arXiv research

Study on heat flow across two half-lines with special boundary conditions.

New method finds precise late-time behavior of wave equations.

Following the recent work on capacity allocation, we formulate the conjecture that the shattering problem in deep neural networks can only be avoided if the capacity propagation through layers has a non-degenerate continuous limit when the number of layers tends to infinity. This allows us to study a number of commonly…

Advanced optimization algorithms such as Newton method and AdaGrad benefit from second order derivative or second order statistics to achieve better descent directions and faster convergence rates. At their heart, such algorithms need to compute the inverse or inverse square root of a matrix whose size is quadratic of …

Study regularity of Schrödinger eigenfunctions with Coulomb-type potentials.

We describe a novel algorithm for noisy global optimisation and continuum-armed bandits, with good convergence properties over any continuous reward function having finitely many polynomial maxima. Over such functions, our algorithm achieves square-root regret in bandits, and inverse-square-root error in optimisation, …

Study Gaussian-process limits of neural networks using tensor programs.

Alpha-based performance evaluation may fail to capture correlated residuals due to model errors. This paper proposes using the Generalized Information Ratio (GIR) to measure performance under misspecified benchmarks. Motivated by the theoretical link between abnormal returns and residual covariance matrix, GIR is deriv…

Adaptive regularization methods pre-multiply a descent direction by a preconditioning matrix. Due to the large number of parameters of machine learning problems, full-matrix preconditioning methods are prohibitively expensive. We show how to modify full-matrix adaptive regularization in order to make it practical and e…

Stability of catenoid in 4D Minkowski space proven without symmetry assumptions.

This paper investigates the impact of pre-existing offline data on online learning, in the context of dynamic pricing. We study a single-product dynamic pricing problem over a selling horizon of $T$ periods. The demand in each period is determined by the price of the product according to a linear demand model with unkn…

Study perturbs Dirac operator on 4D manifolds, proving Kastler-Kalau-Walze theorems.

Polynomial networks converge to Gaussian processes at a rate of O(n^(-1/2)).

Paper solves a key problem in learning from high-dimensional covariance matrices.

Stock prices are known to exhibit non-Gaussian dynamics, and there is much interest in understanding the origin of this behavior. Here, we present a model that explains the shape and scaling of the distribution of intraday stock price fluctuations (called intraday returns) and verify the model using a large database fo…

The covariance matrix is formulated in the framework of a linear multivariate ARCH process with long memory, where the natural cross product structure of the covariance is generalized by adding two linear terms with their respective parameter. The residuals of the linear ARCH process are computed using historical data …

The dueling bandit problem is a variation of the classical multi-armed bandit in which the allowable actions are noisy comparisons between pairs of arms. This paper focuses on a new approach for finding the "best" arm according to the Borda criterion using noisy comparisons. We prove that in the absence of structural a…

In several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-moment estimators requires memory equal to the number of parameters. For …

Bitcoin returns exhibit a distinct inverse cubic law scaling behavior.

New method reduces regret in budgeted learning problems.

Learning rate annealing improves robustness in stochastic optimization.

Identifying behavior that is relatively invariant under different conditions is a challenging task in far-from-equilibrium complex systems. As an example of how the existence of a semi-invariant signature can be masked by the heterogeneity in the properties of the components comprising such systems, we consider the exc…

Let $\mathcal T$ be the Teichmüller space of marked genus $g$, $n$ punctured Riemann surfaces with its bordification $\Tbar$ the {\em augmented Teichmüller space} of marked Riemann surfaces with nodes, \cite{Abdegn, Bersdeg}. Provided with the WP metric $\Tbar$ is a complete CAT(0) metric space, \cite{DW2, Wlcomp, Yam2…

WSqD extends learning rate schedules for large model training without fixed horizons.

The paper accelerates ISTA and FISTA algorithms for composite optimization problems.

Study on helix curves and their Möbius energy asymptotics.

CWGD measures gradient diversity weighted by curvature, improving SGD convergence.

This paper optimizes importance sampling for rare-event options pricing under the Heston model.

LOT improves adversarial robustness by training 1-Lipschitz convolution layers.

Optimal hidden-target learning for online inventory optimization on general convex sets.

The paper analyzes variance reduction in stochastic gradient Langevin dynamics.

Study finds price impact follows a 'double' square-root law, suggesting mechanical origin.

We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial $L_α(G)=D-\sum_{r=1}^dα_rD(D^{-1}A)^r$ where $A$ is the adjacency matrix of a weighted, undirected graph, $D$ is the diagonal matrix of weighted degrees, and $α=(α_1...α_d)$ are nonn…

In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if t…

Develops potential theory for WZW equation in Kähler potentials space.

The paper examines stability of harmonic and symphonic maps with forms and potentials.

The paper examines stability of subelliptic harmonic maps with potential.

The paper describes flat Hessian metrics on surfaces and their potentials.

A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential w…

We consider the problem of learning an interpretable potential energy function from a Hamiltonian system's trajectories. We address this problem for classical, separable Hamiltonian systems. Our approach first constructs a neural network model of the potential and then applies an equation discovery technique to extract…

Article provides Bernstein gradient estimates for heat equations with potential terms.

In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential…

We consider the geodesic equation for the generalized Kahler potential with only mixed second derivatives bounded. We show that given such two generalized Kahler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kahler potential has uniformly bounded mixed second d…

We show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in "Formal Knot Theory" is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potent…

The paper characterizes potential functions whose level sets are orbits in mechanical systems.

The paper studies $m$-quasi Einstein manifolds with convex potential and finds constant scalar curvature.

Estimates classical potential from stock price data using quantum mechanics.

We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator with a nonconvex potential in terms of a distance associated with the potential. The results here can be applied to the double well potential.