Variational Data Assimilation via Sparse Regularization

This paper studies the role of sparse regularization in a properly chosen basis for variational data assimilation (VDA) problems. Specifically, it focuses on data assimilation of noisy and down-sampled observations while the state variable of interest exhibits sparsity in the real or transformed domain. We show that in the presence of sparsity, the $\ell_{1}$-norm regularization produces more accurate and stable solutions than the classic data assimilation methods. To motivate further developments of the proposed methodology, assimilation experiments are conducted in the wavelet and spectral domain using the linear advection-diffusion equation

Matrix completion with queries

In many applications, e.g., recommender systems and traffic monitoring, the data comes in the form of a matrix that is only partially observed and low rank. A fundamental data-analysis task for these datasets is matrix completion, where the goal is to accurately infer the entries missing from the matrix. Even when the data satisfies the low-rank assumption, classical matrix-completion methods may output completions with significant error -- in that the reconstructed matrix differs significantly from the true underlying matrix. Often, this is due to the fact that the information contained in the observed entries is insufficient. In this work, we address this problem by proposing an active version of matrix completion, where queries can be made to the true underlying matrix. Subsequently, we design Order&Extend, which is the first algorithm to unify a matrix-completion approach and a querying strategy into a single algorithm. Order&Extend is able identify and alleviate insufficient information by judiciously querying a small number of additional entries. In an extensive experimental evaluation on real-world datasets, we demonstrate that our algorithm is efficient and is able to accurately reconstruct the true matrix while asking only a small number of queries.Comment: Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Minin

Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices

We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges