Is speckle noise more challenging to mitigate than additive noise?

We study the problem of estimating a function in the presence of both speckle and additive noises. Although additive noise has been thoroughly explored in nonparametric estimation, speckle noise, prevalent in applications such as synthetic aperture radar, ultrasound imaging, and digital holography, has not received as much attention. Consequently, there is a lack of theoretical investigations into the fundamental limits of mitigating the speckle noise. This paper is the first step in filling this gap. Our focus is on investigating the minimax estimation error for estimating a $\beta$-H\"older continuous function and determining the rate of the minimax risk. Specifically, if $n$ represents the number of data points, $f$ denotes the underlying function to be estimated, and $\hat{\nu}_n$ is an estimate of $f$, then $\inf_{\hat{\nu}_n} \sup_f \mathbb{E}_f\| \hat{\nu}_n - f \|^2_2$ decays at the rate $n^{-\frac{2\beta}{2\beta+1}}$. Interestingly, this rate is identical to the one achieved for mitigating additive noise when the noise's variance is $\Theta(1)$. To validate the accuracy of our minimax upper bounds, we implement the minimax optimal algorithms on simulated data and employ Monte Carlo simulations to characterize their exact risk. Our simulations closely mirror the expected behaviors in decay rate as per our theory.