Ph. D. Project
Title:
Uniqueness and Recoverability of Coupled Statistical Factorization Models with Mixed Measurements
Dates:
2024/10/24 - 2027/10/23
Student:
Supervisor(s):
Description:
Coupled statistical factorized models that handle mixed continuous and discrete measurements are vital in numerous applications, including medical
imaging. These models adeptly capture dependencies between varied data types, such as neuroimaging results and patient health condition, which typically include both continuous and discrete variables. To perform statistical estimation, a primary challenge lies in computing joint and conditional probability measures for mixed models, which becomes mathematically intractable for complex scenarios. However, the inherent multivariate nature of these problems aligns well with tensor decompositions, providing a robust framework for examining recoverability and uniqueness properties of factorized models under low-rank assumptions like the canonical polyadic decomposition. However, existing recovery and identifiability results often assume continuous distributions or deterministic data, overlooking the practical scenarios involving discrete variables. This project aims to develop a unified framework to analyze factorized statistical models incorporating both continuous and discrete variables over multiple datasets, leveraging tensor decomposition techniques such as the coupled canonical polyadic decomposition and PARAFAC2, and exploring algebraic and graph-theoretical structures in the model to provide strong results on the uniqueness and stability of the decomposition. The project will further address the challenges of integrating heterogeneous, multimodal datasets, which often contain distinct as well as shared information across different modalities. Coupled tensor decompositions offer a promising approach for this task, yet their theoretical properties are not well understood in statistical scenarios. The PhD candidate will focus on developing new statistical/tensor decomposition methods and establishing their theoretical guarantees. The performance of the developed methods will be validated on publicly available neuroimaging datasets.
imaging. These models adeptly capture dependencies between varied data types, such as neuroimaging results and patient health condition, which typically include both continuous and discrete variables. To perform statistical estimation, a primary challenge lies in computing joint and conditional probability measures for mixed models, which becomes mathematically intractable for complex scenarios. However, the inherent multivariate nature of these problems aligns well with tensor decompositions, providing a robust framework for examining recoverability and uniqueness properties of factorized models under low-rank assumptions like the canonical polyadic decomposition. However, existing recovery and identifiability results often assume continuous distributions or deterministic data, overlooking the practical scenarios involving discrete variables. This project aims to develop a unified framework to analyze factorized statistical models incorporating both continuous and discrete variables over multiple datasets, leveraging tensor decomposition techniques such as the coupled canonical polyadic decomposition and PARAFAC2, and exploring algebraic and graph-theoretical structures in the model to provide strong results on the uniqueness and stability of the decomposition. The project will further address the challenges of integrating heterogeneous, multimodal datasets, which often contain distinct as well as shared information across different modalities. Coupled tensor decompositions offer a promising approach for this task, yet their theoretical properties are not well understood in statistical scenarios. The PhD candidate will focus on developing new statistical/tensor decomposition methods and establishing their theoretical guarantees. The performance of the developed methods will be validated on publicly available neuroimaging datasets.
Keywords:
Uniqueness, mixed statistical models, tensor decompositions, multidimensional data
Department(s):
| Biology, Signals and Systems in Cancer and Neuroscience |