[Paper Review] Projection-Cost-Preserving Sketches: Proof Strategies and Constructions

Cameron Musco, Christopher Musco|arXiv (Cornell University)|Apr 17, 2020

Stochastic Gradient Optimization Techniques11 references4 citations

TL;DR

This paper provides a unified framework for constructing projection-cost-preserving sketches (PCPs) using matrix approximation techniques like subspace embeddings, approximate matrix multiplication, and Frobenius norm preservation. It proves that random projection, ridge leverage score sampling, and deterministic column selection yield PCPs with optimal dimension $ m = \tilde{O}(k/\epsilon^2) $, enabling efficient low-rank approximation and $ k $-means clustering with $ (1\pm\epsilon) $-relative error guarantees.

ABSTRACT

In this note we illustrate how common matrix approximation methods, such as random projection and random sampling, yield projection-cost-preserving sketches, as introduced in [FSS13, CEM+15]. A projection-cost-preserving sketch is a matrix approximation which, for a given parameter $k$, approximately preserves the distance of the target matrix to all $k$-dimensional subspaces. Such sketches have applications to scalable algorithms for linear algebra, data science, and machine learning. Our goal is to simplify the presentation of proof techniques introduced in [CEM+15] and [CMM17] so that they can serve as a guide for future work. We also refer the reader to [CYD19], which gives a similar simplified exposition of the proof covered in Section 2.

Motivation & Objective

• To simplify and systematize proof techniques for constructing projection-cost-preserving sketches (PCPs) from existing matrix approximation guarantees.
• To demonstrate that common dimensionality reduction methods—such as random projection and column sampling—yield PCPs with $ (\epsilon, 0, k) $-guarantee and $ m = \tilde{O}(k/\epsilon^2) $ columns.
• To unify two proof strategies: one based on matrix approximation primitives (subspace embedding, Frobenius norm preservation), and another based on spectral approximation of $ AA^T $ with regularization.
• To provide a guide for future work by clarifying the theoretical underpinnings of PCP constructions in scalable linear algebra and machine learning.

Proposed method

• Uses subspace embedding and approximate matrix multiplication to bound the error in projecting $ A $ onto any $ k $-dimensional subspace via the sketch $ \tilde{A} = AS $.
• Applies a matrix Bernstein inequality to prove spectral approximation of $ AA^T $, ensuring $ (1-\epsilon)AA^T - \lambda I \preceq \tilde{A}\tilde{A}^T \preceq (1+\epsilon)AA^T + \lambda I $ for a properly chosen $ \lambda $.
• Employs a Chernoff bound to establish Frobenius norm preservation for the sketch $ \tilde{A} $, ensuring $ \|A_{\setminus k}S\|_F^2 \approx \|A_{\setminus k}\|_F^2 $.
• Decomposes $ A $ into low-rank (head) and residual (tail) components using SVD, and bounds error contributions from each part separately.
• Uses deterministic column selection via a stable-rank approximation result from [CNW16] to construct a PCP with $ O(k/\epsilon^2) $ columns.
• Applies regularization via $ \lambda = \|A - A_k\|_F^2 / k $ to stabilize the spectral approximation and ensure error control.

Experimental results

Research questions

• RQ1 Can standard matrix approximation techniques like subspace embedding and approximate matrix multiplication be used to prove PCP guarantees?
• RQ2 Do random projection and column sampling methods yield PCPs with $ c = 0 $ and dimension $ m = \tilde{O}(k/\epsilon^2) $?
• RQ3 Can spectral approximation of $ AA^T $ with regularization be used as an alternative proof strategy for PCPs?
• RQ4 How can the two proof strategies—matrix approximation primitives and spectral approximation—be unified or related?
• RQ5 What deterministic algorithms can construct PCPs with optimal dimension and $ c = 0 $?

Key findings

• Random projection yields an $ (\epsilon, 0, k) $-PCP with $ m = O(k/\epsilon^2) $ columns, matching the best-known bounds.
• Ridge leverage score column sampling produces an $ (\epsilon, 0, k) $-PCP with $ m = O(k\log k / \epsilon^2) $ columns, under mild assumptions on score approximation.
• Deterministic column selection yields an $ (\epsilon, 0, k) $-PCP with $ m = O(k/\epsilon^2) $ columns, using a stable-rank matrix multiplication result.
• The proof via spectral approximation shows that if $ \tilde{A}\tilde{A}^T $ spectrally approximates $ AA^T $ with regularization, then $ \tilde{A} $ is a PCP with $ c = 0 $.
• The Frobenius norm preservation and spectral approximation conditions together imply the PCP guarantee, with error bounded by $ (1\pm\epsilon) $ times the original projection cost.
• The framework unifies and simplifies prior proofs from [CEM+15] and [CMM17], offering a clear path for future PCP constructions.