arXiv research

Study proves existence and uniqueness for differential equations with non-Lipschitz coefficients.

Proves existence and uniqueness of solutions for complex stochastic equations.

We consider the approximation of stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence, with a rate. Under some regularity and integrability conditions, we obtain the opt…

Improved MLMC method for barrier options with non-Lipschitz coefficients.

Paper tackles robust control of SDEs with ambiguity, proving value function existence and applying to investment problems.

We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the SDEs and is strongly convergent with rate one. Moreover, we show that this general …

The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^tμX_sds+\int_0^tσ(X^+_s)^pdW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the proc…

New algorithm reduces pricing error by a factor of T^2/3.

Study on non-negative solutions for stochastic Volterra equations with jumps.

We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range of applications in data science, where the objective is used for inducing spars…

New algorithms for online learning without boundedness or Lipschitz loss assumptions.

Paper proves convergence for private FL on non-Lipschitz convex objectives using normalization instead of clipping.

Extends involutivity to non-Lipschitz subbundles and proves the Frobenius Theorem.

A new macroscopic market making model connects market making and optimal execution.

New bounds for online portfolio selection without smoothness assumptions.

Paper analyzes robustness of non-Lipschitz networks, proving powerful adversarial attacks but offering solutions.

The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothn…

New method models fat-tailed distributions with anisotropic tail-adaptive flows.

Study proves existence, uniqueness, and positivity of solutions to a complex volatility model.

The one-dimensional SDE with non Lipschitz diffusion coefficient $dX_{t} = b(X_{t})dt + σX_{t}^γ dB_{t}, \ X_{0}=x, \ γ<1$ is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of this equation, based on a careful i…

We study the use of the multilevel Monte Carlo technique in the context of the calculation of Greeks. The pathwise sensitivity analysis differentiates the path evolution and reduces the payoff's smoothness. This leads to new challenges: the inapplicability of pathwise sensitivities to non-Lipschitz payoffs often makes …

We consider the finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. T…

We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipsc…

New research shows many batch selection methods for training work just as well as full batch training.

We study the robustness of active learning (AL) algorithms against prior misspecification: whether an algorithm achieves similar performance using a perturbed prior as compared to using the true prior. In both the average and worst cases of the maximum coverage setting, we prove that all $α$-approximate algorithms are …

New shuffling methods improve convergence without Lipschitz smoothness.

NES optimizes discrete structured VAEs effectively without gradient propagation.

This work closes the theory-practice gap for distributed optimization methods by introducing a new regularity condition.

The paper relaxes assumptions for analyzing stochastic optimization algorithms.

Deep NURBS improves PINNs for solving PDEs on arbitrary geometries.

Optimization in the presence of sharp (non-Lipschitz), unpredictable (w.r.t. time and amount) changes is a challenging and largely unexplored problem of great significance. We consider the class of piecewise Lipschitz functions, which is the most general online setting considered in the literature for the problem, and …

Meta-learning improves performance across similar tasks in adversarial bandit settings.

In this paper we consider a general matrix factorization model which covers a large class of existing models with many applications in areas such as machine learning and imaging sciences. To solve this possibly nonconvex, nonsmooth and non-Lipschitz problem, we develop a non-monotone alternating updating method based o…

In this short report, we investigate the ability of the DCCA coefficient to measure correlation level between non-stationary series. Based on a wide Monte Carlo simulation study, we show that the DCCA coefficient can estimate the correlation coefficient accurately regardless the strength of non-stationarity (measured b…

Formula connects linking coefficients to Kontsevich integral coefficients.

New algorithm reduces prediction errors across various loss functions.

Computed distortion coefficients for the α-Grushin plane.

The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions g…

Machine learning predicts Kronecker coefficients with high accuracy.

Abstract: Determines thermoelastic coefficients from boundary data.

New filling functions for groups with coefficients show different asymptotic behavior.

Abstract: Determines Lamé coefficients from boundary measurements.

Unified analysis of online optimization with self-concordant barriers, improving regret bounds.

Defines and proves properties of weighted renormalized volume coefficients.

Improved 3D LiDAR data classification using product coefficients.

High-dimensional, large-sample astrophysical databases of galaxy clusters, such as the Chandra Deep Field South COMBO-17 database, provide measurements on many variables for thousands of galaxies and a range of redshifts. Current understanding of galaxy formation and evolution rests sensitively on relationships between…

We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose ele…

Paper calculates third coefficient in Kaehler-Einstein metric expansion.