Low Rank Approximation and Regression in Input Sparsity Time (1207.6365v4)

Abstract: We design a new distribution over $\poly(r \eps^{-1}) \times n$ matrices $S$ so that for any fixed $n \times d$ matrix $A$ of rank $r$, with probability at least 9/10, $\norm{SAx}_2 = (1 \pm \eps)\norm{Ax}_2$ simultaneously for all $x \in \mathbb{R}^d$. Such a matrix $S$ is called a \emph{subspace embedding}. Furthermore, $SA$ can be computed in $\nnz(A) + \poly(d \eps^{-1})$ time, where $\nnz(A)$ is the number of non-zero entries of $A$. This improves over all previous subspace embeddings, which required at least $\Omega(nd \log d)$ time to achieve this property. We call our matrices $S$ \emph{sparse embedding matrices}. Using our sparse embedding matrices, we obtain the fastest known algorithms for $(1+\eps)$-approximation for overconstrained least-squares regression, low-rank approximation, approximating all leverage scores, and $\ell_p$-regression. The leading order term in the time complexity of our algorithms is $O(\nnz(A))$ or $O(\nnz(A)\log n)$. We optimize the low-order $\poly(d/\eps)$ terms in our running times (or for rank-$k$ approximation, the $n*\poly(k/eps)$ term), and show various tradeoffs. For instance, we also use our methods to design new preconditioners that improve the dependence on $\eps$ in least squares regression to $\log 1/\eps$. Finally, we provide preliminary experimental results which suggest that our algorithms are competitive in practice.

• The paper introduces a novel method using sparse embedding matrices to achieve faster algorithms for problems like low-rank approximation and regression in numerical linear algebra.
• Key results include efficient time complexities, such as approximating the best rank-k approximation in O(nnz(A)) + time, and improved leverage score approximation.
• These techniques have significant practical implications, enabling more efficient computation for large-scale data analytics, data mining, and machine learning applications.

Low Rank Approximation and Regression in Input Sparsity Time: An Overview

This paper by Kenneth L. Clarkson and David P. Woodruff addresses efficient computation methods in numerical linear algebra, particularly focusing on low-rank approximation and regression. The authors introduce a novel approach using sparse embedding matrices to enhance time efficiency compared to existing methods.

Summary of Contributions

The paper introduces a new distribution over sparse embedding matrices that achieves subspace embeddings. These embeddings provide more efficient algorithms for computational problems such as overconstrained least-squares regression, low-rank approximation, and leverage scores approximation.

Key Numerical Findings

1. Overconstrained Least-Squares Regression:
   • The algorithm outputs an approximate solution vector x' such that ${Ax'-b}_2 \leq (1+)\min_x {Ax-b}_2$. The algorithm operates in $O(\nnz(A)) + O(d^3^{-2})$ time, and an alternative version completes in $O(\nnz(A)\log(1/)) + O(d^3\log(1/))$.
2. Low-Rank Approximation:
   • The decomposition process for approximating the best rank-k approximation runs in $O(\nnz(A)) + \tilde O(nk^2^{-4} + k^3^{-5})$ time.
3. Leverage Scores Approximation:
   • The method provides constant relative error approximations in $O(\nnz(A) \log n) + O(r^3)$ time, significantly improving efficiency.
4. $\ell_p$-Regression:
   • Produces an approximate solution in $O(\nnz(A) \log n) + (r^{-1})$ time for any constant $1 \leq p < \infty$.

Implications for Research and Practice

The approaches presented have both theoretical and practical significance:

• Theoretical Advances:

The use of sparse embedding matrices with distribution designed for subspace embedding helps reduce time complexity for matrix-related computations. This advancement bridges a gap where traditional methods would be suboptimal, especially in handling large matrices with sparsity.

• Practical Applications:

Fast and efficient computation methods have direct applications in data mining, machine learning, and large-scale data analytics. The ability to quickly approximate solutions to linear algebra problems is crucial in fields like recommendation systems and information retrieval.

Future Directions

• Optimization of Embedding Matrices:

Further investigation into optimizing the polynomial factors in the running times remains open. Such research could potentially lead to even more efficient algorithms.

• Applications Across Different Domains:

Extending these findings to other areas in technology and science where large datasets are prevalent could be beneficial.

• Exploration of New Kind of Embeddings:

Different types of embeddings or alternative matrix transformations may offer additional speed or accuracy improvements.

In conclusion, Clarkson and Woodruff provide a framework for more efficient computations in numerical linear algebra by utilizing sparse embedding matrices. This contribution not only progresses theoretical understanding but also enhances computational techniques necessary for handling vast and complex datasets effectively.