arXiv research

Study 4D steady gradient Ricci solitons reducing to 3D manifolds.

Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations with "appropriate coefficients". The leafwise reduced cohomology is also described…

A new method reduces both input and output dimensions for better goal-oriented analysis.

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike spli…

The dynamics defined by a force field which is positively homogeneous of degree $-3$ can always be reduced, by simply constraining it. The dimension of the phase space is reduced by two dimensions, while it may only be reduced by one dimension if the degree of homogeneity is different from $-3$. This remark is an elega…

A new method reduces dimensionality for better likelihood-free parameter estimation.

Paper reduces turbomachinery CFD simulations by identifying key dimensions.

We study a method of reducing space dimension in multi-dimensional Black-Scholes partial differential equations as well as in multi-dimensional parabolic equations. We prove that a multiplicative transformation of space variables in the Black-Scholes partial differential equation reserves the form of Black-Scholes part…

This research shows that steady solitons in higher dimensions always reduce at infinity.

We discuss Poincaré duality complexes X and the question whether or not their Spivak normal fibration admits a reduction to a vector bundle in the case where the dimension of X is at most 4. We show that in dimensions less than 4 such a reduction always exists, and in dimension 4 such a reduction exists provided X is o…

Paper reduces dimensionality for robust option pricing in 2-asset markets.

The main result of this article states that the (K;N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner…

We consider partially observed multiscale diffusion models that are specified up to an unknown vector parameter. We establish for a very general class of test functions that the filter of the original model converges to a filter of reduced dimension. Then, this result is used to justify statistical estimation for the u…

A new tensor-based layer reduces neural network dimensions without losing important features.

The current study proposes a dimension reduction method, stepwise support vector machine (SVM), to reduce the dimensions of large p small n datasets. The proposed method is compared with other dimension reduction methods, namely, the Pearson product difference correlation coefficient (PCCs), recursive feature eliminati…

Study reduces dimensions for $k$-means clustering for better accuracy.

Mixed dimension embeddings reduce memory usage in recommendation systems.

New method for decomposing high-dimensional parametric domains using PCA and inverse projection.

Two methods preserve tensor structure for reduced dimensionality in tensor regression.

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension …

A new layer, funnel, reduces dimensionality in flows for better performance.

The paper explores how to reduce classification tasks to optimization problems in Euclidean space.

Robust STAP with coprime arrays reduces clutter using sparse modeling.

In this project we further investigate the idea of reducing the dimensionality of datasets using a Borel isomorphism with the purpose of subsequently applying supervised learning algorithms, as originally suggested by my supervisor V. Pestov (in 2011 Dagstuhl preprint). Any consistent learning algorithm, for example kN…

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geode…

We introduce a dimension reduction method for visualizing the clustering structure obtained from a finite mixture of Gaussian densities. Information on the dimension reduction subspace is obtained from the variation on group means and, depending on the estimated mixture model, on the variation on group covariances. The…

Optimizes expensive shape models using Gaussian processes in reduced eigenbases.

Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.

Unsupervised dimension selection is an important problem that seeks to reduce dimensionality of data, while preserving the most useful characteristics. While dimensionality reduction is commonly utilized to construct low-dimensional embeddings, they produce feature spaces that are hard to interpret. Further, in applica…

New algorithm reduces sketching dimension to effective problem size.

Study on $L^p$ cohomology and Hodge decomposition for ALE manifolds.

Sliced Inverse Regression reduces parameter space for estimating complex financial models.

Sparsity helps reduce diffusion model costs.

ML reduces high-dimensional data to reveal its underlying structure.

Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.

The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an algorithm to find a projection that distorts the data as little as possible.

The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "$N$-$\mathcal F$-amenability" or "amenability dimension", and "$d$-BLR condition") and its generalisation, transfer reducibility, which are versions of asymptotic dimension invented for the proofs of the Farrell--Jones and Bo…

Investigates maps and properties in spaces with negative dimensions and curvature.

SkMM selects data for finetuning by balancing bias and variance.

Supervised linear feature extraction can be achieved by fitting a reduced rank multivariate model. This paper studies rank penalized and rank constrained vector generalized linear models. From the perspective of thresholding rules, we build a framework for fitting singular value penalized models and use it for feature …

Holomorphic residue formula for complex supermanifolds.

Reduces function approximation dimensions from high to low with sparse data.

A new method maps high-dimensional Bayesian inverse problems to lower dimensions.

We show how a polar representation of a compact connected Lie group can be linearly determined from its dimension and isotropy subgroup data in the general reducible case.

Study on nilpotent Lie algebras with specific metrics.

We prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far more subtle. We thus analyse the possible self-homeomorphisms of reduc…

Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.

Proves open Riemann surfaces can be embedded into 4D space.