Nonzero Entries Research Articles

Singular matrices that are products of two idempotents or products of two nilpotents

Abstract Over commutative domains we characterize the singular 2 × 2 matrices which are products of two idempotents or products of two nilpotents. The relevant casees are the matrices with zero second row and the singular matrices with only nonzero entries.

Fundamental Limits of Many-User MAC With Finite Payloads and Fading

Consider a (multiple-access) wireless communication system where users are connected to a unique base station over a shared-spectrum radio links. Each user has a fixed number k of bits to send to the base station, and his signal gets attenuated by a random channel gain (quasi-static fading). In this paper we consider the many-user asymptotics of Chen-Chen-Guo'2017, where the number of users grows linearly with the blocklength. Differently, though, we adopt a per-user probability of error (PUPE) criterion (as opposed to classical joint-error probability criterion). Under PUPE the finite energy-per-bit communication is possible, and we are able to derive bounds on the tradeoff between energy and spectral efficiencies. We reconfirm the curious behaviour (previously observed for non-fading MAC) of the possibility of almost perfect multi-user interference (MUI) cancellation for user densities below a critical threshold. Further, we demonstrate the suboptimality of standard solutions such as orthogonalization (i.e. TDMA/FDMA) and treating interference as noise (i.e. pseudo-random CDMA without multi-user detection). Notably, the problem treated here can be seen as a variant of support recovery in compressed sensing for the unusual definition of sparsity with one non-zero entry per each contiguous section of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> coordinates. This identifies our problem with that of the sparse regression codes (SPARCs) and hence our results can be equivalently understood in the context of SPARCs with sections of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">100</sup> . Finally, we discuss the relation of the almost perfect MUI cancellation property and the replica-method predictions.

Information sparsification for visual-inertial odometry by manipulating Bayes tree

A new information sparsification method is developed for visual-inertial odometry (VIO) used for exploratory unmanned aerial vehicles (UAV's). While most of the existing methods are designed for global (and thus postprocessing) bundle adjustment (BA) problems, our method is specifically designed to run incrementally on a local BA formulation in real time. The majority of the existing state-of-the-art methods are designed for pose graph optimization (PGO). Unlike PGO problems where the only variables to be optimized are robot poses, a combination of pose, velocity, and inertial measurement unit (IMU) bias is used in VIO. Existing approaches to sparsification cannot be directly applied to this formulation. Therefore, we design a trick to split the dense information matrix into two parts: IMU information and purely visual information and by doing so, existing methods of sparsification can be reused and applied to the purely visual information, which accounts for most of the nonzero entries within the dense information matrix. In addition, we design a new way of summarizing the dense information matrix: using Bayes tree in an incremental smoothing framework for fast data retrieval. Results using public benchmark dataset EuRoC show that the marginalized and sparsified smoother achieves the primary goal of power saving in terms of shortened pipeline cycle time while maintaining absolute trajectory accuracies comparable to that of the complete formulation and better than those offered by fellow sparsification solutions.

Circular law for random block band matrices with genuinely sublinear bandwidth

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show that there exists τ ∈ (0, 1) so that if the bandwidth of the matrix X is at least n1−τ and the nonzero entries are iid random variables with mean zero and slightly more than four finite moments, then the limiting empirical eigenvalue distribution of X, when properly normalized, converges in probability to the uniform distribution on the unit disk in the complex plane. The key technical result is a least singular value bound for shifted random band block matrices with genuinely sublinear bandwidth, which improves on a result of Cook [Ann. Probab. 46, 3442 (2018)] in the band matrix setting.

A massively parallel interior-point solver for LPs with generalized arrowhead structure, and applications to energy system models

Linear energy system models are a crucial component of energy system design and operations, as well as energy policy consulting. If detailed enough, such models lead to large-scale linear programs, which can be intractable even for the best state-of-the-art solvers. This article introduces an interior-point solver that exploits common structures of energy system models to efficiently run in parallel on distributed-memory systems. The solver is designed for linear programs with doubly-bordered block-diagonal constraint matrix and makes use of a Schur complement based decomposition. In order to handle the large number of linking constraints and variables commonly observed in energy system models, a distributed Schur complement preconditioner is used. In addition, the solver features a number of more generic techniques such as parallel matrix scaling and structure-preserving presolving. The implementation is based on the solver PIPS-IPM. We evaluate the computational performance on energy system models with up to four billion nonzero entries in the constraint matrix—and up to one billion columns and one billion rows. This article mainly concentrates on the energy system model ELMOD, which is a linear optimization model representing the European electricity markets by the use of a nodal pricing market-clearing. It has been widely applied in the literature on energy system analyses in recent years. However, it will be demonstrated that the new solver is also applicable to other energy system models.

