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.
Paper studies efficient function approximation in high-dimensional spaces with low-dimensional structures.
problem Regression of functions varying along a central subspace in high-dimensional spaces.
method Generalized Contour Regression (GCR) algorithm for estimating the central subspace using piecewise polynomials.
result GCR leads to a mean squared estimation error of O(n−1) for the central subspace, improving the mean squared regression error of f to $O(n^{- rac{2s}{2s+d}})$. Paper improves SDR estimation speed and conditions.
problem Improving sufficient dimension reduction for multi-index models.
method Estimating expected smoothed gradient outer product.
result Achieves fast parametric convergence rate of Cd⋅n−1/2. New subspace prototype flag median improves clustering on noisy data.
problem Finding robust prototypes for datasets of images and videos.
method Proposes flag median and introduces FlagIRLS algorithm for its calculation.
result Flag median is robust to outliers and improves cluster purity.
AGOP from KRR recovers central subspace in fewer samples than needed for prediction.
problem Recovering low-dimensional structure in multi-index polynomial functions.
method Fit kernel ridge regression and compute AGOP from the fitted predictor.
result AGOP's top r eigenspace recovers the central subspace in n≍dp+δ samples. Proposes a new algorithm to estimate invariant subspaces across multilayer networks.
problem Estimating invariant subspaces across heterogeneous multiple networks.
method Bias-corrected joint spectral embedding algorithm that recursively calibrates diagonal bias and iteratively updates the subspace estimator.
result Established entrywise subspace perturbation bound and entrywise eigenvector central limit theorem for the algorithm.
New algorithm reduces dimensionality in federated learning.
problem Estimating central dimension reduction subspace and variable selection in federated learning.
method Federated sparse sliced inverse regression, convex optimization, linearized alternating direction method of multipliers.
result Upper bound of statistical error rate established under heterogeneous setting.
We investigate a Gaussian mixture model (GMM) with component means constrained in a pre-selected subspace. Applications to classification and clustering are explored. An EM-type estimation algorithm is derived. We prove that the subspace containing the component means of a GMM with a common covariance matrix also conta…
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.
A multi-step framework tackles online unsupervised domain adaptation with novel mean-target subspace computation.
problem Online unsupervised domain adaptation with unlabelled target data arriving sequentially.
method Multi-step framework with a novel mean-target subspace computation and temporal coherency consideration.
result Improved performance over previous approaches on four datasets.
The main contribution of the paper is a new approach to subspace clustering that is significantly more computationally efficient and scalable than existing state-of-the-art methods. The central idea is to modify the regression technique in sparse subspace clustering (SSC) by replacing the ℓ1 minimization with a g…
Paper proves linear convergence of SCMS algorithm for directional data.
problem Identifying density ridges in directional data.
method Generalized SCMS algorithm to directional data, derived from SCGA with adaptive step size.
result Linear convergence of the proposed directional SCMS algorithm.
We propose a conjugate gradient type optimization technique for the computation of the Karcher mean on the set of complex linear subspaces of fixed dimension, modeled by the so-called Grassmannian. The identification of the Grassmannian with Hermitian projection matrices allows an accessible introduction of the geometr…
LOFT separates subspace rotation and transformation for orthogonal fine-tuning.
problem Conflating subspace rotation and transformation in orthogonal fine-tuning.
method LOFT explicitly separates subspace rotation and transformation, using task-aware support selection.
result LOFT recovers principal-subspace orthogonal adaptation and improves efficiency-performance trade-off.
Proposes a transfer learning method for PCA studies.
problem Enhancing PCA estimation accuracy across multiple studies.
method Two-step algorithm integrating shared subspace information via Grassmannian barycenter.
result Knowledge transfer improves PCA estimation accuracy through enlarged eigenvalue gap.
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.
