Title: Some Modified Matrix Eigenvalue Problems

Abstract: 

This is the title of a well-known numerical linear algebra

survey article by Gene Golub published in 1973. The article

Title: Generating Representative Samples: Neural Networks and More

Abstract: Approximating a probability distribution using a discrete set of points is a fundamental task in modern scientific computation, with applications in uncertainty quantification among other things. We discuss recent advances in this area, including the use of Stein discrepancies and various optimization techniques. In particular, we introduce Stein-Message-Passing Monte Carlo (Stein-MPMC), an extension of the original Message-Passing Monte Carlo model and the first machine-learning algorithm for generating low-discrepancy (space-filling) point sets. Additionally, we present a generalized Subset Selection algorithm, a simpler yet highly effective optimization method.

Title: Log-Sum Regularized Kaczmarz Algorithms for High-Order Tensor Recovery

Abstract: Sparse and low rank tensor recovery has emerged as a significant area of research with applications in many fields such as computer vision. However, minimizing the $\ell_0$-norm of a vector or the rank of a matrix is NP-hard. Instead, their convex relaxed versions are typically adopted in practice due to the computational efficiency, e.g., log-sum penalty. In this presentation, we propose novel log-sum regularized Kaczmarz algorithms for recovering high-order tensors with either sparse or low-rank structures. We present block variants along with convergence analysis of the proposed algorithms. Numerical experiments on synthetic and real-world data sets demonstrate the effectiveness of the proposed methods.

Title: Understanding neutrophil dynamics during COVID-19 infection

Abstract: 

Infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) results in varied clinical outcomes, with virus-induced chronic inflammation and tissue injury being associated with enhanced disease pathogenesis. To determine the role of tissue damage on immune populations recruitment and function a mathematical model of innate immunity following SARS-CoV-2 infection has been proposed. The model was fitted to published longitudinal immune marker data from patients with mild and severe COVID-19 disease and key parameters were estimated for each clinical outcome. Analytical, bifurcation and numerical investigations were conducted to determine the effect of parameters and initial conditions on long-term dynamics. The results were used to suggest changes needed to achieve immune resolution.

Title: Forecasting patient-specific treatment response to neoadjuvant chemotherapy in triple-negative breast cancer via MRI-based digital twins

Abstract: Patients with locally advanced, triple-negative breast cancer (TNBC) typically receive neoadjuvant chemotherapy (NAT) to downstage the tumor and improve the outcome of subsequent breast conservation surgery. In this study, we integrated quantitative magnetic resonance imaging (MRI) data with biology-based mathematical modeling to address the currently unmet need for accurate prediction of TNBC response to NAT on an individual patient basis. Specifically, dynamic contrast-enhanced MRI and diffusion-weighted MRI was acquired in 56 patients before, after two, and after four cycles of Adriamycin/Cytoxan (A/C), and again after Taxol as part of the ARTEMIS (NCT02276443) trial. A biology-based mathematical model was established based on the reaction-diffusion equation to characterize the mobility of tumor cells, tumor proliferation, and treatment-induced cell death. Pre- and mid-treatment images were used for model calibration on a patient-specific basis. Two evaluation Frameworks were built: 1. using images acquired before and after two cycles of A/C for calibration and predicting tumor status after A/C, and 2. using images acquired before, after two cycles, and after four cycles of A/C for calibration and predicting response after NAT. For Framework 1, the Pearson correlation coefficients between the predicted and measured patient-specific, post-A/C changes in tumor cellularity and volume were 0.95 and 0.94, respectively. For Framework 2, the biologically-based model achieved an area under the receiver operator characteristic curve of 0.89 (sensitivity/specificity = 0.72/0.95) for differentiating pathological complete response (pCR) from non-pCR, which is statistically superior (P < 0.05) to the value of 0.78 (sensitivity/specificity = 0.72/0.79) achieved by the tumor volume measured after four cycles of A/C. Overall, our biology-based mathematical model successfully captured the patient-specific, spatiotemporal dynamics of TNBC response to NAT, providing highly accurate predictions of NAT response.  