Ensemble Kalman Filter Updates Based on Regularized Sparse Inverse Cholesky Factors

AbstractThe ensemble Kalman filter (EnKF) is a popular technique for data assimilation in high-dimensional nonlinear state-space models. The EnKF represents distributions of interest by an ensemble, which is a form of dimension reduction that enables straightforward forecasting even for complicated and expensive evolution operators. However, the EnKF update step involves estimation of the forecast covariance matrix based on the (often small) ensemble, which requires regularization. Many existing regularization techniques rely on spatial localization, which may ignore long-range dependence. Instead, our proposed approach assumes a sparse Cholesky factor of the inverse covariance matrix, and the nonzero Cholesky entries are further regularized. The resulting method is highly flexible and computationally scalable. In our numerical experiments, our approach was more accurate and less sensitive to misspecification of tuning parameters than tapering-based localization.

Lower bounds for the bandwidth problem

The bandwidth problem seeks for a simultaneous permutation of the rows and columns of the adjacency matrix of a graph such that all nonzero entries are as close as possible to the main diagonal. This work focuses on investigating novel approaches to obtain lower bounds for the bandwidth problem. In particular, we use vertex partitions to bound the bandwidth of a graph. Our approach contains prior approaches for bounding the bandwidth as special cases. By varying sizes of partitions, we achieve a trade-off between quality of bounds and efficiency of computing them. To compute lower bounds, we derive a Semidefinite Programming relaxation. We evaluate the performance of our approach on several data sets, including real-world instances.

An iterative detector based on sparse bayesian error recovery for uplink large-scale MIMO systems

In uplink large-scale multiple-input and multiple-output (MIMO) systems, the data detection at the receiver is the major challenge due to the considerable increase in dimensions of MIMO systems. In this paper, a Bayesian strategy is investigated for MIMO systems. The main idea of this algorithm is to improve the performance of a detector by finding the incorrectly detected symbols relying on the sparse property of the estimation error. In the proposed method, the conventional massive MIMO model is converted into a sparse model by inducing a sparse constrain on the symbol error vector obtained from a linear detector (minimum mean square error or zero forcing detectors). Then, by recovering the non-zero entries of error (i.e. the incorrectly detected symbols) via the sparse Bayesian approach, the primary estimate of the information vector is corrected. Numerical results reveal that exploiting the sparse characteristic of the estimation error leads to improvement in detection performance and the proposed Bayesian approach achieves better results than conventional and previous state of the art error recovery methods. Meanwhile, computational complexity of the sparse Bayesian error recovery (SBER) algorithm is the same as that of the linear detectors.

Efficient Tuning-Free l1-Regression of Nonnegative Compressible Signals

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements with the general assumption that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the ℓ2-norm, basis pursuit denoising obeys stable and robust recovery guarantees. In the case of unknown noise levels, nonnegative least squares recovers non-negative signals if the measurement matrix fulfills an additional property (sometimes called the M+-criterion). However, if the measurement matrix is the biadjacency matrix of a random left regular bipartite graph it obeys with a high probability the null space property with respect to the ℓ1-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD), which is free of tuning parameters. For these measurement matrices, we prove a uniform, stable and robust recovery guarantee. Such guarantees are important, since binary expander matrices are sparse and thus allow for fast sketching and recovery. We will further present a method to solve the NNLAD numerically and show that this is comparable to state of the art methods. Lastly, we explain how the NNLAD can be used for viral detection in the recent COVID-19 crisis.

A fast multiscale Galerkin method for solving a boundary integral equation in a domain with corners

A fast multiscale Galerkin method is proposed for solving the boundary integral equation derived from the Dirichlet problem of the Laplace equation in a domain with corners. It is well known that the integral operator in the equation can be split into two operators, one is noncompact, the other is compact. We design two truncation strategies for the representation matrices of these operators, respectively, which compress these two dense matrices to sparse ones having only 𝒪(2n) number of nonzero entries, where 2n is the number of the wavelet basis functions used in the method. We prove that the proposed truncation strategies do not ruin the stability and convergence rate of the integral equation. Numerical experiments are presented to verify the theoretical results and demonstrate the effectiveness of the method.

