Statistical Estimation under Nonlinear Algebraic Constraints

机器学习和人工智能就在我们身边：新的算法可以驾驶汽车、检测黑色素瘤、进行自动化手术，还可以用于科学发现。现在，比以往任何时候都迫切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悠久传统可能不再适用。例如，潜变量模型对应于张量中的低秩结构的各种概念；更一般地说，潜在变量图形模型在离散情况下对应更复杂的张量分解，在连续情况下对应其他高度非线性结构。此外，非参数模型，如强正相关随机变量，是通过非线性多项式不等式定义的。这类模型的推理通常具有较高的计算复杂度和样本量复杂度。设计推理和获得收敛保证的方法需要对潜在的非线性结构有透彻的了解。我的研究项目是统计推断，模型选择，以及非线性代数约束下模型的检验。我的每一个长期项目都涉及到基于不同领域的问题：统计、优化、非线性代数和组合学。这个研究计划包括三个主要项目。(1)总正性是一个强大的概念，在统计学（它意味着随机变量的强正依赖形式）和优化（它促进双证书构建-例如在超分辨率成像等问题中）中都很重要。总正性是通过多项式不等式定义的，并在不同的领域提出了大量的开放问题，从统计推断到离散几何。(2)结构化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，然而，它们带来了许多计算和理论挑战。我打算设计快速结构化分解的算法，研究具有一定结构的张量空间的几何性质，探索新的张量分解类型。(3)存在隐变量的因果推理捕获了观测量的真实相互作用，但它提出了各种开放问题。在这里，我建议研究模型描述和等价，以及模型选择问题。在离散和高斯情况下，模型都是半代数集。解决上述问题需要用到代数和组合技术。为了执行这个跨学科的研究项目，我计划与具有不同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打下坚实的基础。

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.