Optimal Reduction

This research area focuses on extending sufficient dimension reduction (SDR) techniques, traditionally applied to vector-valued predictors, to matrix-valued predictors in regression and classification contexts. The key challenge is preserving the inherent structure of the data while reducing dimensionality, enabling effective modeling and interpretation. Methods leverage assumptions such as Kronecker product decompositions of means or covariance structures, and develop computational algorithms that are statistically efficient and scalable.

Reduced-order models (ROMs) are critical for approximating complex high-dimensional systems with lower-dimensional representations. This research investigates how to construct projection operators – mapping full-order systems onto reduced bases – that are optimal with respect to various norms, robust to operator properties (e.g., indefiniteness), and computationally feasible. Specific focus is given to approximations that achieve near-optimality in operator-independent norms, enhancing ROM accuracy for linear systems such as heat conduction and advection–diffusion.

In multicriteria optimization problems, the Pareto set often comprises many alternatives, making final decision-making challenging. This theme investigates techniques to reduce the Pareto set size by incorporating the decision-maker's preference information, modeled as binary preference relations or 'information quanta.' By applying natural axioms on choice procedures, a significant reduction of Pareto optimal alternatives is achieved without losing potentially optimal solutions. The research blends theoretical foundations and practical visualization, enabling more tractable and preference-consistent multicriteria choices.

Lattice reduction algorithms (e.g., LLL, BKZ) play a pivotal role in lattice-based cryptography and cryptanalysis. This research synthesizes extensive empirical studies to predict algorithmic behaviors, bridging the gap between worst-case theoretical bounds and observed practical performance. Key insights target approximation factors achievable, convergence speed, and influencing factors such as lattice structure, ultimately informing cryptosystem parameter selection and security assessments.

Model order reduction is crucial for simplifying high-dimensional linear systems while preserving dynamic characteristics. This theme encompasses methods leveraging hybrid optimization strategies, including metaheuristics like Harmony Search (HS), combined with classical control-theoretic stability criteria (e.g., Routh-Hurwitz). By formulating multi-objective fitness functions involving integral square errors and H-infinity norms, these methods optimize reduced model parameters achieving stability and accuracy better than traditional approaches.

This theory-driven research explores the landscape of polynomial-time reductions among NP-complete sets, distinguishing adaptive (Turing) versus nonadaptive (truth-table) reductions, deterministic versus strong nondeterministic reductions, and many-one reductions restricted to length-increasing functions. The work leverages assumptions like NP not having p-measure zero and concepts from resource-bounded measure theory to prove separations among reduction types, deepen understanding of NP-completeness notions, and resolve open conjectures in complexity theory.

This research area bridges typed lambda-calculus under restrictions like elementary and light affine logic (EAL and LAL) with operational semantics given by optimal reduction techniques, including Lamping's algorithm. It aims to integrate complexity certifications from logic (polynomial or elementary reductions) with concrete local and asynchronous proof-net evaluation, proving soundness, completeness, and complexity bounds via geometry of interaction models, thereby connecting typing discipline to efficient computation.

This theme investigates the implementation of optimal lambda-calculus reduction in parallel and distributed computing environments. Utilizing directed virtual reduction (DVR), a form of graph rewriting, and introducing strategies such as half combustion, research develops parallel algorithms with message aggregation and dynamic load balancing. The goal is to realize fine-grained, local reduction steps that scale efficiently on multiprocessor architectures, improving execution time while preserving correctness and optimality.

Sharing graphs enable local and asynchronous lambda-calculus beta-reduction avoiding duplication inefficiencies. This line of research precisely quantifies the overhead (in time complexity) introduced by sharing operators within the frameworks of elementary linear logic (ELL) and light linear logic (LLL), both guaranteeing bounded computational complexity. Establishing that overheads are at most quadratic with respect to naive reductions strengthens foundational understanding and supports practical adoption of sharing-based implementations.