Useful Compact Representations for Data-Fitting: Methods and Applications

แนวคิดหลัก

Effective limited-memory methods using compact representations for data-fitting tasks.

บทคัดย่อ

This article discusses the development of new compact representations parameterized by vectors for large-scale data fitting problems. It explores their effectiveness in eigenvalue computations, tensor factorizations, and nonlinear regressions. The limited-memory approach reduces memory usage and enables efficient operations on large matrices.

Introduction to large-scale data fitting problems.
Unconstrained optimization methods like Newton's method and gradient-based methods.
Compact representation formulas for Hessian matrix approximations.
Implications of compact representations on eigendecomposition and updating techniques.
Numerical experiments demonstrating the scalability and efficacy of compact representations in optimization algorithms.
Comparison of eigenfactorization using thin QR factorization versus eig function in MATLAB.

สถิติ

Limited-memory parameter: l = 5
Dimensions tested: d ∈ {23, 24, ..., 213}
Rosenbrock function components: f(w) = 100(w2i-1 - w2i)^2 + (w2i-1 - 1)^2

คำพูด

ข้อมูลเชิงลึกที่สำคัญจาก

Useful Compact Representations for Data-Fitting

by Johannes J. ... ที่ arxiv.org 03-20-2024

https://arxiv.org/pdf/2403.12206.pdf

สอบถามเพิ่มเติม

How do limited-memory methods impact computational efficiency in large-scale optimization

Limited-memory methods have a significant impact on computational efficiency in large-scale optimization tasks. By storing only a small subset of vectors and matrices, these methods reduce the memory requirements for computations. This is particularly beneficial when dealing with high-dimensional problems where storing all necessary information can be prohibitive. The limited-memory approach allows for linear complexity in terms of problem dimension, making it feasible to handle large datasets efficiently. Additionally, updating techniques such as column updates and product updates enable seamless integration of new data without recomputing entire matrices or products from scratch.

What are the implications of compact representations on traditional optimization algorithms

The implications of compact representations on traditional optimization algorithms are profound. These representations provide an efficient way to store and update essential matrices while maintaining the accuracy needed for optimization tasks. By using low-rank approximations and clever updating strategies, compact representations streamline computations in algorithms like quasi-Newton methods, trust-region strategies, and stochastic gradient descent approaches. The ability to compute eigenfactorizations implicitly through thin QR factorizations further enhances the applicability of compact representations in various optimization scenarios.

How can the concept of compact representations be applied to other mathematical domains beyond data-fitting

The concept of compact representations extends beyond data-fitting applications into other mathematical domains where efficient storage and computation are crucial. In fields like machine learning, image processing, signal processing, and scientific computing, compact representations can revolutionize algorithm implementations by reducing memory overheads without sacrificing accuracy or performance. For example, in tensor decompositions or nonlinear regressions where large-scale optimizations are common, leveraging compact representations can lead to faster convergence rates and improved scalability across diverse mathematical problems requiring iterative solutions.