Background. Floods are the most common natural disaster in the world, affecting the lives of hundreds of millions. Flood forecasting is therefore a vitally important endeavor, typically achieved using physical water flow simulations, which rely on accurate terrain elevation maps. However, such simulations, based on solving partial differential equations, are computationally prohibitive on a large scale. This scalability issue is commonly alleviated using a coarse grid representation of the elevation map, though this representation may distort crucial terrain details, leading to significant inaccuracies in the simulation.\Contributions. We train a deep neural network to perform physics-informed downsampling of the terrain map: we optimize the coarse grid representation of the terrain maps, so that the flood prediction will match the fine grid solution. For the learning process to succeed, we configure a dataset specifically for this task. We demonstrate that with this method, it is possible to achieve a significant reduction in computational cost, while maintaining an accurate solution. A reference implementation accompanies the paper as well as documentation and code for dataset reproduction.

We consider minimum variance estimation within the sparse linear Gaussian model (SLGM). A sparse vector is to be estimated from a linearly transformed version embedded in Gaussian noise. Our analysis is based on the theory of reproducing kernel Hilbert spaces (RKHS). After a characterization of the RKHS associated with the SLGM, we derive a lower bound on the minimum variance achievable by estimators with a prescribed bias function, including the important special case of unbiased estimation. This bound is obtained via an orthogonal projection of the prescribed mean function onto a subspace of the RKHS associated with the SLGM. It provides an approximation to the minimum achievable variance (Barankin bound) that is tighter than any known bound. Our bound holds for an arbitrary system matrix, including the overdetermined and underdetermined cases. We specialize it to compressed sensing measurement matrices and express it in terms of the restricted isometry constant. For the special case of the SLGM given by the sparse signal in noise model, we derive closed-form expressions of the Barankin bound and of the corresponding locally minimum variance estimator. Finally, we compare our bound with the variance of several well-known estimators, namely, the maximum-likelihood estimator, the hard-thresholding estimator, and compressive reconstruction using orthogonal matching pursuit and approximate message passing.

In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific sampling approach. Essential differences between the noisy and noise-free cases arise from this analysis. In particular, we identify settings in which noise-free recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related to a specific type of structure, which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels for linear reconstruction, based on a generalization of the Karhunen-Loève transform. The results are illustrated for several types of time-delay estimation problems.

The problem studied in this paper is unbiased estimation of a sparse nonrandom vector corrupted by additive white Gaussian noise. It is shown that while there are infinitely many unbiased estimators for this problem, none of them has uniformly minimum variance. Therefore, the focus is placed on locally minimum variance unbiased (LMVU) estimators. Simple closed-form lower and upper bounds on the variance of LMVU estimators or, equivalently, on the Barankin bound (BB) are derived. These bounds allow an estimation of the threshold region separating the low-signal-to-noise ratio (SNR) and high-SNR regimes, and they indicate the asymptotic behavior of the BB at high SNR. In addition, numerical lower and upper bounds are derived; these are tighter than the closed-form bounds and thus characterize the BB more accurately. Numerical studies compare the proposed characterizations of the BB with established biased estimation schemes, and demonstrate that while unbiased estimators perform poorly at low SNR, they may perform better than biased estimators at high SNR. An interesting conclusion of this analysis is that the high-SNR behavior of the BB depends solely on the value of the smallest nonzero entry of the sparse vector, and that this type of dependence is also exhibited by the performance of certain practical estimators.

We consider the problem of estimating a deterministic sparse vector x0from underdetermined measurements Ax0+ w, where w represents white Gaussian noise and A is a given deterministic dictionary. We analyze the performance of three sparse estimation algorithms: basis pursuit denoising, orthogonal matching pursuit, and thresholding. These approaches are shown to achieve near-oracle performance with high probability, assuming that x0is sufficiently sparse. Our results are non-asymptotic and are based only on the coherence of A, so that they are applicable to arbitrary dictionaries.

We consider the estimation of a sparse parameter vector from measurements corrupted by white Gaussian noise. Our focus is on unbiased estimation as a setting under which the difficulty of the problem can be quantified analytically. We show that there are infinitely many unbiased estimators but none of them has uniformly minimum mean-squared error. We then provide lower and upper bounds on the Barankin bound, which describes the performance achievable by unbiased estimators. These bounds are used to predict the threshold region of practical estimators.

The problem considered in this letter is to bound the performance of estimators of a deterministic parameter which satisfies given constraints. Specifically, previous work on the constrained Cramer-Rao bound (CRB) is generalized to include singular Fisher information matrices and biased estimators. A necessary and sufficient condition for the existence of a finite CRB is obtained. A closed form for an estimator achieving the CRB, if one exists, is also provided, as well as a necessary and sufficient condition for the existence of such an estimator. It is shown that biased estimators achieving the CRB can be constructed in situations for which no unbiased technique exists.

A lower bound on the minimum mean-squared error (MSE) in a Bayesian estimation problem is proposed in this paper. This bound utilizes a well-known connection to the deterministic estimation setting. Using the prior distribution, the bias function which minimizes the Cramer-Rao bound can be determined, resulting in a lower bound on the Bayesian MSE. The bound is developed for the general case of a vector parameter with an arbitrary probability distribution, and is shown to be asymptotically tight in both the high and low signal-to-noise ratio (SNR) regimes. A numerical study demonstrates several cases in which the proposed technique is both simpler to compute and tighter than alternative methods.

The Weiss-Weinstein bound (WWB) provides a lower limit on the mean-squared error (MSE) achievable by an estimator of an unknown random parameter. In this correspondence, it is shown that some previously proposed simplified versions of the bound do not always hold for constrained parameters, i.e., parameters whose distribution has finite support. These simplifications can produce results which are no longer lower bounds on the MSE. Sufficient conditions are provided for the reductions to be valid.

We consider the linear regression problem of estimating an unknown, deterministic parameter vector based on measurements corrupted by colored Gaussian noise. We present and analyze blind minimax estimators (BMEs), which consist of a bounded parameter set minimax estimator, whose parameter set is itself estimated from measurements. Thus, our approach does not require any prior assumption or knowledge, and the proposed estimator can be applied to any linear regression problem. We demonstrate analytically that the BMEs strictly dominate the least-squares (LS) estimator, i.e., they achieve lower mean-squared error (MSE) for any value of the parameter vector. Both Stein's estimator and its positive-part correction can be derived within the blind minimax framework. Furthermore, our approach can be readily extended to a wider class of estimation problems than Stein's estimator, which is defined only for white noise and nontransformed measurements. We show through simulations that the BMEs generally outperform previous extensions of Stein's technique.

We present several analytical and numerical results demonstrating the superiority of minimax estimators over least-squares (LS) estimation. We show that, for any bounded parameter set, a linear minimax estimator achieves lower mean-squared error than the LS estimator, over the entire parameter set. When a parameter set is unknown, we propose to estimate the parameter set from the data, and show that in many cases, the obtained blind minimax estimator still dominates the LS estimator. The results are related to and compared with other LS-dominating estimators, such as the James-Stein estimator.