OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings (1211.1002v1)

Abstract: An "oblivious subspace embedding (OSE)" given some parameters eps,d is a distribution D over matrices B in R^{m x n} such that for any linear subspace W in R^n with dim(W) = d it holds that Pr_{B ~ D}(forall x in W ||B x||_2 in (1 +/- eps)||x||_2) > 2/3 We show an OSE exists with m = O(d^2/eps^2) and where every B in the support of D has exactly s=1 non-zero entries per column. This improves previously best known bound in [Clarkson-Woodruff, arXiv:1207.6365]. Our quadratic dependence on d is optimal for any OSE with s=1 [Nelson-Nguyen, 2012]. We also give two OSE's, which we call Oblivious Sparse Norm-Approximating Projections (OSNAPs), that both allow the parameter settings m = ~O(d/eps^2) and s = polylog(d)/eps, or m = O(d^{1+gamma}/eps^2) and s=O(1/eps) for any constant gamma>0. This m is nearly optimal since m >= d is required simply to no non-zero vector of W lands in the kernel of B. These are the first constructions with m=o(d^2) to have s=o(d). In fact, our OSNAPs are nothing more than the sparse Johnson-Lindenstrauss matrices of [Kane-Nelson, SODA 2012]. Our analyses all yield OSE's that are sampled using either O(1)-wise or O(log d)-wise independent hash functions, which provides some efficiency advantages over previous work for turnstile streaming applications. Our main result is essentially a Bai-Yin type theorem in random matrix theory and is likely to be of independent interest: i.e. we show that for any U in R^{n x d} with orthonormal columns and random sparse B, all singular values of BU lie in [1-eps, 1+eps] with good probability. Plugging OSNAPs into known algorithms for numerical linear algebra problems such as approximate least squares regression, low rank approximation, and approximating leverage scores implies faster algorithms for all these problems.

• The paper introduces OSNAPs, a novel sparse subspace embedding method that reduces computational overhead while preserving Euclidean norms.
• It achieves near-optimal sparsity by limiting nonzero entries per column, enabling significant speed-ups in tasks like least squares regression.
• Analytical improvements and tighter dimensionality reduction bounds highlight OSNAP’s practical efficiency in large-scale numerical computations.

Overview of "OSNAP: Faster Numerical Linear Algebra Algorithms via Sparser Subspace Embeddings"

The paper "OSNAP: Faster Numerical Linear Algebra Algorithms via Sparser Subspace Embeddings," authored by Jelani Nelson and Huy L. Nguyen, addresses the field of oblivious subspace embedding (OSE) for numerical linear algebra problems. The focus is on developing computational methods that enhance the efficiency of numerical linear algebra through innovative uses of sparse subspace embeddings.

Background and Problem Definition

Subspace embeddings are crucial in reducing the dimensionality of datasets while preserving certain properties of the data, which can significantly accelerate downstream computational processes. This paper builds upon previous work, such as the Johnson-Lindenstrauss Lemma, to specifically handle the embedding of subspaces with enhanced sparsity. An OSE aims to map a linear subspace into a lower dimension in a manner that approximately preserves Euclidean norms with high probability.

Contributions and Approach

This work introduces Oblivious Sparse Norm-Approximating Projections (OSNAPs), a novel formulation that leverages subspace embeddings to achieve optimal sparsity. The chief technical advancement involves constructing an OSE where each matrix from the distribution supports non-zero entries that are exceptionally sparse—just a single non-zero entry per column in some cases. The authors extend the constructions to achieve reduced embedding sizes m, compared to previous methods, without sacrificing the efficacy of the subspace mapping.

The authors also derive improved analytical guarantees on the preservation of Euclidean norms in embedded spaces and detail methods that circumvent the computational challenges typically encountered in naive OSE constructions. The demonstrated improvements include tighter bounds on dimensionality reduction and reduced computational overhead, particularly beneficial in sparse data scenarios.

Analytical and Numerical Results

Numerically, the proposed methods deliver substantial improvements over prior approaches. For instance, the embeddings provide significant speed-ups for classical numerical tasks including least squares regression, low-rank approximation, and leverage score approximation. For the approximate least squares problem, the paper reports a running time bound of $\tilde{O}(nnz(A) + r^{\omega})$, improving the dependence on the rank of the matrix, r, compared to predecessors.

The theoretical backbone of these results is extensively discussed, utilizing random matrix theory to derive concentration inequalities that bind the singular values of the transformed matrices. The authors progress from these mathematical foundations to robust empirical validations, illustrating the efficacy and efficiency of their methods.

Implications and Future Work

The implications of this work are multifaceted. Primarily, the advancements in sparse embedding techniques can be directly applied to enhance computational algorithms across various large-scale data challenges. This includes streaming settings, where efficient real-time processing is crucial. Additionally, the theoretical insights contribute to a deeper understanding of the limits and capabilities of dimensionality reduction approaches.

Future directions likely include expanding upon these techniques to develop even more efficient embeddings and to explore their application in broader contexts, such as distributed computing and machine learning. The versatility and improved performance of OSNAPs pave the way for further integration into numerical computing libraries and software systems, where they could become standard practice for dimensionality reduction and sketching methods.

In summary, "OSNAP: Faster Numerical Linear Algebra Algorithms via Sparser Subspace Embeddings" provides significant contributions to the area of sparse numerical embeddings. By optimizing both the theoretical and practical aspects of subspace embeddings, it sets a new standard in efficiency and serves as a foundation for future innovation in numerical linear algebra.