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 explores tradeoffs in classification using tensor subspaces.
problem Supervised classification with sample, computation, and storage complexities.
method Use of tensor subspaces, particularly hierarchical Kronecker structured subspaces.
result Hierarchical Kronecker structured subspaces improve classification tradeoffs.
A new method for faster optimization in high dimensions.
problem Slow convergence in high-dimensional optimization problems.
method Subspace cubic regularized Newton method within Krylov subspace.
result Achieves a dimension-independent convergence rate of O(1/mk + 1/k^2).
Detects missing tensor signals in a KS subspace with high probability.
problem Detecting tensor signals with many missing entities in a KS subspace.
method Projecting the signal onto the KS subspace and bounding residual energy.
result Reliable detection is possible if the missing signal cardinality exceeds KS subspace dimensions.
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.
We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial.
We consider the problem of clustering a set of high-dimensional data points into sets of low-dimensional linear subspaces. The number of subspaces, their dimensions, and their orientations are unknown. We propose a simple and low-complexity clustering algorithm based on thresholding the correlations between the data po…
Paper shows affine constraint is unnecessary for high-dimensional data.
problem The necessity of an affine constraint in affine subspace clustering.
method Theoretical and empirical analysis of conditions for correctness of affine subspace clustering methods.
result Affine constraint has negligible effect on clustering performance for high-dimensional data.
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.
Proposes a method to enforce nestedness in subspace learning methods.
problem Consistency between data representations when choosing different subspaces dimensions.
method Lifts Grassmannian optimization criteria to flag manifolds via nested projectors.
result Successfully addresses the nestedness issue in several machine learning methods.
Rare data in a large-scale database are called outliers that reveal significant information in the real world. The subspace-based outlier detection is regarded as a feasible approach in very high dimensional space. However, the outliers found in subspaces are only part of the true outliers in high dimensional space, in…
Paper develops a method to identify feature subspaces contributing to local data complexity.
problem Identifying feature subspaces that contribute to local data complexity.
method Develops an estimator of Local Intrinsic Dimension (LID) along axis projections to identify feature subspaces.
result Preliminary evidence suggests LID decomposition can indicate axis-aligned data subspaces supporting cluster formation.
This paper covers robust subspace learning and tracking methods.
problem Learning and tracking subspaces in the presence of outliers.
method Robust PCA, Robust Subspace Tracking, Robust Subspace Recovery.
result Effective methods for handling outliers in subspace learning and tracking.
The paper generalizes PCA to manifolds using barycentric subspaces.
problem Generalizing PCA to non-Euclidean spaces.
method Introducing barycentric subspaces and optimizing AUV criterion.
result Barycentric Subspaces Analysis (BSA) generalizes PCA to Riemannian manifolds.
Neural networks simplify SDR in regression tasks.
problem Sufficient dimension reduction in regression problems.
method Applying neural networks with rank regularization to estimate the central mean subspace.
result Neural networks effectively perform SDR, consistent with theoretical estimations.
EigenGAN discovers interpretable dimensions in GAN layers for semantic control.
problem Lack of explicit dimensions to control semantic attributes in GAN layers.
method EigenGAN embeds linear subspaces with orthogonal bases into each generator layer, learning eigen-dimensions corresponding to semantic attributes via adversarial training.
result EigenGAN can produce samples with continuous changes corresponding to specific semantic attributes.
Paper analyzes impact of dimensionality reduction on subspace clustering algorithms.
problem Subspace clustering of high-dimensional data points into low-dimensional subspaces.
method Three subspace clustering algorithms (TSC, SSC, SSC-OMP) analyzed with random projections.
result Dimensionality reduction to subspace dimensions is feasible without significant performance loss.
The concept of pure spinor is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the…
Sparse OSEs achieve optimal embedding dimension of O(d).
problem Achieving optimal embedding dimension for sparse OSEs.
method Random sparsified matrix with m≥(1+θ)d non-zeros per column. result Sparse OSEs can achieve embedding dimension m=O(d), improving on previous m=O(dlog(d)). A new geometry-preserving method for interpreting compositional data.
problem Statistical challenges in high-dimensional compositional data.
method Geometry-preserving framework for dimension reduction of compositional data.
result Identification of a central compositional subspace for compositional predictors.
Paper finds necessary condition for logarithmic Minkowski problem in higher dimensions.
problem Logarithmic Minkowski problem in higher dimensions.
method Established a necessary condition through generalization and refinement of previous work.
result Generalizes and refines necessary condition for logarithmic Minkowski problem.
Algorithm finds a subspace minimizing distances to inliers with outliers.
problem Finding a k-dimensional subspace minimizing distances to inliers with outliers. method Extends dimension reduction techniques and bi-criteria approximations based on sampling.
result Efficient algorithm for multiplicative (1+ε)-approximation of optimal solution. GMM with constrained component means in pre-selected subspaces for classification and clustering.
problem Efficiently modeling data with constrained component means in subspaces.
method EM-type estimation algorithm, weighted PCA, multiple kernel densities, maximum likelihood selection.
result Subspace containing component means also contains modes and class means, leading to improved classification/clustering.
We study sparse principal components analysis in high dimensions, where p (the number of variables) can be much larger than n (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We introduce two complementary not…
Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.
problem Understanding correlation and mixing in high-dimensional linear systems with Gaussian noise.
method Sampling from sub-trajectories, using Talagrand's inequality, and analyzing invariant subspaces.
result Large discrepancy between algebraic and geometric multiplicity leads to bottlenecks between invariant subspaces.
