Bidirectional Random Projections for OLS Regression: Theoretical Analysis and Bounds
Researchers have developed an expected excess loss bound for ordinary least squares (OLS) regression using bidirectional random projections applied simultaneously to both the sample and feature dimensions. The work extends prior theory, which only projected along one dimension, and characterizes the performance gap as approximately O(p₁ + C·(1/p₁)), where C scales with the ratio of projected to original sample size. The findings matter because they clarify when compressing data in both directions is beneficial or harmful, with numerical experiments on real-world data confirming the theoretical predictions.
This paper, submitted to arXiv and accepted at Statistics & Probability Letters (Elsevier), analyzes the statistical consequences of applying two random projection matrices simultaneously in OLS regression: one projecting the sample dimension (n to n₁) and one projecting the feature dimension (p to p₁). The authors derive an expected excess loss bound for the doubly-projected estimator and compare it to an established bound for the singly-projected case. The performance gap between the two approaches is characterized as approximately O(p₁ + C·(1/p₁)), where the constant C scales with n₁/n and can turn negative when n₁/n is small, meaning bidirectional projection can actually outperform unidirectional projection in certain regimes. This nuanced result suggests that aggressively reducing the sample dimension relative to the original can offset the cost of also reducing the feature dimension. The theoretical bounds are validated through numerical experiments on real-world datasets, lending empirical support to the analytical findings.
What's missing
The paper does not specify the datasets used in numerical experiments, the range of n₁/n ratios explored, or how the results compare to other dimensionality-reduction methods such as sketching or PCA-based approaches. The precise conditions under which C becomes negative—and thus bidirectional projection is strictly preferable—are not detailed in the abstract. Additionally, computational cost trade-offs between the two projection strategies are not discussed.
What different sources said
- arXiv cs.LGCenter
Bidirectional Random Projections
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