Paper uses non-linear dimension reduction for better economic forecasting.
problem Analyzing economic effects of shocks in large datasets.
method Non-linear dimension reduction in factor-augmented vector autoregressions.
result Non-linear dimension reduction techniques improve forecasting, especially in volatile data.
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.
Efficient method estimates intrinsic dimension for big data.
problem Estimating intrinsic dimension for large datasets is costly and complex.
method Proposes a matrix-vector product-based approach for efficient intrinsic dimension estimation.
result Demonstrates superior performance compared to state-of-the-art methods.
ML reduces high-dimensional data to reveal its underlying structure.
problem Handling large, high-dimensional data sets.
method Non-linear dimension reduction techniques.
result Reveals the geometric shape of high-dimensional data.
FPP creates interpretable 2D embeddings for high-dimensional data.
problem Discovering interpretable relationships in high-dimensional data.
method Function preserving projections (FPP) for scalable linear embeddings.
result FPP reveals non-linear patterns of user-selected response functions.
PEA improves PCA and k-means for non-linear data and complex clusters.
problem Non-linear dimensionality reduction and clustering challenges.
method Principal Elliptical Analysis (PEA) for efficient non-linear approximation.
result PEA outperforms k-means in complex data clustering.
This paper addresses overfitting in dimension reduction methods by calibrating hyperparameters considering noise.
problem Overfitting in dimension reduction methods, especially t-SNE and UMAP, when data contains noise.
method Present a framework to calibrate hyperparameters in the presence of noise for t-SNE and UMAP.
result Recommended hyperparameter values for t-SNE and UMAP are too small and overfit the noise.
New insights into tSNE for large datasets.
problem Limitations of tSNE in handling large datasets.
method Identified continuum limit of tSNE objective function, proposed rescaled model.
result Rescaled model has a consistent limit for large datasets.
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.
Detects adversarial examples with non-linear dimensionality reduction.
problem Vulnerability of deep neural networks to adversarial examples.
method Combining non-linear dimensionality reduction and density estimation.
result Effective detection of adversarial examples by non-adaptive attackers.
Optimal persuasion involves projecting state vectors onto lower-dimensional 'optimal information manifolds'.
problem Optimal persuasion of another agent observing multi-dimensional data.
method Performing non-linear dimension reduction by projecting state vectors onto the 'optimal information manifold'.
result Optimal information design splits information into 'good' and 'bad' components, revealing only the direction of good information.
Suppose the data consist of a set S of points xj,1≤j≤J, distributed in a bounded domain D⊂RN, where N and J are large numbers. In this paper an algorithm is proposed for checking whether there exists a manifold M of low dimension near which many of the points of S lie and fin…
Molecular simulations produce very high-dimensional data-sets with millions of data points. As analysis methods are often unable to cope with so many dimensions, it is common to use dimensionality reduction and clustering methods to reach a reduced representation of the data. Yet these methods often fail to capture the…
This work improves understanding of dimension reduction algorithms and their probabilistic embeddings.
problem Improving theoretical understanding of non-linear dimension reduction algorithms.
method Analytical investigation of a generalized multidimensional scaling optimization problem.
result Probabilistic formulation of the problem leads to deterministic embeddings, contrary to standard implementations.
This paper improves HSIC-based dimensionality reduction for non-linear kernels.
problem Non-convexity of HSIC objective function for non-linear kernels limits optimization efficiency.
method Spectral optimization algorithm with local guarantees and principled initialization.
result Empirical improvements by a factor of 105 in runtime with lower errors. A new method bypasses regularization for disentangled latent variables without tuning.
problem Learning disentangled latent variables in unsupervised settings.
method Projection strategy to modify Gaussian encoder, ensuring zero cross-correlation among latent sub-coordinates.
result The method achieves maximal disentanglement theoretically and without loss in expressiveness.
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.
The paper constructs special Lagrangian n-folds in arbitrary dimensions.
problem Developing a construction for special Lagrangian n-folds in arbitrary dimensions.
method Reduction of special Lagrangian condition to a quasilinear elliptic system of 2D non-linear Cauchy-Riemann equations.
result The structure and multiplicity of singularities are governed by an associated polynomial.
Bayesian neural networks improve uncertainty quantification in non-linear dimensionality reduction.
problem Current neural network models lack adequate uncertainty quantification.
method Deploy Markov chain Monte Carlo sampling algorithms for Bayesian inference in ANN models with latent variables.
result New research directions are needed due to fundamental challenges in neural networks with latent variables.
A new method for fair representation learning using PLS.
problem Fairness in representation learning for data reduction.
method Proposes Fair Partial Least Squares (PLS) components with fairness constraints.
result The new method outperforms standard fair PCA methods on various datasets.
WeSpeR speeds up non-linear shrinkage for high-dimensional weighted covariance.
problem Computing non-linear shrinkage formulas for high-dimensional weighted sample covariance.
method Derive extit{WeSpeR} algorithm using asymptotic sample spectrum properties.
result Significantly speeds up non-linear shrinkage in dimensions higher than 1000.
The Ryu-Takayanagi conjecture connects the entanglement entropy in the boundary CFT to the area of open co-dimension two minimal surfaces in the bulk. Especially in AdS(4), the latter are two-dimensional surfaces, and, thus, solutions of a Euclidean non-linear sigma model on a symmetric target space that can be reduced…
A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifo…
The un-reduction procedure introduced previously in the context of Mechanics is extended to covariant Field Theory. The new covariant un-reduction procedure is applied to the problem of shape matching of images which depend on more than one independent variable (for instance, time and an additional labelling parameter)…
The quadric ansatz solves dKP equations in arbitrary dimensions, leading to Einstein-Weyl structures.
problem Characterizing solutions of the dispersionless KP equation in arbitrary dimensions.
method Quadric ansatz for the dKP equation, constructing Einstein-Weyl spaces.
result Explicit new family of Einstein-Weyl spaces constructed and characterized.