This work tackles phaseless subspace tracking, recovering time-varying signals from phaseless projections.
problem Recovering time-varying signals from phaseless linear projections under gradual subspace change.
method Dynamic subspace tracking approach, leveraging gradual subspace change over time.
result Demonstrates feasibility of phaseless subspace tracking with gradual subspace change.
UDS measures multivariate correlations across any dimensionality.
problem Discovering significant correlations in multi-dimensional data.
method UDS based on cumulative entropy, normalized for comparison across subspaces.
result UDS efficiently captures both linear and non-linear correlations.
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…
We consider the problem of clustering noisy high-dimensional data points into a union of low-dimensional subspaces and a set of outliers. The number of subspaces, their dimensions, and their orientations are unknown. A probabilistic performance analysis of the thresholding-based subspace clustering (TSC) algorithm intr…
We describe ways to define and calculate L1-norm signal subspaces which are less sensitive to outlying data than L2-calculated subspaces. We focus on the computation of the L1 maximum-projection principal component of a data matrix containing N signal samples of dimension D and conclude that the general proble…
Paper uses random projection to preserve subspace structure for efficient data analysis.
problem Efficiently analyzing data with low-dimensional structure.
method Compressed Subspace Learning (CSL) framework based on Johnson-Lindenstrauss property.
result Random projection preserves the UoS structure of data, enabling efficient analysis.
This work presents GROUSE (Grassmanian Rank-One Update Subspace Estimation), an efficient online algorithm for tracking subspaces from highly incomplete observations. GROUSE requires only basic linear algebraic manipulations at each iteration, and each subspace update can be performed in linear time in the dimension of…
The problem of clustering noisy and incompletely observed high-dimensional data points into a union of low-dimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple low-complexity subspace clustering algorithm, w…
CobBO optimizes expensive functions in high dimensions by using a two-stage kernel approach.
problem Bayesian optimization struggles in high dimensions due to computational inefficiency.
method Coordinate backoff Bayesian Optimization with two-stage kernels.
result CobBO finds solutions comparable to or better than other methods in high dimensions.
Motivated by vision tasks such as robust face and object recognition, we consider the following general problem: given a collection of low-dimensional linear subspaces in a high-dimensional ambient (image) space, and a query point (image), efficiently determine the nearest subspace to the query in ℓ1 distance. In…
New bounds on singularities for flow in higher dimensions.
problem Understanding singularities in higher-dimensional flows.
method Entropy and codimension bounds for generic singularities.
result Uniform entropy and dimension bounds for singularities.
Projective DP-SGD reduces privacy error by identifying low-dimensional gradient subspaces.
problem Differentially private SGD's error rate scales with model's dimensionality, problematic for over-parameterized models.
method Projective DP-SGD, projecting noisy gradients to a low-dimensional subspace identified from a public dataset.
result The method reduces the dependence on model dimensionality, improving accuracy in high privacy regimes.
We analyze a non-convex landscape for robust subspace recovery and prove exact recovery conditions.
problem Analyzing the robustness of subspace recovery in non-convex energy landscapes.
method Mathematical analysis and proof of conditions for exact recovery of the underlying subspace.
result A geodesic gradient descent method can exactly recover the underlying subspace under specific conditions.
Unified approach to studying hyperbolic groups using stable subspaces and Morse boundaries.
problem Understanding the geometric and algebraic properties of hyperbolic groups.
method Unified approach to viewing geodesic metric spaces as unions of stable subspaces, using quasi-convex subsets and direct limits of Gromov boundaries.
result Unified understanding of stable subgroups and Morse boundaries, leading to new quasi-isometry invariant dimensions.
A new method for Bayesian inference in high dimensions using projected Stein variational gradient descent.
problem Bayesian inference challenges in high-dimensional data.
method Adapting Stein variational gradient descent to exploit intrinsic low dimensionality of data.
result pSVGD is more accurate and efficient than SVGD, especially in high-dimensional settings.
Study on estimating covariance and precision matrices along specific subspaces.
problem Estimating covariance and precision matrices along prescribed subspaces or directions.
method Analysis of finite sample covariance, focusing on components corresponding to desired subspaces or directions.
result Estimation accuracy depends almost exclusively on components corresponding to desired subspaces or directions.
We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the to…
ALPCAH improves PCA for noisy data samples.
problem Heteroscedastic data with varying noise levels.
method Subspace learning method estimating sample-wise noise variances.
result Improves subspace basis for low-rank data.
We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold…
Finite spaces can be or not coproducts of subspaces.
problem When finite-dimensional diffeological vector spaces are coproducts of their subspaces.
method Reviewing the question in diffeological vector spaces and comparing with other categories.
result Finite-dimensional spaces can be coproducts, but not always.
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.
Proposes methods to compute optimal transport maps via subspace projections.
problem Computing optimal transport in high dimensions is challenging due to the curse of dimensionality.
method Develops two methods to extrapolate optimal transport plans from subspace projections to the full space.
result The best optimal transport plan is a generalization of the Knothe-Rosenblatt transport.
Paper proposes new metrics to quantify CCA estimation loss and characterizes minimax rates.
problem Estimating the loss of canonical correlation analysis (CCA) in learning low-dimensional representations.
method Proposes a new error metric based on excess prediction loss and views CCA as subspace estimation.
result Characterizes non-asymptotic minimax rates for CCA under proposed metrics.