arXiv research

Neural networks simplify SDR in regression tasks.

Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This paper studies regression of a $s$-Hölder function $f$ in $\mathbb{R}^D$ which varies along a central subspace of dimension $d$ while $d\ll D$. A direct approximation of $f$ in $\mathbb{R}^D$ with an $\varepsilon$ acc…

Paper improves SDR estimation speed and conditions.

New subspace prototype flag median improves clustering on noisy data.

AGOP from KRR recovers central subspace in fewer samples than needed for prediction.

Proposes a new algorithm to estimate invariant subspaces across multilayer networks.

New algorithm reduces dimensionality in federated learning.

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.

A multi-step framework tackles online unsupervised domain adaptation with novel mean-target subspace computation.

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 $\ell_1$ minimization with a g…

Paper proves linear convergence of SCMS algorithm for directional data.

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.

Proposes a transfer learning method for PCA studies.

This paper explores the impact of metric choice on Fréchet regression.

A new method tackles bilevel optimization using Lanczos process for efficient hyper-gradient computation.

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.

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 $\hat{U}$ 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 $\mathcal{J}(X)$ and its commuting associated operator $\mathcal{J}(X^{\per…

Generalizes k-means to graphs using PageRank.

Paper proposes distributed sparse multicategory discriminant analysis for classification.

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.

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.

KSS method converges and recovers correct clustering under certain conditions.

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.

Fast robust subspace tracking in sparse data-dependent noise with near-optimal delay.

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.

EpiMer merges models by solving Fréchet mean on a Riemannian manifold.

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 $\leq C\,(1+γ)$ 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.

The conformal Laplacian's algebraic structure is explored in 2D, revealing a central charge.

New algorithm updates eigenvectors of evolving graphs efficiently.

We characterize H-like Lie algebras in terms of subspaces of cones over conjugacy classes in $\mathfrak{so}(\mathbb{R}^q)$, 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…

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 …

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.

New algorithms identify invariant features for domain generalization.

For an ancient solution of the mean curvature flow, we show that each time slice M_t is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient solution is a family M_t that evolves under mean curvature flow for all negative ti…

It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.