Hierarchical Negative Binomial Factorization for Recommender Systems on Implicit Feedback

When exposed to an item in a recommender system, a user may consume it (known as success exposure) or neglect it (known as failure exposure). The recently proposed methods that consider both success and failure exposure merely regard failure exposure as a constant prior, thus being capable of neither modeling various user behavior nor adapting to overdispersed data. In this paper, we propose a novel model, hierarchical negative binomial factorization, which models data dispersion via a hierarchical Bayesian structure, thus alleviating the effect of data overdispersion to help with performance gain for recommendation. Moreover, we factorize the dispersion of zero entries approximately into two low-rank matrices, thus reducing the updating time linear to the number of nonzero entries. The experiment shows that the proposed model outperforms state-of-the-art Poisson-based methods merely with a slight loss of inference speed.

Approximate Multiplication of Sparse Matrices with Limited Space

Approximate matrix multiplication with limited space has received ever-increasing attention due to the emergence of large-scale applications. Recently, based on a popular matrix sketching algorithm---frequent directions, previous work has introduced co-occuring directions (COD) to reduce the approximation error for this problem. Although it enjoys the space complexity of O((m_x+m_y)l) for two input matrices X∈ℝ^{m_x ╳ n} and Y∈ℝ^{m_y ╳ n} where l is the sketch size, its time complexity is O(n(m_x+m_y+l)l), which is still very high for large input matrices. In this paper, we propose to reduce the time complexity by exploiting the sparsity of the input matrices. The key idea is to employ an approximate singular value decomposition (SVD) method which can utilize the sparsity, to reduce the number of QR decompositions required by COD. In this way, we develop sparse co-occuring directions, which reduces the time complexity to Õ((nnz(X)+nnz(Y))l+nl^2) in expectation while keeps the same space complexity as O((m_x+m_y)l), where nnz(X) denotes the number of non-zero entries in X and the Õ notation hides constant factors as well as polylogarithmic factors. Theoretical analysis reveals that the approximation error of our algorithm is almost the same as that of COD. Furthermore, we empirically verify the efficiency and effectiveness of our algorithm.

Ordered multiplicity inverse eigenvalue problem for graphs on six vertices

For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ is governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\min(m,n)\ge 3$ attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum.

Threshold function of ray nonsingularity for uniformly random ray pattern matrices

A complex square matrix is a ray pattern matrix if each of its nonzero entries has modulus 1. A ray pattern matrix naturally corresponds to a weighted-digraph. A ray pattern matrix A is ray nonsingular if for each entry-wise positive matrix K, is nonsingular. A random model of ray pattern matrices with order n is introduced, where a uniformly random ray pattern matrix B is defined to be the adjacency matrix of a simple random digraph whose arcs are weighted with i.i.d. random variables uniformly distributed over the unit circle in the complex plane. It is shown that is a threshold function for the random matrix M = I−B to be ray nonsingular, which is the same as the threshold function of the appearance of giant strong components in .

Sparse representation of vectors in lattices and semigroups

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the ell _0-norm of the vector. Our main results are new improved bounds on the minimal ell _0-norm of solutions to systems Avarvec{x}=varvec{b}, where Ain mathbb {Z}^{mtimes n}, {varvec{b}}in mathbb {Z}^m and varvec{x} is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with ell _0-norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over mathbb {R}, to other subdomains such as mathbb {Z}. We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.

Design of Multi-Port With Desired Reference Impedances Using Y-Matrix and Matching Networks

Two synthesis algorithms of multiport network with desired unequal complex port-impedances are developed considering the multiport network as an interconnection of two-port networks (TPNs). The first one is direct-synthesis using Y-matrix, and the second one is port-impedance transformation method. The techniques are single-frequency methods, as the scattering parameters of known multiport networks are defined at the design frequency. In direct-synthesis, the desired S- matrix of the multiport network is converted into the desired Y-matrix, considering the reference port-impedance matrix. Then the multiport network structure or topology can be predicted using the non-zero off-diagonal entries of Y-matrix. The Y-matrix of the predicted multiport is calculated in terms of unknown design parameters or sub-network parameters. Finally, by equating the predicted Y-matrix with the desired Y- matrix, the design equations are obtained. In the port impedance transformation method, a core network with intermediate port-impedances is designed using the direct-synthesis method. Then the intermediate port-impedances are converted into desired port impedances using external matching networks. The impedance matching networks are designed using the desired phased impedance matching theory. Two different designs of a four-port network are provided to validate the proposed concepts. Two pseudocodes or algorithms are developed considering lumped Π networks as building blocks.

