arXiv research

The paper introduces gapped scale-sensitive dimensions to improve learning rate bounds.

New research shows that the dimension gap between intrinsic and ambient dimensions affects adversarial vulnerability of machine learning models.

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

Local gaps in Ricci shrinkers depend only on dimension.

New estimates show spectral gap stability in RCD spaces, close to Beta distribution.

Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which functions f in F can separate finite sets of points at a predefined resolution gamma > 0. We establish a connection between the g…

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

Study proves rigidity and gap theorems for specific metrics.

In this note, we prove the magnetic spectral gap-labelling conjecture as stated in [arXiv:1508.01064], in all dimensions, for principal solenoidal tori.

Formula derived for FUP exponent in quasi-Fuchsian groups.

In this paper several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result c…

We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap wh…

Study simplicial volume for fixed fundamental groups, finding gaps.

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a un…

We use the energy gap result of pure Yang-Mills equation [Feehan P.M.N., Adv. Math. 312 (2017), 547-587, arXiv:1502.00668] to prove another energy gap result of complex Yang-Mills equations [Gagliardo M., Uhlenbeck K., J. Fixed Point Theory Appl. 11 (2012), 185-198, arXiv:1401.7366], when Riemannian manifold $X$ of dim…

In this paper, we extend the sharp lower bounds of spectal gap, due to Chen- Wang [10, 11], Bakry-Qian [6] and Andrews-Clutterbuck [5], from smooth Riemaniannian manifolds to general metric measure spaces with Riemannian curvature-dimension condition RCD*(K;N).

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…

This paper describes a method for clustering data that are spread out over large regions and which dimensions are on different scales of measurement. Such an algorithm was developed to implement a robotics application consisting in sorting and storing objects in an unsupervised way. The toy dataset used to validate suc…

Proves weak cosmic censorship for a specific Einstein-scalar field system in 2+1 dimensions.

A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this …

We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.

The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.

Estimates intrinsic dimension of data sets robustly to noise.

New metric explains neural network performance, simplifying generalization error calculation.

New symmetry dimensions for higher order ODEs are identified.

The article proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.

Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.

Improved gap-dependent bounds for reinforcement learning with linear approximations.

Optimizes dimension estimate for holomorphic functions on Kähler manifolds.

We prove a \emph{query complexity} lower bound for approximating the top $r$ dimensional eigenspace of a matrix. We consider an oracle model where, given a symmetric matrix $\mathbf{M} \in \mathbb{R}^{d \times d}$, an algorithm $\mathsf{Alg}$ is allowed to make $\mathsf{T}$ exact queries of the form $\mathsf{w}^{(i)} =…

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and a diameter bound f…

C-projective structures are analogues of projective structures in the complex setting. The maximal dimension of the Lie algebra of c-projective symmetries of a complex connection on an almost complex manifold of C-dimension $n>1$ is classically known to be $2n^2+4n$. We prove that the submaximal dimension is equal to $…

This paper evaluates fractal dimension and persistent homology for neural network generalization.

We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.

Improved homological dimension for certain subgroups in Lie groups.

Solves approximation problems for zonoids and neural networks, closing gaps in dimensions 2 and 3.

The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.

For an Alexandrov space (with curvature bounded below), we determine the maximal dimension of its isometry group and show that the space is isometric to a Riemannian manifold, provided the dimension of its isometry group is maximal. We also determine a gap in the possible dimensions of the isometry groups and show that…

New theorem improves spectral gap for sampling from mixture distributions.

New algorithm optimizes convex functions with noisy evaluations in one dimension.

The study proves a gap theorem for CAT(0) spaces with a constant below 1/(6√π).

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…

Study spectral properties on manifolds with conical singularities, proving new inequalities.

We prove that if $G = G_1\times\dots\times G_n$ acts essentially, properly and cocompactly on a CAT(0) cube complex X, then the cube complex splits as a product. We use this theorem to give various examples of groups for which the minimal dimension of a cube complex the group acts on is strictly larger than that of the…

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor,…

Price gap, defined as the logarithmic price difference between the first two occupied price levels on the same side of a limit order book (LOB), is a key determinant of market depth, which is one of the dimensions of liquidity. However, the properties of price gaps have not been thoroughly studied due to the less avail…

Flat minimal tori counterexamples refute Lu's second-gap conjecture.

New lower bounds show challenges in clustering in moderate dimensions.