arXiv research

The total duration of drawdowns is shown to provide a moment-free, unbiased, efficient and robust estimator of Sharpe ratios both for Gaussian and heavy-tailed price returns. We then use this quantity to infer an analytic expression of the bias of moment-based Sharpe ratio estimators as a function of the return distrib…

Sharp decay found for solutions of a specific equation in Lie groups.

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

Sharp stability threshold found for deep residual architectures.

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.

The paper proves Zimmer's conjecture for non-uniform lattices by controlling mass escape and Lyapunov exponents.

Estimates for eigenfunctions and quasimodes on compact manifolds.

New method stabilizes deep neural networks by setting Lyapunov exponent to zero.

Classifies regularity for Lagrangian mean curvature type equations.

We empirically analyze the most volatile component of the electricity price time series from two North-American wholesale electricity markets. We show that these time series exhibit fluctuations which are not described by a Brownian Motion, as they show multi-scaling, high Hurst exponents and sharp price movements. We …

Study free energy in spherical spin glasses, proving universality dichotomy.

Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.

Let $β$ be a braid on $n$ strands, with exponent sum $w$. Let $Δ$ be the Garside half-twist braid. We prove that the coefficient of $v^{w-n+1}$ in the Homfly polynomial of the closure of $β$ agrees with $(-1)^{n-1}$ times the coefficient of $v^{w+n^2-1}$ in the Homfly polynomial of the closure of $βΔ^2$. This coinciden…

Research examines coamenable subgroups in higher rank groups.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…

This paper improves traditional Markowitz optimization by considering variance at multiple time scales.

Study efficient estimation of hidden subspaces in Gaussian Multi-index models.

Study on extremizers for Sobolev inequality on curved manifolds.

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent $n \ge 3$, then it has exactly the $n$-dimensional volume growth. As an application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci curvature satisfies th…

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples.…

In this paper, we study the dynamics of absolute return, trading volume and bid-ask spread after the trading halts using high-frequency data from the Shanghai Stock Exchange. We deal with all three types of trading halts, namely intraday halts, one-day halts and inter-day halts, of 203 stocks in Shanghai Stock Exchange…

We respond to the issues discussed by Farmer and Lillo (FL) related to our proposed approach to understanding the origin of power-law distributions in stock price fluctuations. First, we extend our previous analysis to 1000 US stocks and perform a new estimation of market impact that accounts for splitting of large ord…

Online SGD achieves consistent estimation in high-dimensional non-convex inference tasks.

We give sharp $C^{2,α}$ estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application…

Optimal estimates for spectral projection norms on compact manifolds.

Study on finite entropy and energy in Kähler geometry.

Optimal learning rates decay to zero in easy tasks and maintain a warmup phase in hard tasks.

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if…

The study of random positive 3-strand braids reveals patterns in the roots of their Alexander polynomials.

Anosov subgroups' deformations affect limit cones and growth indicators continuously.

Study semilinear equations on weighted manifolds to prove rigidity.

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it …

The price of financial assets are, since Bachelier, considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works. In this letter we investigate the question of sca…

Study shows computational and statistical gaps in Gaussian Single-Index Models.

Study shows a specific Carnot group violates a curvature exponent bound.

New groups found with critical exponents close to but less than max.

Sharp inequality for eigenvalues of convex bodies, proving ellipsoid uniqueness.

Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that {align*} \|u\|_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_g u|^2+c(n)R_gu^2\}dv_g +A\|u\|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all u\in H^1(M), where 2^*=2n/(n-2), c(n)=(n-2)/[4(n-1)], R_g is the scalar curvature, $K^{-1}=\i…

Kurdyka-Lojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of $\frac12$ for a suitable potential function is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. I…

Study detects a specific type of link using annular Khovanov homology.

New proof for certain groups in higher dimensions.

Sharp feature transitions revealed in extensive-width networks.

Extends Nash-Kuiper theorem to higher Hölder exponents.

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

In this paper, we show how the sampling properties of the Hurst exponent methods of estimation change with the presence of heavy tails. We run extensive Monte Carlo simulations to find out how rescaled range analysis (R/S), multifractal detrended fluctuation analysis (MF-DFA), detrending moving average (DMA) and genera…

Study of deep neural networks using finite-time Lyapunov exponents.

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

New similarity measure for covariate shift improves nonparametric regression rates.