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Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these…
We consider a certain tiling problem of a planar region in which there are no long horizontal or vertical strips consisting of copies of the same tile. Intuitively speaking, we would like to create a dappled pattern with two or more kinds…
Recovery error bounds of tail-minimization and the rate of convergence of an efficient proximal alternating algorithm for sparse signal recovery are considered in this article. Tail-minimization focuses on minimizing the energy in the…
Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling…
We consider the problem of recovering an unknown effectively $(s_1,s_2)$-sparse low-rank-$R$ matrix $X$ with possibly non-orthogonal rank-$1$ decomposition from incomplete and inaccurate linear measurements of the form $y = \mathcal A (X) +…
A primary challenge in metagenomics is reconstructing individual microbial genomes from the mixture of short fragments created by sequencing. Recent work leverages the sparsity of the assembly graph to find $r$-dominating sets which enable…
In [BNRR], it was shown that tiling of general regions with two rectangles is NP-complete, except for a few trivial special cases. In a different direction, R\'emila showed that for simply connected regions by two rectangles, the…
We give a polynomial time, $(1+\epsilon)$-approximation algorithm for the traveling repairman problem (TRP) in the Euclidean plane and on weighted trees. This improves on the known quasi-polynomial time approximation schemes for these…
Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…
By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$.…
Sparse signal recovery problems from noisy linear measurements appear in many areas of wireless communications. In recent years, deep learning (DL) based approaches have attracted interests of researchers to solve the sparse linear inverse…
Recently, greedy algorithm has received much attention as a cost-effective means to reconstruct the sparse signals from compressed measurements. Much of previous work has focused on the investigation of a single candidate to identify the…
We give a new algorithm that quickly finds smooth neighbors.
We consider estimation models of the form $Y=X^*+N$, where $X^*$ is some $m$-dimensional signal we wish to recover, and $N$ is symmetrically distributed noise that may be unbounded in all but a small $\alpha$ fraction of the entries. We…
We investigate a trust-region algorithm to solve a nonconvex optimization problem with $L^p$-regularization for $p\in(0,1)$. The algorithm relies on descent properties of a so-called generalized Cauchy point that can be obtained efficiently…
The sparsity in levels model recently inspired a new generation of effective acquisition and reconstruction modalities for compressive imaging. Moreover, it naturally arises in various areas of signal processing such as parallel…
The well-known problem stated by A. Meir and L. Moser consists in tiling the unit square with rectangles (details), whose side lengths equal $1/n\times 1/(n+1)$, where indices~$n$ range from 1 to infinity. Recently, Terence Tao has proved…
This paper provides a bridge between the classical tiling theory and the complex neighborhood self-assembling situations that exist in practice. The neighborhood of a position in the plane is the set of coordinates which are considered…
This paper studies two structured approximation problems: (1) Recovering a corrupted low-rank Toeplitz matrix and (2) recovering the range of a Fourier matrix from a single observation. Both problems are computationally challenging because…
This paper presents and analyses a new family of linear subdivision schemes to refine noisy data given on triangular meshes. The subdivision rules consist of locally fitting and evaluating a weighted least squares approximating first-degree…