arXiv research

Study on Willmore spheres, calculating their index and relating it to minimal surfaces.

Study shows Reeb orbits on starshaped hypersurfaces grow logarithmically with period.

We study Bogomolny equations on $R^2\times S^1$. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using Nahm transform, we show that periodic monop…

Random surfaces with long systoles created from graph theory ideas.

Region detection in Gaussian Markov fields with limited samples.

Geodesics grow infinitely in certain Finsler manifolds.

The problem of existence of solution for the Heath-Jarrow-Morton equation with linear volatility and purely jump random factor is studied. Sufficient conditions for existence and non-existence of the solution in the class of bounded fields are formulated. It is shown that if the first derivative of the Levy-Khinchin ex…

The study of logarithmic fields associated with nilmanifolds and their singularities.

Constructs a Hodge filtration for vector fields of complex reflection groups.

Minimal hyperbolic surface diameter grows logarithmically with genus.

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds wit…

The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are constructed by taking congruence covers of arithmetic surfaces. In this article we p…

Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.

It is well-known that neural networks are universal approximators, but that deeper networks tend in practice to be more powerful than shallower ones. We shed light on this by proving that the total number of neurons $m$ required to approximate natural classes of multivariate polynomials of $n$ variables grows only line…

The paper finds that circles and logarithmic spirals are the only constant-speed ramps for a specific force field.

The paper analyzes competition among fund managers using excess logarithmic returns and constructs games to find optimal allocations.

We present a construction of sequences of closed hyperbolic surfaces that have long systoles which form pants decompositions of these surfaces. The length of the systoles of these surfaces grows logarithmically as a function of their genus.

Random surfaces' diameter grows logarithmically with size.

New algorithms achieve logarithmic regret in learning linear quadratic control systems.

In this work, we give a formula for the logarithmic invariant of knots in terms of certain derivatives of the colored Jones invariant. This invariant is related to the logarithmic conformal field theory, and was defined by using the centers in the radical of the restricted quantum group at root of unity. A relation bet…

Study Higgs bundles on curves with punctures, extending spectral correspondence.

Study shows how many crossings arise in curves on surfaces.

The paper constructs a Saito basis for a specific class of divisors and applies it to logarithmic Poisson geometry.

The length of shortest non-simple geodesics grows logarithmically with surface genus.

Proves a stack of G-bundles with logarithmic connections is finite type.

We present asymptotic and finite-sample results on the use of stochastic blockmodels for the analysis of network data. We show that the fraction of misclassified network nodes converges in probability to zero under maximum likelihood fitting when the number of classes is allowed to grow as the root of the network size …

Finite-time queue peaks in stochastic networks have logarithmic scaling after geometric thresholds.

Optimized bandit algorithms have heavy-tailed regret distributions that can grow faster than expected.

Method calculates systolic length of modular curves.

We present a new anytime algorithm that achieves near-optimal regret for any instance of finite stochastic partial monitoring. In particular, the new algorithm achieves the minimax regret, within logarithmic factors, for both "easy" and "hard" problems. For easy problems, it additionally achieves logarithmic individual…

The paper provides estimates for flows on Riemannian manifolds using truncated expansions.

Logarithmic-time schedules boost large-scale language model training efficiency.

We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for small values of $n$. We also show that this function is greater than a function that…

Logarithmic connections on principal bundles over normal varieties are studied.

PPO-B improves sampling efficiency by using a logarithmic barrier method.

Deep neural networks undergo hierarchical free-energy landscape transitions with increasing data size.

Adaptive sampling results in dramatic improvements in the recovery of sparse signals in white Gaussian noise. A sequential adaptive sampling-and-refinement procedure called Distilled Sensing (DS) is proposed and analyzed. DS is a form of multi-stage experimental design and testing. Because of the adaptive nature of the…

Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As…

The paper analyzes Q-learning in 2-player Markov games and provides gap-dependent logarithmic regret bounds.

This work shows linear convergence for two-layer neural networks in mean-field regime.

We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed -- the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a sequence of subsets $A_n$ of a homogeneous space $G/Γ$ ($G$ a semisimple Lie group…

Improved particle approximation for mean-field neural networks.

We shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a $C^\infty$ logarithmic symplectic structure has unobstruc…

Minimal submanifolds confined in space are highly restricted.

The problem of distributed learning and channel access is considered in a cognitive network with multiple secondary users. The availability statistics of the channels are initially unknown to the secondary users and are estimated using sensing decisions. There is no explicit information exchange or prior agreement amon…

Near-logarithmic regret per switch achieved for mixable/exp-concave losses.

Study on neural networks' performance under different normalizations as N grows.

Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.