Abstract: Recent advances in biotechnology and genome sequencing, resulting in a surge of data, are bringing in new opportunities in the mathematical modeling of biological systems. However, the amount of data that can be practically collected in everyday patients in the clinic is limited due to various reasons including the cost and the patient’s burden. Especially the amount of data that can be collected in the time domain is limited. This motivates us to transfer the mathematical and computational models to meet the challenges in clinical setting, to guide patient therapy via prediction. In this talk, I will discuss modeling approaches on the two ends of the spectrum of data. In the first part, I will discuss a Bayesian information-theoretic approach to determining effective scanning protocols for cancer patients. We propose a modified mutual information function with a temporal penalty term to account for the loss of temporal data. The effectiveness of our framework is demonstrated in determining patient scanning scheduling for prostate cancer patients. In the second part, I will discuss modeling work using single-cell gene sequencing data. Due to the high cost of obtaining gene sequencing data, temporal data is also lacking. We show that our cell state dynamics model can be used to incorporate genetic alteration at low cost, where we provide an example of modeling a hematopoiesis system and simulating abnormal differentiation that corresponds to acute myeloid leukemia.
Abstract: Kernels are efficient in representing nonlocal dependence and are widely used to design operators between function spaces or high-dimensional data. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal dependence, the inverse problem is often severely ill-posed with a data-dependent operator that is nearly singular. Therefore, regularization is necessary. However, little information is available to select a proper regularization norm. We tackle this issue by introducing a data-adaptive RKHS for regularization, penalizing small singular values. It leads to convergent estimators that are robust to noise, outperforming the widely used L2- or l2-regularizers. We will discuss both direct and iterative methods.