arXiv research

Researchers derive a new equation for valuing American options.

Derives stochastic and dissipative dynamics preserving Gibbs measure.

Most known examples of doubly periodic minimal surfaces in $\mathbb{R}^3$ with parallel ends limit as a foliation of $\mathbb{R}^3$ by horizontal noded planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a correspo…

Study geodesic equation on mixed-volume forms on balanced manifolds, proving existence of solutions.

We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied bet…

The paper explores various option pricing models by considering the volume of transactions and its impact on volatility.

The paper models CBF dynamics using queueing theory and insurance risk models.

We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit toward…

SGD in DLNs reveals feature learning dynamics.

By using numerical simulation, we confirm that Takayasu--Sato--Takayasu (TST) model which leads Pareto's law satisfies the detailed balance under Gibrat's law. In the simulation, we take an exponential tent-shaped function as the growth rate distribution. We also numerically confirm the reflection law equivalent to the…

We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of $u$. In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanced split sets, identifying the cases where the only balanced split loc…

In this paper, we propose a novel technique to implement stochastic gradient methods, which are beneficial for learning from large datasets, through accelerated stochastic dynamics. A stochastic gradient method is based on mini-batch learning for reducing the computational cost when the amount of data is large. The sto…

New metrics solve complex equations on special 3D shapes.

Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a balanced metric which is hermitian Ricci--flat. This can be viewed as a differential f…

A new sampling method called Restart improves both speed and quality of generative processes.

Improved GFlowNets learn more efficiently with trajectory balance.

Noise balance theory explains SGD's behavior in neural networks.

By and large the behavior of stochastic gradient is regarded as a challenging problem, and it is often presented in the framework of statistical machine learning. This paper offers a novel view on the analysis of on-line models of learning that arises when dealing with a generalized version of stochastic gradient that …

Model uses Navier-Stokes equations to assess liquidity and systemic risk.

BalLOT uses optimal transport for balanced k-means clustering.

We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced…

Due to the limited predictability of wind power and other stochastic generation, trading this energy in competitive electricity markets is challenging. This paper derives revenue-maximising and risk-constrained strategies for stochastic generators participating in electricity markets with a single-price balancing mecha…

Model predicts insolvency risks in banks due to liquidity and credit risks.

New minimal surfaces derived from helicoids.

We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.

New method finds balanced clusters in graphs using auxiliary information.

A new algorithm balances fairness in clustering to avoid discrimination.

Proposes unbiased estimators for training mixture of experts models.

We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle $F$ over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on $F$ coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson.…

We derive a class of macroscopic differential equations that describe collective adaptation, starting from a discrete-time stochastic microscopic model. The behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior. We show tha…

A scalable framework uses Langevin sampling to approximate neural network models of evolving processes.

A twisted Higgs bundle on a Kähler manifold $X$ is a pair $(E,φ)$ consisting of a holomorphic vector bundle $E$ and a holomorphic bundle morphism $φ\colon M\otimes E \to E$ for some holomorphic vector bundle $M$. Such objects were first considered by Hitchin when $X$ is a curve and $M$ is the tangent bundle of $X$, and…

In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian pa…

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

Study shows different price correlations in European electricity markets.

A new method called MCLMC avoids dissipation in sampling from canonical distributions.

A number of optimization approaches have been proposed for optimizing nonconvex objectives (e.g. deep learning models), such as batch gradient descent, stochastic gradient descent and stochastic variance reduced gradient descent. Theory shows these optimization methods can converge by using an unbiased gradient estimat…

Noise in linear networks minimizes sharpness and leads to shrinkage-thresholding.

Abstract: Necessary and sufficient conditions for gradient flows of relative entropy in Lindblad equations.

We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving "closed" sections of a vector bundle, generalizing a method used recently by Bryant and Xu for studying the Laplacian flow in G_2-geometry. We apply this theorem in balanced geometry introducin…

Paper introduces multitask neural networks for efficient stochastic control problems.

Using Traizet's regeneration method, we prove the existence of many new 3-dimensional families of embedded, doubly periodic minimal surfaces. All these families have a foliation of 3-dimensional Euclidean space by vertical planes as a limit. In the quotient, these limits can be realized conformally as noded Riemann sur…

From the work of Dervan-Keller, there exists a quantization of the critical equation for the J-flow. This leads to the notion of J-balanced metrics. We prove that the existence of J-balanced metrics has a purely algebro-geometric characterization in terms of Chow stability, complementing the result of Dervan-Keller. We…

In this work we apply the Poincare-Cartan formalism of the Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles of the configurational bundle and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constituti…

Neural networks can approximate complex stochastic equations well.

New method efficiently computes gradients for stochastic differential equations.

Pathwise uniqueness shown for specific stochastic equations.

New method reveals insights about stochastic optimization methods using modified equations.