Max Pfeffer

We study energy minimization problems in quantum chemistry through the lens of computational algebraic geometry. We focus on minimizing the Rayleigh quotient of a Hamiltonian over a tensor train variety. The complex critical points of this problem approximate eigenstates of the quantum system, with the global minimum approximating the ground state. We call the number of critical points the Rayleigh-Ritz degree. After introducing tensor train varieties, we identify instances when they are Segre products of projective spaces. We also report what we know about the defining ideals of tensor trains. We present a birational parametrization of them from products of Grassmannians. Along the way, we study the Rayleigh-Ritz degree, and we introduce the Rayleigh-Ritz discriminant, which describes Hamiltonians that lead to deficient number of critical points. We use homotopy continuation to compute all critical points of this optimization problem over various tensor train and determinantal varieties. Finally, we use these results to benchmark state-of-the-art methods, the Alternating Linear Scheme and Density Matrix Renormalization Group.

Tensor factorizations have been widely used for the task of uncovering patterns in various domains. Often, the input is time-evolving, shifting the goal to tracking the evolution of the underlying patterns instead. To adapt to this more complex setting, existing methods incorporate temporal regularization but they either have overly constrained structural requirements or lack uniqueness which is crucial for interpretation. In this paper, in order to capture the underlying evolving patterns, we introduce t(emporal)PARAFAC2, which utilizes temporal smoothness regularization on the evolving factors. Previously, Alternating Optimization and Alternating Direction Method of Multipliers-based algorithmic approach has been introduced to fit the PARAFAC2 model to fully observed data. In this paper, we extend this algorithmic framework to the case of partially observed data and use it to fit the tPARAFAC2 model to complete and incomplete datasets with the goal of revealing evolving patterns. Our numerical experiments on simulated datasets demonstrate that tPARAFAC2 can extract the underlying evolving patterns more accurately compared to the state-of-the-art in the presence of high amounts of noise and missing data. Using two real datasets, we also demonstrate the effectiveness of the algorithmic approach in terms of handling missing data and tPARAFAC2 model in terms of revealing evolving patterns. The paper provides an extensive comparison of different approaches for handling missing data within the proposed framework, and discusses both the advantages and limitations of tPARAFAC2 model.

To determine the optimal set of hyperparameters of a Gaussian process based on a large number of training data, both a linear system and a trace estimation problem must be solved. In this paper, we focus on establishing numerical methods for the case where the covariance matrix is given as the sum of possibly multiple Kronecker products, i.e., can be identified as a tensor. As such, we will represent this operator and the training data in the tensor train format. Based on the AME n method and Krylov subspace methods, we derive an efficient scheme for computing the matrix functions required for evaluating the gradient and the objective function in hyperparameter optimization.

We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses proximal maps defined on the manifold and therefore does not require or work in the embedding. We investigate its convergence properties and illustrate its numerical performance, particularly for nonconvex or nonembedded problems that are hence out of reach for other methods.

Uveal melanoma (UM) is the most common primary intraocular malignancy with a limited five-year survival for metastatic patients. Limited therapeutic treatments are currently available for metastatic disease, even if the genomics of this tumor has been deeply studied using next-generation sequencing (NGS) and functional experiments. The profound knowledge of the molecular features that characterize this tumor has not led to the development of efficacious therapies, and the survival of metastatic patients has not changed for decades. Several bioinformatics methods have been applied to mine NGS tumor data in order to unveil tumor biology and detect possible molecular targets for new therapies. Each application can be single domain based while others are more focused on data integration from multiple genomics domains (as gene expression and methylation data). Examples of single domain approaches include differentially expressed gene (DEG) analysis on gene expression data with statistical methods such as SAM (significance analysis of microarray) or gene prioritization with complex algorithms such as deep learning. Data fusion or integration methods merge multiple domains of information to define new clusters of patients or to detect relevant genes, according to multiple NGS data. In this work, we compare different strategies to detect relevant genes for metastatic disease prediction in the TCGA uveal melanoma (UVM) dataset. Detected targets are validated with multi-gene score analysis on a larger UM microarray dataset.

To determine the optimal set of hyperparameters of a Gaussian process based on a large number of training data, both a linear system and a trace estimation problem must be solved. In this paper, we focus on establishing numerical methods for the case where the covariance matrix is given as the sum of possibly multiple Kronecker products, i.e., can be identified as a tensor. As such, we will represent this operator and the training data in the tensor train format. Based on the AMEn method and Krylov subspace methods, we derive an efficient scheme for computing the matrix functions required for evaluating the gradient and the objective function in hyperparameter optimization.

There is a growing number of multi-domain genomic datasets for human tumors. Multi-domain data are usually interpreted after separately analyzing single-domain data and integrating the results post hoc. Data fusion techniques allow for the real integration of multi-domain data to ideally improve the tumor classification results for the prognosis and prediction of response to therapy. We have previously described the joint singular value decomposition (jSVD) technique as a means of data fusion. Here, we report on the development of these methods in open source code based on R and Python and on the application of these data fusion methods. The Cancer Genome Atlas (TCGA) Skin Cutaneous Melanoma (SKCM) dataset was used as a benchmark to evaluate the potential of the data fusion approaches to improve molecular classification of cancers in a clinically relevant manner. Our data show that the data fusion approach does not generate classification results superior to those obtained using single-domain data. Data from different domains are not entirely independent from each other, and molecular classes are characterized by features that penetrate different domains. Data fusion techniques might be better suited for response prediction, where they could contribute to the identification of predictive features in a domain-independent manner to be used as biomarkers.

In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular, a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show near-optimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the special case of unknown sparsity.

In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show near-optimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the special case of unknown sparsity.