arXiv research

NWoS solves high-dimensional Poisson equations using neural networks.

Spheres' spectral structure converges to Gaussian space's as dimensions grow.

The study constructs and shows isotopy of high-dimensional Legendrian spheres.

Minimal triangulations of spheres map almost linearly to boundaries of high-dimensional polytopes.

We investigate the existence of homotopy comoment maps (comoments) for high-dimensional spheres seen as multisymplectic manifolds. Especially, we solve the existence problem for compact effective group actions on spheres and provide explicit constructions for such comoments in interesting particular cases.

New method explains high-dimensional sphere data with latent factors.

The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.

Conditions for integer signatures of high-dimensional knots.

Variational Auto-Encoder (VAE) has been widely applied as a fundamental generative model in machine learning. For complex samples like imagery objects or scenes, however, VAE suffers from the dimensional dilemma between reconstruction precision that needs high-dimensional latent codes and probabilistic inference that f…

We present a rigidity theorem for the action of the mapping class group $π_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a great number of cases, for example to simply connected $6$-manifolds and high dimensi…

A classical result of Milman roughly states that every Lipschitz function on $\mathbb{S}^n$ is almost constant on a sufficiently high-dimensional sphere $\mathbb{S}^m\subset \mathbb{S}^n$. In this paper we extend the result by proving that any Lipschitz function on a positively curved homogeneous space is almost consta…

It is known that any target function is realized in a sufficiently small neighborhood of any randomly connected deep network, provided the width (the number of neurons in a layer) is sufficiently large. There are sophisticated theories and discussions concerning this striking fact, but rigorous theories are very compli…

New MCMC methods map high-dimensional problems to spheres for better mixing.

Two-parameter models can learn high-dimensional targets via gradient flow.

We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homo…

SRCA reduces high-dimensional data to lower dimensions while preserving geometric structures.

Proves triviality of inertia groups in high-dimensional manifolds.

Similar simplices can be inscribed in most smoothly embedded spheres.

The paper solves the Nirenberg problem on high-dimensional half spheres with pinching conditions.

We consider pairs (X,Y) where X is a compact, locally CAT(-1) space, and Y is a totally geodesic subspace. The inclusion induces an embedding of the boundaries at infinity of the universal covers; we focus on the case where these are spheres whose dimensions differ by 2. We show that if the embedding is tame, then it i…

We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there…

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements of infinite order in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (o…

We present a method for training multi-label, massively multi-class image classification models, that is faster and more accurate than supervision via a sigmoid cross-entropy loss (logistic regression). Our method consists in embedding high-dimensional sparse labels onto a lower-dimensional dense sphere of unit-normed …

We show that, for certain families $φ_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $φ^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find example…

We propose and evaluate alternative ensemble schemes for a new instance based learning classifier, the Randomised Sphere Cover (RSC) classifier. RSC fuses instances into spheres, then bases classification on distance to spheres rather than distance to instances. The randomised nature of RSC makes it ideal for use in en…

New simplicial complexes show unavoidable link of spheres in high dimensions.

Paper proves embedding theorem for conformally compact manifolds.

New families of embeddings in 4-manifolds, topologically trivial but smoothly non-trivial.

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high dimensional manifolds. To be more specific, we study gauge groups of bundles over $(n-1)$-connected closed $2n$-manifolds, the classification of which was …

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in th…

New method solves PDEs on spheres using physics-informed convolutional neural networks.

While matrix factorisation models are ubiquitous in large scale recommendation and search, real time application of such models requires inner product computations over an intractably large set of item factors. In this manuscript we present a novel framework that uses the inverted index representation to exploit struct…

Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kahler manifold. Differing f…

A new Gaussian process regression method infers implicit manifold structure from data.

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

Completes classification of high-dimensional manifolds up to exotic sums.

Reflective Hamiltonian Monte Carlo struggles with high-dimensional sampling.

The hypercube's perimeter is significantly larger than expected near half volume.

New insights into simple kernel smoothing reveal surprising asymptotics.

The study examines how gamma positivity and PL homeomorphism types affect simplicial spheres.

New formula for curvatures of curves in n-dimensional space.

Classifies certain high-dimensional manifolds with specific cohomology properties.

Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to b…

Analyzes optimal learning rate schedules in high-dimensional non-convex optimization problems.

Gradient descent solves rank-one matrix estimation problem with detailed time evolution analysis.

We present a novel view of nonlinear manifold learning using derivative-free optimization techniques. Specifically, we propose an extension of the classical multi-dimensional scaling (MDS) method, where instead of performing gradient descent, we sample and evaluate possible "moves" in a sphere of fixed radius for each …

A novel method relaxes binary constraints to non-negative spheres for multi-matching and clustering.

Study learns a projection and function in Gaussian models.