Modulated Sparse Superposition Codes for the Complex AWGN Channel

This paper studies a generalization of sparse superposition codes (SPARCs) for communication over the complex additive white Gaussian noise (AWGN) channel. In a SPARC, the codebook is defined in terms of a design matrix, and each codeword is a generated by multiplying the design matrix with a sparse message vector. In the standard SPARC construction, information is encoded in the locations of the non-zero entries of the message vector. In this paper we generalize the construction and consider modulated SPARCs, where information in encoded in both the locations and the values of the non-zero entries of the message vector. We focus on the case where the non-zero entries take values from a phase-shift keying (PSK) constellation. We propose a computationally efficient approximate message passing (AMP) decoder, and obtain analytical bounds on the state evolution parameters which predict the error performance of the decoder. Using these bounds we show that PSK-modulated SPARCs are asymptotically capacity achieving for the complex AWGN channel, with either spatial coupling or power allocation. We also provide numerical simulation results to demonstrate the error performance at finite code lengths. These results show that introducing modulation to the SPARC design can significantly reduce decoding complexity without sacrificing error performance.

Strong structural controllability of colored structured systems

This paper deals with strong structural controllability of linear structured systems in which the system matrices are given by zero/nonzero/arbitrary pattern matrices. Instead of assuming that the nonzero and arbitrary entries of the system matrices can take their values completely independently, this paper allows equality constraints on these entries, in the sense that a priori given entries in the system matrices are restricted to take arbitrary but identical values. To formalize this general class of structured systems, we introduce the concepts of colored pattern matrices and colored structured systems. The main contribution of this paper is that it generalizes both the classical results on strong structural controllability of structured systems as well as recent results on controllability of systems defined on colored graphs. In this paper we will establish both algebraic and graph-theoretic conditions for strong structural controllability of this more general class of structured systems.

A fast Fourier-Galerkin method solving boundary integral equations for the Helmholtz equation with exponential convergence

A boundary integral equation in general form will be considered, which can be used to solve Dirichlet problems for the Helmholtz equation. The goal of this paper is to develop a fast Fourier-Galerkin method solving these boundary integral equations. To this aim, a scheme for splitting integral operators is presented, which splits the corresponding integral operator into a convolution operator and a compact operator. A truncation strategy is presented, which can compress the dense coefficient matrix to a sparse one having only $\mathcal {O}(n)$ nonzero entries, where n is the order of the Fourier basis functions used in the method. The proposed fast method preserves the stability and optimal convergence order. Moreover, exponential convergence can be obtained under suitable assumptions. Numerical examples are presented to confirm the theoretical results for the approximation accuracy and computational complexity of the proposed method.

Deterministic radiative transfer equation solver on unstructured tetrahedral meshes: Efficient assembly and preconditioning

Due to its integro-differential nature, deriving schemes for numerically solving the radiative transfer equation (RTE) is challenging. Most solvers are efficient in specific scenarios: structured grids, simulations with low-scattering materials... In this paper, a full solver, from the discretization of the steady-state monochromatic RTE to the solution of the resulting system, is derived.Using a mixed matrix-ready/matrix-free approach, our solver is able to discretize and solve a 45.7 billion unknown problem on 27 thousand processes in three minutes for a full physics involving scattering, absorption, and reflection. Because of the high dimensionality of the continuous equation, the linear system would have had more than 6×1015 nonzero entries if assembled explicitly. Our approach allows for large memory gains by only storing lower dimension reference matrices.The finite element-based solver is wrapped around open-source software, FreeFEM for discretization, PETSc for linear algebra, and hypre for the algebraic multigrid infrastructure. Overall, deterministic results are presented on arbitrarily-decomposed unstructured grids for radiative transfer problems with scattering, absorbing, and reflecting heterogeneities on up to 27 thousand processes.