arXiv research

New method for multiway clustering of 3rd order tensors.

Subspace clustering refers to the problem of clustering high-dimensional data into a union of low-dimensional subspaces. Current subspace clustering approaches are usually based on a two-stage framework. In the first stage, an affinity matrix is generated from data. In the second one, spectral clustering is applied on …

A study of proper affine vector fields in plane symmetric static space-times by using the rank of the Rieman matrix and holonomy. Studying proper affine vector fields in each case, It is shown that the special class of the above space-times admit proper affine vector fields.

We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the con…

We propose and study a row-and-column affine measurement scheme for low-rank matrix recovery. Each measurement is a linear combination of elements in one row or one column of a matrix $X$. This setting arises naturally in applications from different domains. However, current algorithms developed for standard matrix rec…

Subspace clustering (SC) refers to the problem of clustering high-dimensional data into a union of low-dimensional subspaces. Based on spectral clustering, state-of-the-art approaches solve SC problem within a two-stage framework. In the first stage, data representation techniques are applied to draw an affinity matrix…

Incorporates matrix exponential into generative flows for improved performance.

Long term optimal investment problems are studied in a factor model with matrix valued state variables. Explicit parameter restrictions are obtained under which, for an isoelastic investor, the finite horizon value function and optimal strategy converge to their long-run counterparts as the investment horizon approache…

Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measur…

The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the affine Grassmannian as a matrix manifold and extend Riemannian optimization algorit…

We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theo…

This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrix-valued affine processes in finance, including multi-asset option pricing with stochastic volatil…

Doubly-stochastic normalization improves robustness to heteroskedastic noise.

Most existing approaches address multi-view subspace clustering problem by constructing the affinity matrix on each view separately and afterwards propose how to extend spectral clustering algorithm to handle multi-view data. This paper presents an approach to multi-view subspace clustering that learns a joint subspace…

Developed moment estimators for affine stochastic volatility models.

New method computes affine normal directions efficiently for sparse polynomials.

This paper classifies reversible and strongly reversible elements in affine groups.

The paper introduces a new class of multivariate mixtures for actuarial applications.

Develops a theory for mth order p-affine capacity for convex bodies containing the origin.

Many applications require recovering a matrix of minimal rank within an affine constraint set, with matrix completion a notable special case. Because the problem is NP-hard in general, it is common to replace the matrix rank with the nuclear norm, which acts as a convenient convex surrogate. While elegant theoretical c…

We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional…

Solitons are special polygon midpoints under affine transformations.

MASC balances dataset representation using affinity clustering and distribution discrepancies.

DKLM learns adaptive kernels for robust nonlinear subspace clustering.

New PSDMF algorithms derived from PR and ARM methods.

Spectrahedral regression fits convex functions via a non-convex optimization problem.

New AMP algorithm estimates signals and latent variables in mixed regression models.

The paper classifies vector fields on 5D nilpotent Lie groups.

In this paper, we consider the connectedness of planar self-affine set $T(A,\mathcal{D})$ arising from an integral expanding matrix $A$ with characteristic polynomial $f(x)=x^2+bx+c$ and a digit set $\mathcal{D}=\{0,1,\dots, m\}v$. The necessary and sufficient conditions only depending on $b,c,m$ are given for the $T(A…

These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Ampère and linearized Monge-Ampère equations. The prescribed affine mean curvature equation is a fully nonlinear, fourth order, geometri…

Let $A$ be an expanding $d\times d$ matrix with integer entries and ${\mathcal D}\subset {\mathbb Z}^d$ be a finite digit set. Then the pair $(A, {\mathcal D})$ defines a unique integral self-affine set $K=A^{-1}(K+{\mathcal D})$. In this paper, by replacing the Euclidean norm with a pseudo-norm $w$ in terms of $A$, we…

Anosov groups study matrix coefficients and orbit counting in symmetric spaces.

Multi-view subspace clustering has been applied to applications such as image processing and video surveillance, and has attracted increasing attention. Most existing methods learn view-specific self-representation matrices, and construct a combined affinity matrix from multiple views. The affinity construction process…

We study the connectedness of the planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det(A)|=3$ and a non-collinear digit set ${\mathcal D}=\{0, v, kAv\}$ where $k\in {\mathbb Z}\setminus\{0\}$ and $v\in {\mathbb Z}^2$ such that $\{v, Av\}$ is linearly independent. By chec…

In this paper we construct a homomorphism of the affine braid group $Br_n^{aff}$ in the convolution algebra of the equivariant matrix factorizations on the space $\overline{\mathcal{X}}_2=\mathfrak{b}_n\times GL_n\times\mathfrak{n}_n$ considered in the earlier paper of the authors. We explain that the pull-back on the …

Affinity propagation is one of the most effective unsupervised pattern recognition algorithms for data clustering in high-dimensional feature space. However, the numerous attempts to test its performance for community detection in complex networks have been attaining results very far from the state of the art methods s…

An $\varepsilon$-coreset for Least-Mean-Squares (LMS) of a matrix $A\in{\mathbb{R}}^{n\times d}$ is a small weighted subset of its rows that approximates the sum of squared distances from its rows to every affine $k$-dimensional subspace of ${\mathbb{R}}^d$, up to a factor of $1\pm\varepsilon$. Such coresets are useful…

Retrieving the most similar objects in a large-scale database for a given query is a fundamental building block in many application domains, ranging from web searches, visual, cross media, and document retrievals. State-of-the-art approaches have mainly focused on capturing the underlying geometry of the data manifolds…

Spectral Clustering(SC) is a prominent data clustering technique of recent times which has attracted much attention from researchers. It is a highly data-driven method and makes no strict assumptions on the structure of the data to be clustered. One of the central pieces of spectral clustering is the construction of an…

A fast method estimates correlations in hybrid systems using observable market data.

In the paper, we focus on the connectedness of planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det (A)|=3$ and a collinear digit set ${\mathcal{D}}=\{0,1,b\}v$, where $b>1$ and $v\in {\mathbb{R}}^2$ such that $\{v, Av\}$ is linearly independent. We discuss the domain of…

Let $T:= T(A, {\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\partial T$ with a sofic system by constructing a ne…

Reconstruction based subspace clustering methods compute a self reconstruction matrix over the samples and use it for spectral clustering to obtain the final clustering result. Their success largely relies on the assumption that the underlying subspaces are independent, which, however, does not always hold in the appli…

Self-affine tiles homeomorphic to a ball proven for a specific digit set.

Viewing a data set such as the clouds of Jupiter, coherence is readily apparent to human observers, especially the Great Red Spot, but also other great storms and persistent structures. There are now many different definitions and perspectives mathematically describing coherent structures, but we will take an image pro…

We provide a proof of backpropagation algorithm in matrix notation.

Image translation with convolutional neural networks has recently been used as an approach to multimodal change detection. Existing approaches train the networks by exploiting supervised information of the change areas, which, however, is not always available. A main challenge in the unsupervised problem setting is to …

No non-product Hessian rank 1 affine homogeneous hypersurfaces exist in dimensions 5 and above.