arXiv research

New method avoids surface self-collision in geometric optimization.

Optimizes shapes of curves using Möbius energy gradients.

The paper discusses fractional Sobolev immersions of flat domains into 3D space.

We study estimation of (semi-)inner products between two nonparametric probability distributions, given IID samples from each distribution. These products include relatively well-studied classical $\mathcal{L}^2$ and Sobolev inner products, as well as those induced by translation-invariant reproducing kernels, for whic…

This work further develops the properties of fractional differential forms. In particular, finite dimensional subspaces of fractional form spaces are considered. An inner product, Hodge dual, and covariant derivative are defined. Coordinate transformation rules for integral order forms are also computed. Matrix order f…

In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $μ$ that naturall…

Sharp fractional Sobolev inequalities on closed manifolds identified.

Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.

This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.

Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.

Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.

Develops nonparametric regression for non-smooth functions using fractional Laplacian.

Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, partly due to a lack of practical estimators. They also include, as special cases, $L^2$ quantities which are used in many applicatio…

Paper introduces new fractional Dirac operator and Q-curvature.

We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It ext…

We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary …

Many procedures in science, engineering and medicine produce data in the form of geometric shapes. Mathematically, a shape can be modeled as an un-parameterized immersed sub-manifold, which is the notion of shape used here. Endowing shape space with a Riemannian metric opens up the world of Riemannian differential geom…

Study higher rank inner products and their tilings to describe tori degenerations.

We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $A\mapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+Δ^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain loc…

Researchers prove inner product recovery is impossible in latent space models.

Improved CR Sobolev inequalities on CR sphere established.

Paper proposes a new method to optimize feature coordinates for better image classification.

Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.

Study of Gaussian distributions using entropic Gromov-Wasserstein and inner product Gromov-Wasserstein.

The paper verifies deep neural networks' ability to approximate functions on spheres.

Let $M$ be a compact smooth manifold equipped with a positive smooth density $μ$ and $H$ be a smooth distribution endowed with a fiberwise inner product $g$. We define the Laplacian $Δ_H$ associated with $(H,μ,g)$ and prove that it gives rise to an unbounded self-adjoint operator in $L^2(M,μ)$. Then, assuming that $H$ …

We propose a quantization based approach for fast approximate Maximum Inner Product Search (MIPS). Each database vector is quantized in multiple subspaces via a set of codebooks, learned directly by minimizing the inner product quantization error. Then, the inner product of a query to a database vector is approximated …

We extend the well-known result that any $f \in W^{1,n}(Ω,\mathbb{R}^n)$, $Ω\subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(Ω)$ for any $s \geq \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distr…

Given a compact manifold $N^n \subset \mathbb{R}^ν$, $s \ge 1$ and $1 \le p < \infty$, we prove that the class of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when $N^n$ is $\lfloor sp \rfloor$ simply connected. For $sp$ integer, we prove weak den…

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves $\operatorname{Imm}(S^1,\mathbb{R}^d)$ and on its Sobolev completions $\mathcal{I}^{q}(S^1,\mathbb{R}^{d})$. We prove local well-posedness of the ge…

Study on kernel regression risk in high dimensions using Pinsker bound.

We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively n…

We study one extremal problem on the product of power of generalized inner radii of non-overlapping domains in $\mathbb{C}^{n}$.

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…

We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinct…

The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold $M$ of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric $H^s$ of order $0\le s<\tfrac12$ on the Lie algebra $\mathfrak X_c(M)$ of vector fields with compact …

Convex learning for diverse invariances in semi-inner-product space.

We present the first provably sublinear time algorithm for approximate \emph{Maximum Inner Product Search} (MIPS). Our proposal is also the first hashing algorithm for searching with (un-normalized) inner product as the underlying similarity measure. Finding hashing schemes for MIPS was considered hard. We formally sho…

Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.

Study on well-posedness of EPDiff equations with pseudo-differential inertia.

Develops Poisson structures on weak Sobolev loop spaces for integrable systems.

New tensorization theorem for Sobolev spaces on product spaces.

Being E a vector space with inner product and S the sphere of E, will be given a demonstration that every application of the sphere S itself it such that preserve inner product is the restriction of a linear isometry in E.

This study approximates neural network features for modeling relations and attention mechanisms.

The Bergman kernels of holomorphic vector bundles are studied to extend the Fubini-Study map.

Classifies metrics on specific Lie groups.

Study optimal partitions on spheres using fractional Q-curvature and variational methods.