A new method tackles bilevel optimization using Lanczos process for efficient hyper-gradient computation.
problem Efficiently solving large-scale bilevel optimization problems with gradient-based methods.
method Constructing low-dimensional approximate Krylov subspaces with the Lanczos process to approximate the Hessian inverse vector product.
result Demonstrates a O(ε−1) convergence rate and efficiency in synthetic and deep learning tasks. In this letter, we consider two sets of observations defined as subspace signals embedded in noise and we wish to analyze the distance between these two subspaces. The latter entails evaluating the angles between the subspaces, an issue reminiscent of the well-known Procrustes problem. A Bayesian approach is investigat…
Centralized exchanges influence staking behavior and decentralization in Proof of Stake blockchain ecosystems.
problem How do centralized exchanges affect staking behavior and decentralization in Proof of Stake blockchain ecosystems?
method Formulate a continuous-time mean field model of miners as validators and traders in a centralized market.
result Centralized trading activities enhance staking participation and promote decentralization through market incentives.
We consider the problem of subspace estimation in a Bayesian setting. Since we are operating in the Grassmann manifold, the usual approach which consists of minimizing the mean square error (MSE) between the true subspace U and its estimate U^ may not be adequate as the MSE is not the natural metric in the Gra…
Many modern datasets can be represented as graphs and hence spectral decompositions such as graph principal component analysis (PCA) can be useful. Distinct from previous graph decomposition approaches based on subspace projection of a single topological feature, e.g., the Fiedler vector of centered graph adjacency mat…
We consider four-dimensional Riemannian manifolds with commuting higher order Jacobi operators defined on two-dimensional orthogonal subspaces (polygons) and on their orthogonal subspaces. More precisely, we discuss higher order Jacobi operator J(X) and its commuting associated operator $\mathcal{J}(X^{\per…
Generalizes k-means to graphs using PageRank.
problem Clustering nodes in directed and undirected graphs.
method Utilizes PageRank to compute node centrality in graphs.
result Robustly computes centrality in graphs and metric spaces.
Paper proposes distributed sparse multicategory discriminant analysis for classification.
problem Sparse multicategory classification with distributed data.
method Convex formulation, distributed setting, invariant discriminant subspace recovery.
result Distributed sparse multicategory linear discriminant analysis performs as good as centralized version after a few rounds of communications.
We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the ma…
Geometric framework for SPD matrices preserving subspace structures.
problem Processing SPD-valued data with preserved subspace structures.
method Thompson geometry of the semidefinite cone, extreme generalized eigenvalues, geodesic space structure.
result Novel inductive mean of SPD matrices based on Thompson geometry.
This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points.…
Study Sp(n)-orbits in complex and Σ-complex subspaces of Hermitian quaternionic vector spaces.
problem Characterize Sp(n)-orbits in Grassmannians of complex and Σ-complex subspaces. method Decompose subspaces into 4-dimensional complex addends and 2-dimensional totally complex subspace. Use properties of isoclinic subspaces and principal angles.
result Determine full set of invariants for Sp(n)-orbits in GrR(2k,4n). KSS method converges and recovers correct clustering under certain conditions.
problem Subspace clustering for semi-randomly sampled data.
method Local convergence analysis and recovery guarantee for KSS method.
result KSS method converges superlinearly and finds correct clustering within loglog N iterations.
We consider the problem of subspace clustering: given points that lie on or near the union of many low-dimensional linear subspaces, recover the subspaces. To this end, one first identifies sets of points close to the same subspace and uses the sets to estimate the subspaces. As the geometric structure of the clusters …
The paper finds Koopman invariant subspaces using personalized PageRank.
problem Selecting a finite dictionary of observables for Koopman-invariant span.
method Exploiting zero-block structure in EDMD matrices and applying PageRank.
result Personalized PageRank can detect Koopman invariant subspaces.
Proposes a new regularizer for semi-supervised learning on multilayer graphs.
problem Semi-supervised learning on multilayer graphs with labeled and unlabeled data.
method Generalized matrix mean regularizer and matrix-free numerical scheme.
result The regularizer outperforms state-of-the-art methods numerically.
