Faster Least Squares Approximation (0710.1435v4)

Abstract: Least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution. In a typical setting, one lets $n$ be the number of constraints and $d$ be the number of variables, with $n \gg d$. Then, existing exact methods find a solution vector in $O(nd^2)$ time. We present two randomized algorithms that provide very accurate relative-error approximations to the optimal value and the solution vector of a least squares approximation problem more rapidly than existing exact algorithms. Both of our algorithms preprocess the data with the Randomized Hadamard Transform. One then uniformly randomly samples constraints and solves the smaller problem on those constraints, and the other performs a sparse random projection and solves the smaller problem on those projected coordinates. In both cases, solving the smaller problem provides relative-error approximations, and, if $n$ is sufficiently larger than $d$, the approximate solution can be computed in $O(nd \log d)$ time.

• The paper presents randomized algorithms that leverage the Hadamard Transform to efficiently approximate least squares solutions.
• It achieves a computational speedup from O(nd²) to O(nd ln d) by employing strategic data sampling and projection techniques.
• Empirical results and theoretical analysis confirm high-probability relative-error guarantees for large, overconstrained systems.

Faster Least Squares Approximation

The paper discusses advancements in randomized algorithms for efficiently solving overconstrained least squares approximation problems. Conventional methods require $O(nd^2)$ computational time for exact solutions, where $n$ represents constraints, and $d$ signifies variables. By contrast, the algorithms proposed in this paper utilize techniques from randomized numerical linear algebra to deliver approximations more quickly.

Key Contributions

1. Algorithms Overview: Two randomized algorithms are introduced:
   • Sampling-Based Algorithm: Preprocesses the data using the Randomized Hadamard Transform, then randomly samples constraints and solves the smaller problem.
   • Projection-Based Algorithm: Applies the Randomized Hadamard Transform and then uses a sparse random projection to solve a reduced-sized problem.
2. Computational Efficiency: The approximate solution can be computed in $O(nd \ln d)$ time if $n$ is sufficiently larger than $d$. This represents a significant improvement in computational efficiency over traditional exact methods.
3. Relative-Error Guarantees: Both algorithms provide accurate relative-error approximations to the optimal solution, significantly reducing processing time without sacrificing precision for large systems of equations.
4. Strong Numerical Results: With $n \gg d$, these algorithms consistently deliver acceptable results with high probability. The first algorithm, simpler to implement, requires $n = \Omega(d^2)$, whereas the second algorithm operates under the less restrictive assumption that $n = O(e^d)$.
5. Theoretical Implications: The statistical and geometric interpretations of the mathematical groundwork are discussed, emphasizing conditions where vectors lie in effectively lower-dimensional subspaces.

Theoretical Underpinnings

The theorem central to their findings establishes that for matrices $A \in \mathbb{R}^{n \times d}$ and vectors $b \in \mathbb{R}^{n}$, the algorithms generate an approximate least-squares solution $\tilde{x}_{opt}$ efficiently. They leverage the Randomized Hadamard Transform to spread out the data energy and apply randomized sampling and projection strategies to ensure the effective subset of the equations is solved.

Related Work

The work is built on prior efforts in fast Johnson-Lindenstrauss transforms and sparse random projections. It plays a crucial role in advancing randomized reductions and approximations in linear algebra, a domain applicable in numerous scientific and engineering computations.

Empirical Evaluation

Empirical tests demonstrated the effectiveness of these algorithms even on matrices of substantial size. Variants of the proposed algorithms have proven beneficial in large-scale data tasks and suggest improvements over standard mechanisms like LAPACK.

Future Directions

While the proposed algorithms improve efficiency, the paper suggests avenues for future research, including relaxing constraints on $n$ and enhancing practical implementations tailored to specific matrix properties. The theoretical framework opens new lines of inquiry in random linear transformations and sampling complexity.

Conclusions

This paper contributes to the pursuit of computational efficiency in solving least squares problems. By leveraging randomization, it provides innovative strategies to meet the demanding processing requirements of large-scale systems. Such developments hold potent applications in data science, machine learning, and statistical analysis, offering scalable solutions to traditionally computationally intensive problems.