Statistical Estimation under Nonlinear Algebraic Constraints

机器学习和人工智能无处不在：新的算法驾驶汽车，检测黑色素瘤，执行自动化手术，并用于科学发现。现在，比以往任何时候都更迫切地需要严格的理解和性能保证，以便我们能够依赖这些新技术。机器学习中的许多推理问题都是非线性的，统计学中线性模型的长期传统可能不再适用。例如，潜变量模型对应于张量中低阶结构的各种概念；更一般地，潜变量图形模型对应于离散情况下更复杂的张量分解，以及连续情况下的其他高度非线性结构。此外，非参数模型，如强正相依随机变量，是通过非线性多项式不等式定义的。对于这类模型的推断通常会遇到较高的计算和样本量复杂性。设计推理方法和获得收敛保证需要对潜在的非线性结构有透彻的了解。我的研究项目是对由非线性代数约束给出的模型进行统计推断、模型选择和检验。我的每个长期项目都涉及不同领域的问题：统计学、最优化、非线性代数和组合学。这项研究计划包括三个主要项目。(1)完全正性是一个强大的概念，在统计学(它意味着随机变量的一种强形式的正相关性)和优化(它促进双重证书构造-例如，在超分辨率成像等问题中)都是重要的。全正性是通过多项式不等式来定义的，它提出了从统计推断到离散几何等不同领域的大量未决问题。(2)结构张量分解是许多统计和科学问题(如离散潜变量模型和盲源分离)的核心，但它们提出了许多计算和理论挑战。我打算设计快速结构分解的算法，研究具有一定结构的张量空间的几何性质，并探索新类型的张量分解。(3)隐变量存在下的因果推理捕捉到了真实世界中观察量的相互作用，但它提出了各种开放问题。在这里，我建议致力于模型描述和等价性，以及模型选择问题。在离散和高斯两种情况下，模型都是半代数集。解决上述问题需要代数和组合技术。为了执行这一跨学科研究计划，我计划与具有各种技能和兴趣的学生和年轻研究人员合作。我们的合作将为他们每个人在学术界或工业界的职业生涯奠定坚实的基础。

Machine learning and artificial intelligence are all around us: new algorithms drive cars, detect melanoma, perform automated surgeries, and are used for scientific discovery. Now, more than ever, there is an urgent need for rigorous understanding and performance guarantees so that we can rely on these new technologies. Many inference questions in machine learning are nonlinear, and the long tradition of linear models in statistics may no longer be applicable.  For example, latent variable models correspond to various notions of low-rank structure in tensors; more generally, latent variable graphical models correspond to more complicated tensor decompositions in the discrete case, and other highly nonlinear structure in the continuous case. Additionally, non-parametric models, such as strongly positively dependent random variables, are defined via nonlinear polynomial inequalities. Inference for such models often suffers from high computational and sample size complexity. Designing methods for inference and obtaining convergence guarantees requires a thorough understanding of the underlying nonlinear structure. My research program is on statistical inference, model selection, and testing for models given by nonlinear algebraic constraints. Each of my long-term projects involves problems that are grounded in different fields: statistics, optimization, nonlinear algebra, and combinatorics. This research program consists of three main projects. (1) Total positivity is a powerful notion important in both statistics (where it implies a strong form of positive dependence of random variables) and optimization (where it facilitates dual certificate construction - in problems like super-resolution imaging, for example). Total positivity is defined via polynomial inequalities and raises a multitude of open problems across diverse fields, from statistical inference to discrete geometry. (2) Structured tensor decompositions lie at the core of many statistical and scientific problems (such as discrete latent variable models and blind source separation), however, they pose many computational and theoretical challenges. I intend to design algorithms for fast structured decompositions, to study geometric properties of spaces of tensors with certain structure, and to explore new types of tensor decompositions. (3) Causal inference in the presence of hidden variables captures real-world interaction of observed quantities, however it poses a variety of open problems. Here I propose to work on model description and equivalence, as well as model selection problems. In both the discrete and Gaussian cases, the models are semialgebraic sets. Solving the above problems will call for algebraic and combinatorial techniques. To execute this interdisciplinary research program I plan to work with students and young researchers with various skills and interests. Our work together will give each of them a strong foundation for a career in academia or industry.