arXiv research

Study on sets with positive reach in Euclidean and Riemannian spaces.

We give new characterisations of sets of positive reach and show that a closed hypersurface has positive reach if and only if it is of class $C^{1,1}$. These results are then used to prove new alternating Steiner formulæ for hypersurfaces of positive reach. Furthermore, it will turn out that every hypersurface that sat…

The paper extends submanifold reach to less regular $C^{1,α}$ classes.

Discuss folklore statements about manifolds with curvature bounds.

Study finds a non-locally contractible $r$-convex set.

Extends mapping results to non-compact Riemannian manifolds with positive reach.

New curvature measures characterize non-convex Wulff shapes in normed spaces.

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive $μ$-reach can be estimated by the same curvature measures of the offset of a compact set…

Total curvatures of certain hypersurfaces are continuous.

In this paper, we propose a deep reinforcement learning (DRL) solution to the grasping problem using 2.5D images as the only source of information. In particular, we developed a simulated environment where a robot equipped with a vacuum gripper has the aim of reaching blocks with planar surfaces. These blocks can have …

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving…

Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space, it is possible to recover the homology groups of $X$ by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set $Y$. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…

Study on covering probability of random balls in bounded open sets.

A new method for robot manipulation tasks using imagined object goals.

We study, using Mean Curvature Flow methods, 2+1 dimensional cosmologies with a positive cosmological constant and matter satisfying the dominant and the strong energy conditions. If the spatial slices are compact with non-positive Euler characteristic and are initially expanding everywhere, then we prove that the spat…

This paper proposes new get-rich-quick schemes that involve trading in a financial security with a non-degenerate price path. For simplicity the interest rate is assumed zero. If the price path is assumed continuous, the trader can become infinitely rich immediately after it becomes non-constant (if it ever does). If i…

We consider here a Fokker--Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, e…

In this paper, we study the problem of escaping from saddle points in smooth nonconvex optimization problems subject to a convex set $\mathcal{C}$. We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set $\mathcal{C}$ is simple for a quadratic objectiv…

In this note we clarify the structure of the moduli space of constant scalar curvature Kaehler metrics as one approaches the boundary of the Kaehler cone on cscK manifolds blown up at finite set of points, in the spirit of the previous work arXiv:math/0504115. Results about which Kaehler classes can be reached and abou…

To convert standard Brownian motion $Z$ into a positive process, Geometric Brownian motion (GBM) $e^{βZ_t}, β>0$ is widely used. We generalize this positive process by introducing an asymmetry parameter $α\geq 0$ which describes the instantaneous volatility whenever the process reaches a new low. For our new process, …

Computes tube formulas for valuations in complex space forms.

Study calculates reach and curvature of a specific geometric variety.

Computes bounds on reach and r-convexity from point cloud data.

Study improves understanding of submanifold reach in Riemannian geometry.

Proves existence of special 2-spheres in curved 3-spaces.

Defining the $m$-th stratum of a closed subset of an $n$ dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least $n-m$ linearly independent directions, we establish that the $m$-th stratum is second-order rectifiable of dimension $m$ and a Borel set. This was known for co…

Paper estimates manifold reach using convexity defect function.

Improved bounds for function approximation in nonlinear sets.

We determine the optimal strategy for investing in a Black-Scholes market in order to maximize the probability that wealth at death meets a bequest goal $b$, a type of goal-seeking problem, as pioneered by Dubins and Savage (1965, 1976). The individual consumes at a constant rate $c$, so the level of wealth required fo…

Various problems in manifold estimation make use of a quantity called the reach, denoted by $τ\_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. W…

Graph neural controlled differential equations learn graph dynamics from vertex observations.

Novel unsupervised feature selection method using multi-step Markov transition probability.

Diffusion reach probability between two nodes on a network is defined as the probability of a cascade originating from one node reaching to another node. An infinite number of cascades would enable calculation of true diffusion reach probabilities between any two nodes. However, there exists only a finite number of cas…

We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 v…

Optimizes leveraged staking strategies in decentralized finance.

Current reinforcement learning (RL) algorithms can be brittle and difficult to use, especially when learning goal-reaching behaviors from sparse rewards. Although supervised imitation learning provides a simple and stable alternative, it requires access to demonstrations from a human supervisor. In this paper, we study…

This work models financial market returns with asymmetric Tsallis distributions, improving fit over symmetric q-Gaussians.

We find the optimal investment strategy for an individual who seeks to minimize one of four objectives: (1) the probability that his wealth reaches a specified ruin level {\it before} death, (2) the probability that his wealth reaches that level {\it at} death, (3) the expectation of how low his wealth drops below a sp…

AdaDetectGPT improves text authorship detection with statistical guarantees.

In this paper we show that an expanding or steady gradient Ricci soliton warped product $B^n\times_f F^m$, $m>1$, whose warping function $f$ reaches both maximum and minimum must be a Riemannian product. Moreover, we present a necessary and sufficient condition for constructing a gradient Ricci soliton warped product. …

Quantifying the importance and power of individual nodes depending on their position in socio-economic networks constitutes a problem across a variety of applications. Examples include the reach of individuals in (online) social networks, the importance of individual banks or loans in financial networks, the relevance …

Active learning tackles cold start and imbalanced data issues.

Study shows spheres in high dimensions have maximum volume if they are smooth and have a specific reach.

We establish a quantitative lower bound on the reach of flat norm minimizers for boundaries in $\mathbb{R}^2$.

Recently, sentiment analysis has received a lot of attention due to the interest in mining opinions of social media users. Sentiment analysis consists in determining the polarity of a given text, i.e., its degree of positiveness or negativeness. Traditionally, Sentiment Analysis algorithms have been tailored to a speci…

Geometric finiteness theory for essential surfaces in knot exteriors with geometric bounds.

Affine spiking neural networks learn efficiently and generalize well.

It has been shown that injecting noise into the neural network weights during the training process leads to a better generalization of the resulting model. Noise injection in the distributed setup is a straightforward technique and it represents a promising approach to improve the locally trained models. We investigate…