arXiv research

In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume ${\rm Vol}^{(2,2)}$ for 5-dimensional and 6-dimensional spin manifolds with boundary and we also get the Kastler-Kalau-Walze type theorem in this case.

New model estimates higher-order interactions in stochastic processes using lower-dimensional projections.

In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential …

Lower-dimensional video discriminators improve GAN performance.

In this paper, we define lower dimensional volumes associated to sub-Dirac operators for foliations. In some cases, we compute these lower dimensional volumes. We also prove the Kastler-Kalau-Walze type theorems for foliations with or without boundary. As a corollary, we give an explanation of the gravitational action …

A novel GP architecture, Thin and Deep GP, learns lower-dimensional representations without losing interpretability.

In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume ${\rm Vol}_{6}^{(1,3)}$ for 6-dimensional spin manifolds with boundary and the gravity on boundary is derived by the noncommutative residue associated with Dirac operators.For 6-dimensional manifo…

Generative Topological Networks (GTNs) simplify latent space generation.

Many features of dimensional reduction schemes are determined by the breaking of higher dimensional general covariance associated with the selection of a particular subset of coordinates. By investigating residual covariance we introduce lower dimensional tensors --generalizing to one side Kaluza-Klein gauge fields and…

The results of this paper are outdated. Finer versions of them will appear elsewhere.

Bayesian optimization (BO) is a powerful approach for seeking the global optimum of expensive black-box functions and has proven successful for fine tuning hyper-parameters of machine learning models. However, BO is practically limited to optimizing 10--20 parameters. To scale BO to high dimensions, we usually make str…

We mostly determine which closed smooth oriented 4-manifolds fibering over lower dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds.

RS-HDMR-GPR simplifies complex functions with machine-learned lower-dimensional terms.

In the present paper, we determine the topologies of three-dimensional closed Alexandrov spaces which converge to lower dimensional spaces in the Gromov-Hausdorff topology.

We explore algebraic characterizations of 2-knots whose associated knot manifolds fibre over lower-dimensional orbifolds, and consider also some issues related to the groups of higher-dimensional fibred knots.

We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold.

Two methods monitor high-dimensional processes via manifold fitting or learning.

In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a l…

Suppose that two large, multi-dimensional data sets are each noisy measurements of the same underlying random process, and principle components analysis is performed separately on the data sets to reduce their dimensionality. In some circumstances it may happen that the two lower-dimensional data sets have an inordinat…

New algorithm balances spatial data approximation and prediction accuracy.

Optimal persuasion involves projecting state vectors onto lower-dimensional 'optimal information manifolds'.

In this paper, we define lower dimensional volumes of compact Riemannian manifolds with boundary. For five dimensional spin manifolds with boundary, we prove a Kastler-Kalau-Walze type theorem associated with one-form perturbations of Dirac operators in this case.

A method models nonlinear dynamics from data using barycentric coordinates and memory.

Estimates high-dimensional posterior densities by marginal distributions and neural networks.

LDAdam optimizes large models with low memory by adapting to lower-dimensional subspaces.

New method reconstructs manifolds from data using Gaussian processes.

Study on symmetries in wide neural networks' dynamics without bias.

The decomposability of a Cartesian product of two nondecomposable manifolds into products of lower dimensional manifolds is studied. For 3-manifolds we obtain an analog of a result due to Borsuk for surfaces, and in higher dimensions we show that similar analogs do not exist unless one imposes further restrictions such…

With the increasing adoption of AI, inherent security and privacy vulnerabilities formachine learning systems are being discovered. One such vulnerability makes itpossible for an adversary to obtain private information about the types of instancesused to train the targeted machine learning model. This so-called model i…

Study geometric properties of loss functions to understand neural network performance.

Manifold-valued data naturally arises in medical imaging. In cognitive neuroscience, for instance, brain connectomes base the analysis of coactivation patterns between different brain regions on the analysis of the correlations of their functional Magnetic Resonance Imaging (fMRI) time series - an object thus constrain…

In statistical dimensionality reduction, it is common to rely on the assumption that high dimensional data tend to concentrate near a lower dimensional manifold. There is a rich literature on approximating the unknown manifold, and on exploiting such approximations in clustering, data compression, and prediction. Most …

We consider Riemannian 4-manifolds that Gromov-Hausdorff converge to a lower dimensional limit space, with the Ricci tensor going to zero. Among other things, we show that if the limit space is two dimensional then under some mild assumptions, the limiting four dimensional geometry away from the curvature blowup region…

A new method improves generative models by learning lower-dimensional representations.

We show that if a co-dimension two knot is deform-spun from a lower-dimensional co-dimension 2 knot, there are constraints on the Alexander polynomials. In particular this shows, for all n, that not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}.

A new matrix factorization method for high-dimensional data.

This paper is the first of a 3-part series that classifies the 5-dimensional Thurston geometries. The present paper (part 1 of 3) summarizes the general classification, giving the full list, an outline of the method, and some illustrative examples. This includes phenomena that have not appeared in lower dimensional geo…

New examples of Calabi-Yau 3-folds with unique properties.

The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integ…

We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix repres…

KPCA-BO improves BO for high-dimensional optimization problems by learning a non-linear sub-manifold.

Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.

GenSDR tackles SDR by leveraging generative models to fully recover lower-dimensional structures.

Bayesian neural networks improve uncertainty quantification in non-linear dimensionality reduction.

This work reduces the dimensionality of text data using SVD, improving performance and computational efficiency.

Study shows large extra dimensions are hidden from view by black holes.

The paper calculates transformation operators and proves a theorem on 4D manifolds.

In this paper, we study the collapsing behaviour of negative Kähler-Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space $\mathbb{Q}$-factorial, the Kähler-Einstein metrics on fibers collapse to a lower dim…