A Unified Approach to Statistical Estimation Under Nonlinear Observations: Tensor Estimation and Matrix Factorization
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| Format: | Preprint |
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2025
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| author | Chen, Junren Ding, Lijun Xia, Dong Yuan, Ming |
| author_facet | Chen, Junren Ding, Lijun Xia, Dong Yuan, Ming |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16965 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Unified Approach to Statistical Estimation Under Nonlinear Observations: Tensor Estimation and Matrix Factorization Chen, Junren Ding, Lijun Xia, Dong Yuan, Ming Statistics Theory Information Theory Methodology 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. |
| title | A Unified Approach to Statistical Estimation Under Nonlinear Observations: Tensor Estimation and Matrix Factorization |
| topic | Statistics Theory Information Theory Methodology |
| url | https://arxiv.org/abs/2510.16965 |