Bidirectional Random Projections for OLS Regression: Theoretical Analysis and Bounds
Researchers have developed theoretical bounds for ordinary least squares regression when both input features and samples are compressed using random projections. The analysis shows that applying projections in both directions introduces a performance gap compared to single-direction projection, with the gap depending on projection dimensions and sample ratios. This work provides theoretical foundations for understanding computational trade-offs in high-dimensional regression problems.
A new theoretical analysis examines bidirectional random projections applied to ordinary least squares regression under fixed design settings. The study develops expected excess loss bounds for OLS estimators built on doubly-projected data (WXR, WY), where R and W are random projection matrices applied to samples and features respectively. The key finding is that the performance gap compared to single-direction projection (XR, Y) scales approximately as O(p₁ + C/p₁), where C depends on the ratio of projected to original sample size and can be negative for small ratios. The theoretical results are validated through numerical experiments on real-world datasets. This work contributes to understanding computational efficiency in regression by quantifying the cost of dimensionality reduction in both feature and sample spaces.
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- arXiv cs.LGCenter
Bidirectional Random Projections
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