Low-distortion Subspace Embeddings in Input-sparsity Time and Applications to Robust Linear Regression (1210.3135v3)

Abstract: Low-distortion embeddings are critical building blocks for developing random sampling and random projection algorithms for linear algebra problems. We show that, given a matrix $A \in \R^{n \times d}$ with $n \gg d$ and a $p \in [1, 2)$, with a constant probability, we can construct a low-distortion embedding matrix $\Pi \in \R^{O(\poly(d)) \times n}$ that embeds $\A_p$, the $\ell_p$ subspace spanned by $A$'s columns, into $(\R^{O(\poly(d))}, | \cdot |p)$; the distortion of our embeddings is only $O(\poly(d))$, and we can compute $\Pi A$ in $O(\nnz(A))$ time, i.e., input-sparsity time. Our result generalizes the input-sparsity time $\ell_2$ subspace embedding by Clarkson and Woodruff [STOC'13]; and for completeness, we present a simpler and improved analysis of their construction for $\ell_2$. These input-sparsity time $\ell_p$ embeddings are optimal, up to constants, in terms of their running time; and the improved running time propagates to applications such as $(1\pm \epsilon)$-distortion $\ell_p$ subspace embedding and relative-error $\ell_p$ regression. For $\ell_2$, we show that a $(1+\epsilon)$-approximate solution to the $\ell_2$ regression problem specified by the matrix $A$ and a vector $b \in \R^n$ can be computed in $O(\nnz(A) + d^3 \log(d/\epsilon) /\epsilon^2)$ time; and for $\ell_p$, via a subspace-preserving sampling procedure, we show that a $(1\pm \epsilon)$-distortion embedding of $\A_p$ into $\R^{O(\poly(d))}$ can be computed in $O(\nnz(A) \cdot \log n)$ time, and we also show that a $(1+\epsilon)$-approximate solution to the $\ell_p$ regression problem $\min{x \in \R^d} |A x - b|_p$ can be computed in $O(\nnz(A) \cdot \log n + \poly(d) \log(1/\epsilon)/\epsilon^2)$ time. Moreover, we can improve the embedding dimension or equivalently the sample size to $O(d^{3+p/2} \log(1/\epsilon) / \epsilon^2)$ without increasing the complexity.

• The paper introduces a scalable algorithm that constructs low-distortion subspace embeddings in input-sparsity time for ℓₚ-norm subspaces.
• It generalizes Clarkson and Woodruff’s ℓ₂-based method to ℓₚ spaces, ensuring a (1±ε)-distortion guarantee with efficient computation.
• The approach improves robust linear regression and matrix approximations in high-dimensional, sparse data, offering significant computational gains.

Low-distortion Subspace Embeddings in Input-sparsity Time and Applications to Robust Linear Regression

The paper "Low-distortion Subspace Embeddings in Input-sparsity Time and Applications to Robust Linear Regression" addresses the computational efficiency of embedding techniques in subspace representation and robust regression problems. Low-distortion subspace embeddings serve as foundational components for optimizing random sampling and projection algorithms, critical to linear algebra concerns such as regression and matrix approximations.

Key Contributions

The authors present a method for constructing a low-distortion embedding matrix, denoted as $\Pi$, for a given matrix $A \in \mathbb{R}^{n \times d}$ where $n \gg d$ and $p \in [1, 2)$. This embedding guarantees that $\Pi$ maps the $\ell_p$-norm subspace spanned by the columns of $A$ into $$(\mathbb{R}^{\mathcal{O}(\text{poly}(d))}, \|\cdot\|_p)$$ with distortion bound $\mathcal{O}(\text{poly}(d))$. Crucially, this mapping is achieved in $\mathcal{O}(\text{nnz}(A))$ time, aligning with input-sparsity time, which is optimal up to constant factors for the general case.

The advancements in this paper stem from generalizing the $\ell_2$-based input-sparsity time subspace embedding introduced by Clarkson and Woodruff. Moreover, this research offers a refined and simplified analysis of the $\ell_2$ construction, showing iterative improvements encompassing $\ell_p$ spaces for $p \in [1, 2)$.

Numerical Results and Claims

The paper asserts significant numerical and theoretical performance enhancements, including:

• A $(1+\epsilon)$-approximate solution to the $\ell_2$ regression problem can be computed in $$\mathcal{O}(\text{nnz}(A) + d^3 \log(d/\epsilon) /\epsilon^2)$$ time.
• For $\ell_p$ with $p \in [1, 2)$, the $(1\pm\epsilon)$-distortion embedding can be computed in $\mathcal{O}(\text{nnz}(A) \cdot \log n)$ time.
• The paper guarantees an embedding dimension improvement, reducing it to $$\mathcal{O}(d^{3+p/2} \log(1/\epsilon) / \epsilon^2)$$ under certain conditions without augmenting the computational complexity.

Implications and Future Directions

The insights from this paper have practical ramifications in large-scale data scenarios, offering computational models that efficiently handle high-dimensional sparse data. The methodologies not only promise enhancements in robust regression computations but also extend to low-rank matrix approximation and leveraging score computations, promising broader applicability in machine learning and data analytics.

This research invites future exploration into algorithmic scaling, particularly focusing on reducing computational overhead while maintaining or enhancing accuracy. The intersection of such subspace embeddings with emerging data processing architectures, such as distributed and streaming environments, could be pivotal.

Overall, this paper deepens our understanding of subspace embeddings and their computational reach, heralding new avenues for tackling complex linear algebra problems with sparse inputs efficiently.