Optimal Linear Estimation under Unknown Nonlinear Transform

Linear regression studies the problem of estimating a model parameter $\beta^* \in \R^p$, from $n$ observations $\{(y_i, x_i)\}_{i=1}^n$ from linear model $y_i = \langle \x_i, \beta^* \rangle + \epsilon_i$. We consider a significant generalization in which the relationship between $\langle x_i, \beta^* \rangle$ and $y_i$ is noisy, quantized to a single bit, potentially nonlinear, noninvertible, as well as unknown. This model is known as the single-index model in statistics, and, among other things, it represents a significant generalization of one-bit compressed sensing. We propose a novel spectral-based estimation procedure and show that we can recover $\beta^*$ in settings (i. e. , classes of link function $f$) where previous algorithms fail. In general, our algorithm requires only very mild restrictions on the (unknown) functional relationship between $y_i$ and $\langle x_i, \beta^* \rangle$. We also consider the high dimensional setting where $\beta^*$ is sparse, and introduce a two-stage nonconvex framework that addresses estimation challenges in high dimensional regimes where $p \gg n$. For a broad class of link functions between $\langle x_i, \beta^* \rangle$ and $y_i$, we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.

Authors

• Xinyang Yi
• Zhaoran Wang
• Constantine Caramanis University of Texas - Austin
• Han Liu Dalian University of Technology

Keywords

No keywords are indexed for this paper.