{"ID":2853231,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.16965","arxiv_id":"2510.16965","title":"A Unified Approach to Statistical Estimation Under Nonlinear Observations: Tensor Estimation and Matrix Factorization","abstract":"We consider the estimation of some parameter $\\mathbf{x}$ living in a cone from the nonlinear observations of the form $\\{y_i=f_i(\\langle\\mathbf{a}_i,\\mathbf{x}\\rangle)\\}_{i=1}^m$. We develop a unified approach that first constructs a gradient from the data and then establishes the restricted approximate invertibility condition (RAIC), a condition that quantifies how well the gradient aligns with the ideal descent step. We show that RAIC yields linear convergence guarantees for the standard projected gradient descent algorithm, a Riemannian gradient descent algorithm for low Tucker-rank tensor estimation, and a factorized gradient descent algorithm for asymmetric low-rank matrix estimation. Under Gaussian designs, we establish sharp RAIC for the canonical statistical estimation problems of single index models, generalized linear models, noisy phase retrieval, and one-bit compressed sensing. Combining the convergence guarantees and the RAIC, we obtain a set of optimal statistical estimation results, including, to our knowledge, the first minimax-optimal and computationally efficient algorithms for tensor single index models, tensor logistic regression, (local) noisy tensor phase retrieval, and one-bit tensor sensing. Moreover, several other results are new or match the best known guarantees. We also provide simulations and a real-data experiment to illustrate the theoretical results.","short_abstract":"We consider the estimation of some parameter $\\mathbf{x}$ living in a cone from the nonlinear observations of the form $\\{y_i=f_i(\\langle\\mathbf{a}_i,\\mathbf{x}\\rangle)\\}_{i=1}^m$. We develop a unified approach that first constructs a gradient from the data and then establishes the restricted approximate invertibility...","url_abs":"https://arxiv.org/abs/2510.16965","url_pdf":"https://arxiv.org/pdf/2510.16965v1","authors":"[\"Junren Chen\",\"Lijun Ding\",\"Dong Xia\",\"Ming Yuan\"]","published":"2025-10-19T19:10:08Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"cs.IT\",\"stat.ME\"]","methods":"[]","has_code":false}