Sasaki structures via projective geometry.
method Establishing a connection between 3-Sasaki structures and projective structures with specific holonomy reductions.
result 3-Sasaki structures are described as projective structures with a particular holonomy reduction to the unitary quaternionic group.
An algorithm preserves topological features in dimensionality reduction.
problem Preserving topological features in dimensionality reduction.
method Simulated annealing for finding a linear projection preserving persistent homology.
result Measures of topological equivalence between filtrations.
A deep learning approach for efficient multidimensional projections.
problem Computational inefficiency and stability issues in existing projection methods.
method Train a deep neural network on sample projections to infer new ones.
result Generates projections similar to learned ones, faster and more stable.
Develops precise expressions for random projections for better machine learning tasks.
problem Improving the accuracy of dimensionality reduction in machine learning tasks.
method Exploits recent developments in spectral analysis of random matrices to derive accurate expressions for random projection matrices.
result Provides precise expressions that reflect the practical performance of sketching methods, including Gaussian and Rademacher sketches.
Reductive quotients preserve klt singularities in algebraic geometry.
problem Preserving klt singularities in quotients of klt singularities.
method Proving that the quotient of a klt type singularity by a reductive group is of klt type.
result The quotient of a klt variety by a reductive group results in a klt variety with a suitable boundary.
Authors adapt Tannakian approach to reduce equivariant principal bundles.
problem Dimensional reduction of holomorphic principal bundles over complex projective manifolds.
method Adapt Tannakian approach to equivariant principal bundles.
result Established Hitchin--Kobayashi type correspondence for dimensional reduction.
Recent theoretical work has identified random projection as a promising dimensionality reduction technique for learning mixtures of Gausians. Here we summarize these results and illustrate them by a wide variety of experiments on synthetic and real data.
SRP efficiently learns class-aware embeddings for large datasets.
problem High computational complexity in supervised dimensionality reduction for large datasets.
method Supervised random projections (SRP) for direct class-aware embedding learning.
result SRP achieves 1-2 orders of magnitude better computational performance.
Optimizes t-SNE for high-dimensional data with random projections.
problem High computational cost of t-SNE for high-dimensional data.
method Use random projections to reduce high-dimensional data to a few dimensions, then apply t-SNE.
result Random projections preserve clustering while significantly reducing t-SNE runtime.
It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al. \ci…
A faster Wasserstein k-means algorithm for histogram data reduces computation and maintains clustering quality.
problem Efficiently clustering histogram data with reduced computation time.
method Sparse simplex projection to reduce data samples, centroids, and ground cost matrix, dynamically removing lower-valued samples.
result Significant reduction in computational complexity without compromising clustering quality.
SDSPCAAN combines supervised and local data structures for better dimensionality reduction.
problem Preserving both global and local data structures for noisy high-dimensional data.
method Supervised discriminative sparse PCA with adaptive neighbors (SDSPCAAN).
result SDSPCAAN improves classification accuracy on high-dimensional datasets.
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. The paper proves convexity results for a specific type of Lie groups.
problem Convexity results for non-compact real reductive Lie groups.
method Proves convexity results through orbit projection.
result Convexity results for quasi-hermitian Lie groups.
Optimizes differentially private kernel learning with random projection.
problem Privacy-preserving learning algorithms with optimal performance.
method Differentially private kernel ERM algorithm based on random projection in reproducing kernel Hilbert space.
result Achieves minimax-optimal excess risk rates for various loss functions.
Proposes a method to learn optimal neighbors and projection matrix in low-dimensional space.
problem Difficulty in precisely measuring similarity and selecting optimal neighbors in high-dimensional space.
method Models similarity and neighbors as variables, optimizing a unified objective function with nonnegative and sum-to-one constraints.
result Optimal similarity and projection matrix learned simultaneously, with adaptive regularization parameter.
The Johnson-Lindenstrauss Lemma allows for the projection of n points in p−dimensional Euclidean space onto a k−dimensional Euclidean space, with k≥3ε2−2ε324lnn, so that the pairwise distances are preserved within a factor of 1±ε. Here, working directly with the distributions of the …
A novel method reduces dimensionality for filtering SRNs with observed variables.
problem Challenges in estimating hidden state variables in SRNs with limited observations.
method Filtered Markovian Projection (Filtered MP) for dimensionality reduction in filtering.
result Filtered MP guarantees consistency and superior computational efficiency in high dimensions.
Develops an oblique projection technique to approximate a foliation for non-normal dynamics.
problem Modeling dynamics far from a primary Spectral Submanifold (SSM) in non-normal systems.
method Oblique projection technique based on experimental data.
result Approximates a stable invariant foliation for non-normal dynamics efficiently.
Dimensionality reduction helps analyze molecular simulations data.
problem High-dimensional molecular simulation data is hard to analyze.
method Various dimensionality reduction methods (k-means, autoencoder, PCA, tICA) applied to molecular simulation data.
result Methods learned different conformations of molecular processes.
Efficiently reduces data dimensionality with guaranteed geometry preservation.
problem Efficiently reducing high-dimensional data while preserving its geometric structure.
method Random subspace method with Johnson-Lindenstrauss guarantees, densifying preprocessing for sparse data.
result Random subspace method achieves geometry preservation with logarithmic dimensionality in data points.
FlowSDR learns a low-dimensional projection preserving the response's conditional distribution.
problem Learning a low-dimensional projection that captures the response's conditional distribution.
method FlowSDR uses conditional log-likelihood maximization with monotone rational-quadratic spline flows to learn the projection and conditional density.
result FlowSDR outperforms existing SDR methods in various simulation settings and a face-age prediction task.
Paper reduces turbomachinery CFD simulations by identifying key dimensions.
problem Reducing computational cost in turbomachinery 3D CFD simulations.
method Statistical sufficient dimension reduction methods and polynomial variable projection.
result Polynomial variable projection accurately identifies dimension reducing subspaces at lower cost.
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show tha…