Related papers: Sparser Johnson-Lindenstrauss Transforms
We study network loss tomography based on observing average loss rates over a set of paths forming a tree -- a severely underdetermined linear problem for the unknown link loss probabilities. We examine in detail the role of sparsity as a…
We consider the problem of learning the underlying graph of a sparse Ising model with $p$ nodes from $n$ i.i.d. samples. The most recent and best performing approaches combine an empirical loss (the logistic regression loss or the…
Looking for sparsity is nowadays crucial to speed up the training of large-scale neural networks. Projections onto the $\ell_{1,2}$ and $\ell_{1,\infty}$ are among the most efficient techniques to sparsify and reduce the overall cost of…
This paper studies the performance of sparse regression codes for lossy compression with the squared-error distortion criterion. In a sparse regression code, codewords are linear combinations of subsets of columns of a design matrix. It is…
The well-known M-P (Moore-Penrose) pseudoinverse is used in several linear-algebra applications; for example, to compute least-squares solutions of inconsistent systems of linear equations. It is uniquely characterized by four properties,…
Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing. Computations over sparse structures are less complex compared to their dense counterparts, and their storage consumes less space. This…
A new non-linear optimization approach is proposed for the sparse reconstruction of log-conductivities in current density impedance imaging. This framework comprises of minimizing an objective functional involving a least squares fit of the…
Johnson-Lindenstrauss (JL) matrices implemented by sparse random synaptic connections are thought to be a prime candidate for how convergent pathways in the brain compress information. However, to date, there is no complete mathematical…
The article concerns compressed sensing methods in the quaternion algebra. We prove that it is possible to uniquely reconstruct - by $\ell_1$ norm minimization - a sparse quaternion signal from a limited number of its real linear…
This paper is concerned with estimating the column space of an unknown low-rank matrix $\boldsymbol{A}^{\star}\in\mathbb{R}^{d_{1}\times d_{2}}$, given noisy and partial observations of its entries. There is no shortage of scenarios where…
In this note, we develop fast and deterministic dimensionality reduction techniques for a family of subspace approximation problems. Let $P\subset \mathbbm{R}^N$ be a given set of $M$ points. The techniques developed herein find an $O(n…
The optimization problem with sparsity arises in many areas of science and engineering such as compressed sensing, image processing, statistical learning and data sparse approximation. In this paper, we study the dual-density-based…
In sparse optimization, enforcing hard constraints using the $\ell_0$ pseudo-norm offers advantages like controlled sparsity compared to convex relaxations. However, many real-world applications demand not only sparsity constraints but also…
This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on…
We give an algorithm for $\ell_2/\ell_2$ sparse recovery from Fourier measurements using $O(k\log N)$ samples, matching the lower bound of \cite{DIPW} for non-adaptive algorithms up to constant factors for any $k\leq N^{1-\delta}$. The…
We consider the compressed sensing problem, where the object $x_0 \in \bR^N$ is to be recovered from incomplete measurements $y = Ax_0 + z$; here the sensing matrix $A$ is an $n \times N$ random matrix with iid Gaussian entries and $n < N$.…
The Johnson-Lindenstrauss transform is a fundamental method for dimension reduction in Euclidean spaces, that can map any dataset of $n$ points into dimension $O(\log n)$ with low distortion of their distances. This dimension bound is tight…
Sparse linear regression is one of the most basic questions in machine learning and statistics. Here, we are given as input a design matrix $X \in \mathbb{R}^{N \times d}$ and measurements or labels ${y} \in \mathbb{R}^N$ where ${y} = {X}…
Signal models formed as linear combinations of few atoms from an over-complete dictionary or few frame vectors from a redundant frame have become central to many applications in high dimensional signal processing and data analysis. A core…
Analysis sparsity is a common prior in inverse problem or machine learning including special cases such as Total Variation regularization, Edge Lasso and Fused Lasso. We study the geometry of the solution set (a polyhedron) of the analysis…