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Random Hessians and Jacobians: theory and applications

随机 Hessians 和 Jacobian：理论与应用

基本信息

批准号：
EP/V002473/1
负责人：
Yan Fyodorov
金额：
$ 105.08万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV002473%2F1
关键词：
Random Hessians Jacobians theory applications

项目摘要

Properties of complicated 'landscapes', i.e. randomfunctions defined on very high dimensional spaces, have recently attracted considerable attention, e.g. in theory of Deep Machine Learning and Optimization. In particular, one may be interested in number of 'valleys' (i.e. local minima) at a given 'height', 'ridges' or barriers separating them, and more generally 'critical points' (saddles and maxima). An important role in characterising geometry of the landscapes, especially close to the critical points, is played by the matrix of second derivatives known as the Hessian. It determines e.g. the gradient descent dynamics within these landscapes, which has many practical applications for search algorithms. Depending on the context, the landscape can correspond to the energy of a physical system, to the loss function of a machine-learning algorithm, to the cost function of an optimization problem, or tothe fitness function of a biological system. In the analysis of critical points the (modulus of) the characteristic polynomial of the Hessian appears naturally. Similarly, to characterize equilibria in complicated dynamical systems (e.g. communities of many interacting species) requires investigating properties of more general, asymmetric, Jacobian matrices, for which Hessians are only a special case. Jacobians are deeply related to questions of stability of systems under small perturbations, and as such are very fundamental. Note that in contrast to Hessians whose spectra are real and eigenvectors form an orthogonal set, the Jacobians have in general complex eigenvalues and bi-orthogonal set of left and right eigenvectors. The studies of the associated 'eigenvector non-orthogonality' in random setting turn out to be relevant both for complex systems stability as well as to chaotic wave scattering and random lasing. The present research proposal is mainly centred around analysis of various properties of random matrices and operators, mostly arising via Hessians of random landscapes, or random Jacobians of various origin.

复杂景观的性质，即定义在高维空间上的随机函数，最近引起了相当大的关注，例如在深度机器学习和优化理论中。具体地说，人们可能感兴趣的是在给定的“高度”、“山脊”或分隔它们的障碍物以及更一般的“临界点”(鞍点和最大点)处的一些“谷”(即局部极小值)。二阶导数矩阵在刻画景观的几何特征，特别是在临界点附近扮演着重要的角色，被称为黑森矩阵。例如，它确定了这些景观内的梯度下降动态，这对于搜索算法具有许多实际应用。根据背景，景观可以对应于物理系统的能量、机器学习算法的损失函数、优化问题的成本函数或生物系统的适应度函数。在临界点分析中，黑森的特征多项式(模)自然出现。类似地，要刻画复杂动力系统(例如，许多相互作用物种的群落)中的平衡，需要研究更一般的、不对称的雅可比矩阵的性质，对于这些性质，Hessians只是一个特例。雅可比与系统在小扰动下的稳定性问题密切相关，因此是非常基本的。请注意，与谱为实数且特征向量形成正交集的黑森人不同，雅可比人通常具有复本征值和左右特征向量的双正交集。研究随机环境中伴随的“特征向量非正交性”不仅与复杂系统的稳定性有关，而且与混沌波散射和随机激光有关。目前的研究建议主要集中在分析随机矩阵和算子的各种性质，这些性质主要是通过随机景观的黑森或各种来源的随机雅可比产生的。