arXiv research

The paper tackles noisy combinations of continuous and step functions, providing conditions for their identification.

A Levi-Malcev type decomposition for $2$-step solvable Lie algebras with a complex structure

New methods for scoring function decomposition improve forecast evaluation.

We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. In the appendix of this paper, we show an abstract Nash-Moser normal form theorem, which generalizes our Levi decomposition result and which may be helpful in the study of other smooth nor…

Smooth 4-manifolds have simple horizontal decompositions.

A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step t…

We construct a decomposition of the identity operator on a Riemannian manifold $M$ as a sum of smooth orthogonal projections subordinate to an open cover of $M$. This extends a decomposition of the real line by smooth orthogonal projection due to Coifman, Meyer and Auscher, Weiss, Wickerhauser, and a similar decomposit…

This study proposes methods for multi-step-ahead stock price prediction using decomposition and neural networks.

In the present paper, a fuzzy logic based method is combined with wavelet decomposition to develop a step-by-step dynamic hybrid model for the estimation of financial time series. Empirical tests on fuzzy regression, wavelet decomposition as well as the new hybrid model are conducted on the well known $SP500$ index fin…

Decomposes smooth manifolds into algebraic submanifolds.

Dual energy computed tomography (DECT) imaging plays an important role in advanced imaging applications due to its material decomposition capability. Image-domain decomposition operates directly on CT images using linear matrix inversion, but the decomposed material images can be severely degraded by noise and artifact…

New examples of Calabi-Yau metrics on cones with irregular smooth links.

A new algorithm solves constrained optimization problems with stochastic gradients.

We present an approach for penalized tensor decomposition (PTD) that estimates smoothly varying latent factors in multi-way data. This generalizes existing work on sparse tensor decomposition and penalized matrix decompositions, in a manner parallel to the generalized lasso for regression and smoothing problems. Our ap…

A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.

The paper studies asymptotic dimensions of manifolds and spaces, proving key results about their geometric decompositions.

EnCF improves data assimilation for implicit, non-smooth observations.

Paper proves Whitney stratified spaces can be given a conically smooth structure.

New gradient methods solve multiscale optimization problems efficiently.

We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-man…

Classifies 4-manifolds with genus one horizontal handlebody decomposition.

Improved state estimation in nonlinear models using amortized backward variational inference.

Gay and Kirby introduced the notion of a trisection of a smooth 4-manifold, which is a decomposition of the 4-manifold into three elementary pieces. Rubinstein and Tillmann later extended this idea to construct multisections of piecewise-linear (PL) manifolds in all dimensions. Given a PL manifold $Y$ of dimension $n$,…

We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts each of which h…

On a compact, oriented, Riemannian manifold, the Hodge decomposition theorem associates a smooth primitive to any exact smooth form omega. In this paper, we show that given a smooth family of exact smooth forms omega(t), the family of associated primitives is also a smooth family with respect to t.

Improved convergence for Polyak steps with momentum in smooth convex optimization.

We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…

We propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. Our methods possess three main advantages compared to current state-of-the-art accelerated first-order methods: (1) they allow using two different extrapolation points to evaluate the grad…

Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and tensors analogs of much of the matrix algebra toolkit are unlikely to exist because …

Develops PRPCA for smooth image recovery combining low-rank and smoothness.

TATD predicts missing entries in time-evolving tensors by exploiting temporal dependency and sparsity.

Paper proves optimal decomposition for matrix fields, reducing convex integration steps.

Sparse Polyak improves high-dimensional statistical estimation.

Concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index <=2 (``Heegard'' decompositions).Sample result: Two 2-complexes are (up to 2-deformation) dual spines of a Heegard decomposition of the 4-sphere if and only if they satisfy the conclusions of the Alexander-Lefshet…

The paper studies foliations on smooth projective varieties and their properties.

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open b…

Multisections generalize trisections for 4-manifolds, allowing complex operations and explicit diagrams.

We extend the theory of relative trisections of smooth, compact, oriented $4$-manifolds with connected boundary given by Gay and Kirby to include $4$-manifolds with an arbitrary number of boundary components. Additionally, we provide sufficient conditions under which relatively trisected $4$-manifolds can be glued to o…

Improved sampling for Diffusion Models by accounting for covariance.

Let $M$ be a closed, oriented and smooth manifold of dimension $d$. Let $ŁM$ be the space of smooth loops in $M$. Chas and Sullivan introduced loop product, a product of degree $-d$ on the homology of $LM$. In this paper we show how for three manifolds the ``nontriviality'' of the loop product relates to the ``hyperbol…

Inverted file and asymmetric distance computation (IVFADC) have been successfully applied to approximate nearest neighbor search and subsequently maximum inner product search. In such a framework, vector quantization is used for coarse partitioning while product quantization is used for quantizing residuals. In the ori…

Step decay schedules improve convergence in non-convex optimization.

Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …

Study on spin random fields using chaos decomposition for cosmic microwave background modeling.

We propose an efficient ADMM method with guarantees for high-dimensional problems. We provide explicit bounds for the sparse optimization problem and the noisy matrix decomposition problem. For sparse optimization, we establish that the modified ADMM method has an optimal convergence rate of $\mathcal{O}(s\log d/T)$, w…

New SPS variant improves non-smooth optimization without small gradients.

The paper shows how gradient flow on over-parametrized tensor decomposition behaves like deflation.

We propose a novel sparse tensor decomposition method, namely Tensor Truncated Power (TTP) method, that incorporates variable selection into the estimation of decomposition components. The sparsity is achieved via an efficient truncation step embedded in the tensor power iteration. Our method applies to a broad family …