arXiv research

Consider a generic $r$-dimensional subspace of $\mathbb{R}^d$, $r<d$, and suppose that we are only given projections of this subspace onto small subsets of the canonical coordinates. The paper establishes necessary and sufficient deterministic conditions on the subsets for subspace identifiability.

Due to diverse nature of data acquisition and modern applications, many contemporary problems involve high dimensional datum $\x \in \R^\d$ whose entries often lie in a union of subspaces and the goal is to find out which entries of $\x$ match with a particular subspace $\sU$, classically called \emph {matched subspace…

CobBO optimizes expensive functions in high dimensions by using a two-stage kernel approach.

In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by d…

A general class of Newton algorithms on Graßmann and Lagrange-Graßmann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergenc…

EAGC boosts GCD by regulating gradient entanglement, improving known and novel category separability.

We describe the Fast Greedy Sparse Subspace Clustering (FGSSC) algorithm providing an efficient method for clustering data belonging to a few low-dimensional linear or affine subspaces. The main difference of our algorithm from predecessors is its ability to work with noisy data having a high rate of erasures (missed e…

We study the linear subspace fitting problem in the overparameterized setting, where the estimated subspace can perfectly interpolate the training examples. Our scope includes the least-squares solutions to subspace fitting tasks with varying levels of supervision in the training data (i.e., the proportion of input-out…

LOFT separates subspace rotation and transformation for orthogonal fine-tuning.

The paper improves conditions for unique recovery in homomorphic sensing of subspaces.

SLMC improves sampling efficiency for high-dimensional distributions.

Develops accelerated methods for optimization using low-dimensional projected-gradient information.

We study representation stability in the sense of Church and Farb of sequences of cohomology groups of complements of arrangements of linear subspaces in real and complex space as $S_n$-modules. We consider arrangement of linear subspaces defined by sets of diagonal equalities $x_i = x_j$ and invariant under the action…

Space of hyperbolic surfaces is path-connected.

GDMaps reduces high-dimensional data to lower dimensions for better classification.

This work further develops the properties of fractional differential forms. In particular, finite dimensional subspaces of fractional form spaces are considered. An inner product, Hodge dual, and covariant derivative are defined. Coordinate transformation rules for integral order forms are also computed. Matrix order f…

Finding relationships between multiple views of data is essential both for exploratory analysis and as pre-processing for predictive tasks. A prominent approach is to apply variants of Canonical Correlation Analysis (CCA), a classical method seeking correlated components between views. The basic CCA is restricted to ma…

Paper develops new patterns for unique matrix completions.

Efficiently recovers data corrupted by adversarial noise in structured settings.

Study optimizes submatrices in 2D spaces, linking to polygon geometry.

Method selects interpretable circular coordinates from data.

Orthogonal Matching Pursuit (OMP) plays an important role in data science and its applications such as sparse subspace clustering and image processing. However, the existing OMP-based approaches lack of data adaptiveness so that the data cannot be represented well enough and may lose the accuracy. This paper proposes a…

Adaptive stochastic gradient algorithms in the Euclidean space have attracted much attention lately. Such explorations on Riemannian manifolds, on the other hand, are relatively new, limited, and challenging. This is because of the intrinsic non-linear structure of the underlying manifold and the absence of a canonical…

Ankle sprains and instability are major public health concerns. Up to 70% of individuals do not fully recover from a single ankle sprain and eventually develop chronic ankle instability (CAI). The diagnosis of CAI has been mainly based on self-report rather than objective biomechanical measures. The goal of this study …

For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components …

GAME improves matrix completion by considering subgroup-specific latent structures.

A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,ω)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $Ψ: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E.…

TOFU-POV tackles partially observed linear bandits, achieving sublinear regret with low-dimensional action vectors.

Let $G(n)={\rm Sp}(n,1)$ or ${\rm SU}(n,1)$. We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$. Such pairs, called `non-singular', were introduced by Gongopadhyay and Parsad for ${\rm SU}(3,1)$. We extend this notion and classify $G(n)$-conjugation orbits of such elements in arbitrary dim…

A geometric string solution has background fields in overlapping coordinate patches related by diffeomorphisms and gauge transformations, while for a non-geometric background this is generalised to allow transition functions involving duality transformations. Non-geometric string backgrounds arise from T-duals and mirr…

Memory-efficient optimizers fail to track a subspace, leading to unpredictable model performance.

New method identifies latent components in PNL mixtures without strong assumptions.

We construct a C-space associated with every closed 3-form on a spacetime $M$ and show that it depends on the class of the form in $H^3(M, Z)$. We also demonstrate that C-spaces have a relation to generalized geometry and to gerbes. C-spaces are constructed after introducing additional coordinates at the open sets and …

Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[S_r] of a symmetric group S_r. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W^{\bot} of W within the dual space of K[S_r] yield linear identities needed for a t…

In the double field theory, gauge symmetries are realized as generalized diffeomorphisms in the doubled spacetime. By consistency of the theory, dependence of tensor fields on the doubled coordinates is strongly constrained. This causes finite transformation law highly complicated, both technically and conceptually. In…

We consider the moduli space M_r of polygons with fixed side lengths in five-dimensional eucledian space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between M_r and a weighted quotient of the n-fold product of the quaternionic projective line HP^1 by the diagonal PSL(2,H…

Convex iso-Delaunay regions found in flat surface strata.

There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial diffferential equations in n variables, the so called U/K-system. Let G_{m,n} denote the Grassmannian of n-d…

This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. We consider the diffeological dual formally checking that the assignment to each space of its dual defines a covariant functor from the category of finite-dimensional diffeological vector …

New framework assesses neural sensitivity to small perturbations.

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise in the form of Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are comm…

We obtain the following version of Lidskii theorem. Let L, M, N be p-dimensional subspaces in R^n. Let ψ_j be the angles between L and M, let φ_j be the angles between M and N, and let θ_j be the angles between L and N. Consider the orbit of the vector ψwith respect to permutations of coordinates and inversions of axis…

The paper introduces a method for interpretable principal component analysis of high-dimensional time series.

New algorithm for nonconvex optimization on constrained Riemannian manifolds converges quickly.

We propose a general technique for improving alternating optimization (AO) of nonconvex functions. Starting from the solution given by AO, we conduct another sequence of searches over subspaces that are both meaningful to the optimization problem at hand and different from those used by AO. To demonstrate the utility o…

The paper improves alignment methods for deep neural networks using geometric and spectral analysis.

A new method solves the projection robust Wasserstein distance problem efficiently.

The investment on the stock market is prone to be affected by the Internet. For the purpose of improving the prediction accuracy, we propose a multi-task stock prediction model that not only considers the stock correlations but also supports multi-source data fusion. Our proposed model first utilizes tensor to integrat…