Global Convergence of Adaptive Sensing for Principal Eigenvector Estimation

Computer Science > Machine Learning [Submitted on 16 May 2025 (v1), last revised 30 May 2026 (this version, v2)] Title:Global Convergence of Adaptive Sensing for Principal Eigenvector Estimation View PDF HTML (experimental)Abstract:Principal component analysis classically requires full $d$-dimensional samples, yet in various applications hardware limits acquisition to a few scalar measurements per sample. We analyze a compressed variant of Oja's algorithm for estimating the principal eigenvector of the data covariance matrix using only two adaptive measurements per sample. At each iteration, we observe one measurement along the current estimate and one in a random orthogonal direction. We prove that after $t$ iterations, the expected sine-squared error to the true eigenvector is $\mathcal{O}(\lambda_1\lambda_2 d^2 / (\Delta^2 t))$, where $d$ is the ambient dimension, $\lambda_1, \lambda_2$ are the leading eigenvalues, and $\Delta = \lambda_1 - \lambda_2$ is the eigengap. We complement this with a matching information-theoretic lower bound of $\Omega(\lambda_1\lambda_2 d^2 / (\Delta^2 t))$ -- the first for compressed eigenvector estimation -- proving that the $d^2$ factor, an additional factor of $d$ compared to the fully-observed minimax rate $\Theta(\lambda_1\lambda_2 d / (\Delta^2 t))$, is the fundamental cost of compression and cannot be improved. In contrast, any non-adaptive scheme with two measurements per iteration suffers $\Omega(\lambda_2^2 d^3 / (\Delta^2 t))$, an additional power of $d$. This separates fully-observed, adaptive-compressed, and non-adaptive-compressed PCA across three powers of $d$. Our analysis handles the noisy setting where the covariance has nonzero trailing eigenvalues, providing the first convergence guarantee for adaptive compressed subspace tracking beyond the noiseless case. Submission history From: Alex Saad-Falcon [view email][v1] Fri, 16 May 2025 05:41:11 UTC (219 KB) [v2] Sat, 30 May 2026 19:58:32 UTC (1,180 KB) Current browse context: cs.LG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) IArxiv Recommender (What is IArxiv?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.