Title: Is a Classification Procedure Good Enough?—A Goodness-of-Fit Assessment Tool for Classification Learning

Abstract: In recent years, many nontraditional classification methods, such as random forest, boosting, and neural network, have been widely used in applications. Their performance is typically measured in terms of classification accuracy. While the classification error rate and the like are important, they do not address a fundamental question: Is the classification method underfitted? To our best knowledge, there is no existing method that can assess the goodness of fit of a general classification procedure. Indeed, the lack of a parametric assumption makes it challenging to construct proper tests. To overcome this difficulty, we propose a methodology called BAGofT that splits the data into a training set and a validation set. First, the classification procedure to assess is applied to the training set, which is also used to adaptively find a data grouping that reveals the most severe regions of underfitting. Then, based on this grouping, we calculate a test statistic by comparing the estimated success probabilities and the actual observed responses from the validation set. The data splitting guarantees that the size of the test is controlled under the null hypothesis, and the power of the test goes to one as the sample size increases under the alternative hypothesis. For testing parametric classification models, the BAGofT has a broader scope than the existing methods since it is not restricted to specific parametric models (e.g., logistic regression). Extensive simulation studies show the utility of the BAGofT when assessing general classification procedures and its strengths over some existing methods when testing parametric classification models.

Computational modeling using a novel continuum approach coupled with Pathway-informed neural networks to optimize Dynein-mediated centrosome positioning in Polarized cells

Microtubules (MTs) are cytoskeletal polymers that interact with motor proteins such as dynein to position the centrosomes and nucleus within a cell. Centrosome positioning specifies the cell’s division plane by determining the location and orientation of the mitotic spindle. In polarized cells, centrosome alignment along the polarity axis causes the cell to divide asymmetrically, producing unequal daughter cells. Proper centrosome positioning is critical during development where it is required for important processes such as cell fate specification. Improper centrosome positioning is implicated in disease processes: cancer cells often exhibit abnormal centrosome positioning prior to division. While many studies have focused on centrosome movement during mitosis, centrosomes are often positioned prior to mitosis. This movement prior to mitosis when the centrosomes are associated with the intact pronuclear envelope is not well understood. Many aspects of dynein-mediated centrosome movement are highly nonlinear and rely on biochemical, mechanical and geometric features in the cell that are difficult to investigate experimentally. Mathematical modeling can easily deal with this complexity, bridging the varying time and space scales, and provide a fundamental understanding of the mechanisms of positioning centrosomes. This model provides the key features required to integrate modeling and experiments on early embryos of the C. elegans to elucidate the interplay between biochemical, mechanical and geometric signals that act to position centrosomes in polarized cells through the following aims. The same non-linear framework for confined geometries is extended to create a comprehensive data driven digital twin of an individual’s mental health profile and analyze spatiotemporal behavior. Although dynamic study and modeling of depression-related behavior exist in literature, we employ a novel digital twin model that combines Sensitive, Exposed, Induced and Excluded models with Disease-informed neural networks to identify progression and intensity of depression related behavior. 

Title: The Model Behind ChatGPT



Abstract: This will be a general talk to introduce a deep learning model called Generative Pre-training Transformer (GPT). First, I will discuss in concept the machine learning approach for the task of question-answering. Then I will describe language modeling and some models such as recurrent neural network (RNN) models for that task. Finally, I will present the Transformer as well as the GPT models that have led to ChatGPT.

Title: The nonlinearity of regulation in biological networks

Abstract: The extent to which the components of a biological system are (non)linearly regulated determines how amenable they are to therapy and control. To better understand this property termed `regulatory nonlinearity', we analyzed a suite of 137 published Boolean network models, containing a variety of complex nonlinear interactions, using a probabilistic generalization of Boolean logic that George Boole himself had proposed. Leveraging the continuous-nature of this formulation, we used Taylor decomposition to approximate the models with various levels of nonlinearity. A comparison of the resulting series of approximations of the biological models with appropriate random ensembles revealed that biological regulation tends to be less nonlinear than expected. A further categorical analysis of the biological models revealed that the nonlinearity of cancer and disease networks could not only be sometimes higher than expected but are also relatively more variable. We show that this variation is caused by differences in the apportioning of information among the various orders of nonlinearity. Taken together, our results suggest, but do not imply, that biological regulation may have evolved to be more linear on average, and certain systems such as cancer may have, on the other hand, evolved to be more nonlinear.