arXiv research

Proves new concentration inequalities for sub-gaussian and sub-exponential variables.

New approach to concentration inequalities for unbounded state space dynamical systems.

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and disc…

Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for …

New inequality on sphere generalizes circle inequality.

We establish an analog Hardy inequality with sharp constant involving exponential weight function. The special case of this inequality (for n=2) leads to a direct proof of Onofri inequality on S^2.

Optimal learning for parametric prophet inequalities with exponential-type distributions

In this paper, we study the CR Yamabe flow with zero CR Yamabe invariant. We use the CR Poincaré inequality and a Gagliardo-Nirenberg type interpolation inequality to show that this flow has long time solution and the solution converges to a contact form with flat pseudo-Hermitian scalar curvature exponentially.

Extends online learning to metric spaces using exponential weights.

Study hyperbolic geometry to find Fibonacci numbers.

New comparison theorem for submanifolds with geometric inequalities.

We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the lo…

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…

We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an "integrated" Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang's Harnack …

The fundamental properties of $J$-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a $J$-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We …

We prove a variant of the Davies-Gaffney-Grigor'yan Lemma for the continuous time heat kernel on graphs. We use it together with the Li-Yau inequality to obtain strong heat kernel estimates for graphs satisfying the exponential curvature dimension inequality.

Paper analyzes sparse aggregation in GLMs with Kullback-Leibler risk bounds.

Improved Cauchy-Schwarz inequality for $L^1$ and $L^2$ norms.

Estimating divergences in a consistent way is of great importance in many machine learning tasks. Although this is a fundamental problem in nonparametric statistics, to the best of our knowledge there has been no finite sample exponential inequality convergence bound derived for any divergence estimators. The main cont…

Improved concentration inequalities for sub-Weibull variables enhance statistical and machine learning applications.

Probability distributions of money, income, and energy consumption per capita are studied for ensembles of economic agents. The principle of entropy maximization for partitioning of a limited resource gives exponential distributions for the investigated variables. A non-equilibrium difference of money temperatures betw…

New energy functional and fields for Yang-Mills theory, proving monotonicity and vanishing theorems.

Study on curvature flow in 4D ball, proving existence and convergence.

Paper proves PI consensus algorithm converges exponentially under restricted secant inequality.

New findings on Helmholtz equation solutions show exponential growth in constant for three ball inequality.

We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $ω_t=Δω+aω\ln ω$ on closed manifolds. We also derive a new interpolated Harnack inequality for the equation $ω_t=Δω-ω\lnω+\varepsilon Rω$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we prove…

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We …

This paper extends the convergence analysis of Langevin Monte Carlo beyond Poincaré inequalities.

Study on Wasserstein gradient flow for MMD between Coulomb measures.

The paper analyzes convergence rates of Langevin dynamics and Proximal Sampler using $Φ$-divergence.

Study of curve evolution in 2D space forms converging to a circle.

Optimization rates improved for manifolds with bounded geometry.

We propose a definition of quasi-local mass based on the Penrose Inequality. Two further definitions are given by measuring distortions of the exponential map.

Deviation inequalities and limit laws for random walks on metric spaces.

We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimens…

We show that the mapping class group of a handlebody of genus at least 2 has a Dehn function of at most exponential growth type.

Uniform Poincaré inequalities established for various metric spaces.

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequa…

Thompson Sampling has been demonstrated in many complex bandit models, however the theoretical guarantees available for the parametric multi-armed bandit are still limited to the Bernoulli case. Here we extend them by proving asymptotic optimality of the algorithm using the Jeffreys prior for 1-dimensional exponential …

Paper provides a new lower bound on MMSE using Poincaré inequality.

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

The paper reviews and improves concentration inequalities for statistical inference.

In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we p…

Sharp Gaussian isoperimetry proven along Ricci flow.

Data processing inequalities link Fisher information to local differential privacy constraints.

New bounds on learning algorithm generalization error derived using information density.

New framework improves EM algorithm convergence under log-Sobolev inequality.

EM algorithm converges exponentially fast for overspecified Gaussian mixtures.