Fast robust subspace tracking in sparse data-dependent noise with near-optimal delay.
problem Robustly tracking time-varying subspaces in the presence of sparse outliers.
method Introduces a fast mini-batch robust ST solution under mild assumptions.
result Provably correct subspace tracking with near-optimal delay and same time complexity as simple PCA.
We formulate and analyze a multi-agent model for the evolution of individual and systemic risk in which the local agents interact with each other through a central agent who, in turn, is influenced by the mean field of the local agents. The central agent is stabilized by a bistable potential, the only stabilizing force…
We study the problem of subspace tracking in the presence of missing data (ST-miss). In recent work, we studied a related problem called robust ST. In this work, we show that a simple modification of our robust ST solution also provably solves ST-miss and robust ST-miss. To our knowledge, our result is the first `compl…
Constructs finite element spaces for (p,q)-forms, excluding one subspace.
problem Constructing finite element spaces for (p,q)-forms. method Piecewise polynomial finite element spaces for all natural subspaces of (p,q)-forms, excluding one subspace. result Recovers known finite element spaces and introduces new ones.
EpiMer merges models by solving Fréchet mean on a Riemannian manifold.
problem Integrating knowledge from multiple models without retraining.
method EpiMer casts model merging as solving the Fréchet mean on a Riemannian manifold, restricting computation to a low-rank subspace.
result EpiMer outperforms flat-geometry methods on image classification tasks.
We show that all closed 2-dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subspace of small dimension. The entropy and dimension of the subspace are both ≤C(1+γ) for some universal constant C and genus γ. Th…
Sparsity-based subspace clustering algorithms have attracted significant attention thanks to their excellent performance in practical applications. A prominent example is the sparse subspace clustering (SSC) algorithm by Elhamifar and Vidal, which performs spectral clustering based on an adjacency matrix obtained by sp…
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.
The conformal Laplacian's algebraic structure is explored in 2D, revealing a central charge.
problem Exploring the algebraic structure of the conformal Laplacian in 2D.
method Using prefactorization algebras and Green functions.
result In 2D, the conformal Laplacian's algebraic structure is revealed through a central charge.
New algorithm updates eigenvectors of evolving graphs efficiently.
problem Updating eigenvectors of dynamic graphs.
method Subspace projection based on Rayleigh-Ritz projections.
result Strong performance in eigenvector approximation and downstream tasks.
We characterize H-like Lie algebras in terms of subspaces of cones over conjugacy classes in so(Rq), translating the classification problem for H-like Lie algebras to an equivalent problem in linear algebra. We study properties of H-like Lie algebras, present new methods for constructing them, in…
Model shows how centralization occurs in cryptocurrency mining.
problem Centralization of reward and computational power in Bitcoin-like cryptocurrencies.
method Mean field game model to study miner competition and reward distribution.
result Heterogeneity of initial wealth leads to greater imbalance in reward distribution.
We study the stability vis a vis adversarial noise of matrix factorization algorithm for matrix completion. In particular, our results include: (I) we bound the gap between the solution matrix of the factorization method and the ground truth in terms of root mean square error; (II) we treat the matrix factorization as …
A new model reduces noise and speeds up subspace segmentation.
problem Subspace segmentation from noisy data.
method Group norm regularized factorization model (GNRFM) with AALM algorithm.
result The method is faster and more robust to noise.
By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a real finite-dimensional linear space. In the paper we give a meticulous and self-contained treatment of this arch-classical subject, which lays a solid mathematical groundwork of Markowitz mean-variance theory of efficient por…
Python package for projecting onto quadratic hypersurfaces.
problem Projections onto non-cylindrical central quadratic hypersurfaces.
method User-friendly Python package with documentation.
result Efficiently projects points onto quadratic hypersurfaces.