Numerical Algebraic Geometry Defines Rayleigh-Ritz Degree for Energy Minimization on Tensor Train Varieties

Energy calculations lie at the heart of many scientific disciplines, yet accurately modelling complex systems remains a significant challenge. Viktoriia Borovik, Hannah Friedman, and Serkan Hoşten, along with Max Pfeffer, investigate this problem through a novel application of computational algebraic geometry, focusing on energy minimization within tensor train varieties. Their work introduces new mathematical tools to understand and compute the critical points of these energy landscapes, which closely correspond to the system’s fundamental states, including its lowest energy ground state. By developing a birational parametrization of tensor trains and introducing the Rayleigh-Ritz discriminant, the team provides a method for identifying problematic Hamiltonians and accurately benchmarking established computational techniques like the Alternating Linear Scheme and Density Matrix Renormalization Group, ultimately advancing the precision and efficiency of energy calculations in complex systems.

Low Rank Tensor Decomposition and Geometry

This research investigates energy calculations within complex systems, addressing a significant challenge in many scientific disciplines. Their work introduces new mathematical tools to understand and compute the critical points of these energy landscapes, which closely correspond to the system’s fundamental states, including its lowest energy ground state. Researchers engineered a method to approximate eigenstates of a system by identifying complex critical points of this optimization problem, with the global minimum representing the ground state. A central innovation lies in the introduction of the Rayleigh-Ritz degree, quantifying the number of these critical points, and the associated Rayleigh-Ritz discriminant, which characterizes Hamiltonians leading to a deficient number of critical points. To facilitate this analysis, the team developed a birational parametrization of tensor train varieties using products of Grassmannians, providing a formula for their dimension.

This parametrization enables the study of the constrained Rayleigh quotient optimization problem from an algebraic geometry perspective. Researchers employed homotopy continuation, a numerical algebraic geometry technique, to compute all critical points over various tensor train and determinantal varieties, effectively mapping the solution space of the optimization problem. The study pioneers a connection between optimization and Euclidean distance, framing the problem as a distance minimization with respect to the Bombieri-Weyl inner product, leading to the definition of the BW correspondence, which the team proved has a parametrization for all tensor train varieties. Numerical experiments, conducted using the HomotopyContinuation. These experiments demonstrate the ability of the ALS method to converge to various local minima, and assess the corresponding energy values relative to the optimal solution.,.

Tensor Train Varieties and Rayleigh-Ritz Degree

This work presents a novel application of algebraic geometry to the study of energy minimization problems, specifically focusing on the Rayleigh quotient of a Hamiltonian constrained to a tensor train variety. Researchers have developed a detailed geometric understanding of these tensor train varieties, demonstrating instances where they can be expressed as Segre products of projective spaces, a key structural result. They have also established a birational parametrization of these varieties using products of Grassmannians, allowing for the calculation of their dimension, which is a crucial step in analyzing the optimization landscape. A central concept introduced is the Rayleigh-Ritz (RR) degree, defined as the number of complex critical points of the energy minimization problem for a generic symmetric matrix.

The team developed methods for computing these critical points and presented an upper bound on the RR degree, providing valuable insights into the complexity of the optimization. Furthermore, they defined the RR discriminant, identifying matrices that lead to a deficient number of critical points, which is essential for understanding the limitations of certain optimization algorithms. Through numerical experiments using the HomotopyContinuation. The results provide a foundation for improving the accuracy and reliability of these widely used algorithms in quantum mechanical calculations and beyond. The work establishes a powerful connection between algebraic geometry and numerical optimization, offering new tools for analyzing and improving energy minimization methods.,.

Rayleigh-Ritz Discriminant and Critical Point Computation

This research presents a novel approach to energy minimization problems, framing them within the context of computational algebraic geometry and specifically examining the Rayleigh quotient of a Hamiltonian over tensor train varieties. The team successfully identified instances where these tensor train varieties align with Segre products of projective spaces and developed a birational parametrization using products of Grassmannians, offering new insights into their underlying structure. A key achievement lies in the introduction of the Rayleigh-Ritz discriminant, which characterizes Hamiltonians potentially leading to an insufficient number of critical points in the optimization process. The authors acknowledge that the number of critical points can be limited by the properties of the Hamiltonian, and that further research is needed to fully explore the behaviour of these optimization problems in more complex scenarios. Future work may focus on extending these techniques to broader classes of tensor networks and investigating the computational efficiency of the proposed methods for large-scale problems.