# Fast and Reliable Parameter Estimation from Nonlinear Observations

arXiv: https://arxiv.org/abs/1610.07108 · HTML: https://mlsanity.com/paper/1610.07108/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2016.

## Built on by (most cited first, 5 of 5)

- [Sharp global convergence guarantees for iterative nonconvex optimization: A Gaussian process perspective](https://mlsanity.com/paper/2109.09859/) (2021)
- [Non-Iterative Recovery from Nonlinear Observations using Generative Models](https://mlsanity.com/paper/2205.15749/) (2022)
- [Misspecified Phase Retrieval with Generative Priors](https://mlsanity.com/paper/2210.05571/) (2022)
- [Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors](https://mlsanity.com/paper/2209.10093/) (2022)
- [Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction](https://mlsanity.com/paper/2310.03956/) (2023)

## Abstract (arXiv)

In this paper we study the problem of recovering a structured but unknown parameter bfθ^* from n nonlinear observations of the form yi=f(langle bfxi,bfθ^*rangle) for i=1,2,ldots,n. We develop a framework for characterizing time-data tradeoffs for a variety of parameter estimation algorithms when the nonlinear function f is unknown. This framework includes many popular heuristics such as projected/proximal gradient descent and stochastic schemes. For example, we show that a projected gradient descent scheme converges at a linear rate to a reliable solution with a near minimal number of samples. We provide a sharp characterization of the convergence rate of such algorithms as a function of sample size, amount of a-prior knowledge available about the parameter and a measure of the nonlinearity of the function f. These results provide a precise understanding of the various tradeoffs involved between statistical and computational resources as well as a-prior side information available for such nonlinear parameter estimation problems.