This paper deals with a new filter algorithm for selecting the smallest subset of features carrying all the information content of a data set (i.e. for removing redundant features). It is an advanced version of the fractal dimension reduction technique, and it relies on the recently introduced Morisita estimator of Int…
TaCo prevents non-linear classifiers from detecting sensitive attributes.
problem Ensuring fairness in NLP models by preventing sensitive attribute detection.
method Targeted Concept Erasure (TaCo) removes sensitive information from final latent representations, even against non-linear classifiers.
result TaCo outperforms state-of-the-art methods in reducing sensitive attribute prediction accuracy while preserving overall task performance.
Diffusion Maps improves on Functional PCA for non-linear functional data.
problem Functional PCA's linear manifold assumption fails for non-linear functional data.
method Extends Diffusion Maps to functional data and compares it to Functional PCA.
result Diffusion Maps outperforms Functional PCA in non-linear functional data analysis.
This paper conditions non-linear infinite-dimensional diffusion processes.
problem Conditioning non-linear and infinite-dimensional diffusion processes.
method Infinite-dimensional Girsanov's theorem to condition function-valued stochastic processes.
result Conditioning of non-linear infinite-dimensional diffusion processes is achieved.
New algorithm improves topological stability in non-linear dimensionality reduction.
problem Topological instability in choosing nearest neighbors in Isomap.
method Uses point and its two nearest neighbors to find subspace and orthogonal complement, then adds new points based on distance and angle.
result Improves topological stability and reduces short-circuit errors.
New method corrects missing data bias in dimension reduction.
problem Missing data complicates high-dimensional data analysis.
method Developed a bias-corrected Gram matrix for heterogeneous missingness.
result Proposed method improves dimension reduction techniques significantly.
Two new algorithms reduce feature space while preserving non-linear relationships.
problem High-dimensional data and overfitting issues.
method Bias-variance analysis for non-linear transformations and generalized linear models.
result Competitive performance on regression and classification tasks.
New algorithm approximates conditional expectations with fast convergence.
problem Approximating conditional expectations in stochastic derivative weights.
method Least-squares Monte Carlo with brute-force SVD truncation.
result Convergence rate is arbitrarily fast polynomial in number of samples.
In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs.
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…
Scalable GPLVM reduces complexity in scRNA-seq data, accounting for technical and biological confounders.
problem Complexity and confounders in scRNA-seq data hamper interpretation.
method Extended Gaussian process latent variable model (GPLVM) to handle large datasets.
result Framework reconstructs latent signatures and captures disease-specific gene expression.
Consensus dimension reduction combines multiple visualizations to identify shared patterns.
problem Conflicting visualizations from different dimension reduction methods.
method Multi-view learning to identify stable patterns across multiple views.
result Consensus visualization effectively identifies shared low-dimensional data structure.
Sequential Monte Carlo techniques are useful for state estimation in non-linear, non-Gaussian dynamic models. These methods allow us to approximate the joint posterior distribution using sequential importance sampling. In this framework, the dimension of the target distribution grows with each time step, thus it is nec…
Unified neural network for linear and nonlinear dimension reduction.
problem Efficiently perform linear and nonlinear sufficient dimension reduction.
method Belted and Ensembled Neural Network (BENN) framework.
result Unified framework for both linear and nonlinear dimension reduction.
Introduces MFVDM for high-dimensional data analysis.
problem Non-linear dimensionality reduction of high-dimensional datasets.
method Combines multiple unitary irreducible representations for nonlinear embeddings.
result Achieves better nearest neighbor search and alignment estimation on noisy data.
In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man ifold geometry using either local or global features of the data. Building on t…
The Gaussian process latent variable model (GP-LVM) is a popular approach to non-linear probabilistic dimensionality reduction. One design choice for the model is the number of latent variables. We present a spike and slab prior for the GP-LVM and propose an efficient variational inference procedure that gives a lower …
A method for clustering small datasets in high dimensions using random projections.
problem Challenges in clustering small datasets in high-dimensional spaces.
method Random projection followed by binary clustering in one-dimensional space.
result Statistically significant clustering structures can be found with as few as 100-200 points.
Manifold embedding algorithms map high-dimensional data down to coordinates in a much lower-dimensional space. One of the aims of dimension reduction is to find intrinsic coordinates that describe the data manifold. The coordinates returned by the embedding algorithm are abstract, and finding their physical or domain-r…
POTD estimates SDR subspace using optimal transport for binary response.
problem Insufficient performance of existing SDR methods for categorical responses.
method Principal optimal transport direction (POTD) using optimal transport coupling.
result POTD exclusively estimates SDR subspace for error-free class labels.
In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a l…
In this paper, we investigate the non-linear Black--Scholes equation: ut+ax2uxx+bx3uxx2+c(xux−u)=0,a,b>0, c≥0. and show that the one can be reduced to the equation ut+(uxx+ux)2=0 by an appropriate point transformation of variables. For the resulting equation, we study the group-theore…
We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension n=2, we prove an optimal dimension reduction theorem for complete translating Kähler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reducti…