Poisson Matrix Research Articles

Empirical Bayes Poisson matrix completion

An empirical Bayes method for the Poisson matrix denoising and completion problems is proposed, and a corresponding algorithm called EBPM (Empirical Bayes Poisson Matrix) is developed. This approach is motivated by the non-central singular value shrinkage prior, which was used for the estimation of the mean matrix parameter of a matrix-variate normal distribution. Numerical experiments show that the EBPM algorithm outperforms the common nuclear norm penalized method in both matrix denoising and completion. The EBPM algorithm is highly efficient and does not require heuristic parameter tuning, as opposed to the nuclear norm penalized method, in which the regularization parameter should be selected. The EBPM algorithm also performs better than others in real-data applications.

Fixing Inventory Inaccuracies at Scale

Problem definition: Inaccurate records of inventory occur frequently and, by some measures, cost retailers approximately 4% in annual sales. Detecting inventory inaccuracies manually is cost-prohibitive, and existing algorithmic solutions rely almost exclusively on learning from longitudinal data, which is insufficient in the dynamic environment induced by modern retail operations. Instead, we propose a solution based on cross-sectional data over stores and stock-keeping units (SKUs), viewing inventory inaccuracies as a problem of identifying anomalies in a (low-rank) Poisson matrix. State-of-the-art approaches to anomaly detection in low-rank matrices apparently fall short. Specifically, from a theoretical perspective, recovery guarantees for these approaches require that nonanomalous entries be observed with vanishingly small noise (which is not the case in our problem and, indeed, in many applications). Methodology/results: So motivated, we propose a conceptually simple entrywise approach to anomaly detection in low-rank Poisson matrices. Our approach accommodates a general class of probabilistic anomaly models. We show that the cost incurred by our algorithm approaches that of an optimal algorithm at a min-max optimal rate. Using synthetic data and real data from a consumer goods retailer, we show that our approach provides up to a 10× cost reduction over incumbent approaches to anomaly detection. Along the way, we build on recent work that seeks entrywise error guarantees for matrix completion, establishing such guarantees for subexponential matrices, a result of independent interest. Managerial implications: By utilizing cross-sectional data at scale, our novel approach provides a practical solution to the issue of inventory inaccuracies in retail operations. Our method is cost-effective and can help managers detect inventory inaccuracies quickly, leading to increased sales and improved customer satisfaction. In addition, the entrywise error guarantees that we establish are of interest to academics working on matrix-completion problems. History: This paper was selected for Fast Track in M&SOM from the 2022 MSOM Supply Chain Management SIG Conference. Funding: Financial support from the National Science Foundation Division of Civil, Mechanical, and Manufacturing Innovation [Grant CMMI 1727239] is gratefully acknowledged. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0146 .

Reduced-order modeling for Ablowitz–Ladik equation

In this paper, reduced-order models (ROMs) are constructed for the Ablowitz–Ladik equation (ALE), an integrable semi-discretization of the nonlinear Schrödinger equation (NLSE) with and without damping. Both ALEs are non-canonical conservative and dissipative Hamiltonian systems with the Poisson matrix depending quadratically on the state variables, and with quadratic Hamiltonian. The full-order solutions are obtained with the energy preserving midpoint rule for the conservative ALE and exponential midpoint rule for the dissipative ALE. The reduced-order solutions are constructed intrusively by preserving the skew-symmetric structure of the reduced non-canonical Hamiltonian system by applying proper orthogonal decomposition (POD) with the Galerkin projection. For an efficient offline–online decomposition of the ROMs, the quadratic nonlinear terms of the Poisson matrix are approximated by the discrete empirical interpolation method (DEIM). The computation of the reduced-order solutions is further accelerated by the use of tensor techniques. Preservation of the Hamiltonian and momentum for the conservative ALE, and preservation of dissipation properties of the dissipative ALE, guarantee the long-term stability of soliton solutions.

Graph link prediction in computer networks using Poisson matrix factorisation

Graph link prediction is an important task in cybersecurity: relationships between entities within a computer network, such as users interacting with computers or system libraries and the corresponding processes that use them, can provide key insights into adversary behaviour. Poisson matrix factorisation (PMF) is a popular model for link prediction in large networks, particularly useful for its scalability. In this article PMF is extended to include scenarios that are commonly encountered in cybersecurity applications. Specifically, an extension is proposed to explicitly handle binary adjacency matrices and include known categorical covariates associated with the graph nodes. A seasonal PMF model is also presented to handle seasonal networks. To allow the methods to scale to large graphs, variational methods are discussed for performing fast inference. The results show an improved performance over the standard PMF model and other statistical network models.

A Bayesian model of dose-response for cancer drug studies

Exploratory cancer drug studies test multiple tumor cell lines against multiple candidate drugs. The goal in each paired (cell line, drug) experiment is to map out the dose-response curve of the cell line as the dose level of the drug increases. We propose Bayesian tensor filtering (BTF), a hierarchical Bayesian model for dose-response modeling in multisample, multitreatment cancer drug studies. BTF uses low-dimensional embeddings to share statistical strength between similar drugs and similar cell lines. Structured shrinkage priors in BTF encourage smoothness in the dose-response curves while remaining adaptive to sharp jumps when the data call for it. We focus on a pair of cancer drug studies exhibiting a particular pathology in their experimental design, leading us to a nonconjugate monotone mixture-of-gammas likelihood. To perform posterior inference, we develop a variant of the elliptical slice sampling algorithm for sampling from linearly-constrained multivariate normal priors with nonconjugate likelihoods. In benchmarks, BTF outperforms state-of-the-art methods for covariance regression and dynamic Poisson matrix factorization. On the two cancer drug studies, BTF outperforms the current standard approach in biology and reveals potential new biomarkers of drug sensitivity in cancer. Code is available at https://github.com/tansey/functionalmf.

Energy preserving reduced-order modeling of the rotating thermal shallow water equation

In this paper, reduced-order models (ROMs) are developed for the rotating thermal shallow water equation (RTSWE) in the non-canonical Hamiltonian form with state-dependent Poisson matrix. The high fidelity full solutions are obtained by discretizing the RTSWE in space with skew-symmetric finite-differences, while preserving the Hamiltonian structure. The resulting skew-gradient system is integrated in time by the energy preserving average vector field (AVF) method. The ROM is constructed by applying proper orthogonal decomposition with the Galerkin projection, preserving the reduced skew-gradient structure, and integrating in time with the AVF method. The nonlinear terms of the Poisson matrix and Hamiltonian are approximated with the discrete empirical interpolation method to reduce the computational cost. The solutions of the resulting linear-quadratic reduced system are accelerated by the use of tensor techniques. The accuracy and computational efficiency of the ROMs are demonstrated for a numerical test problem. Preservation of the energy (Hamiltonian) and other conserved quantities, i.e., mass, buoyancy, and total vorticity, show that the reduced-order solutions ensure the long-term stability of the solutions while exhibiting several orders of magnitude computational speedup over the full-order model. Furthermore, we show that the ROMs are able to accurately predict the test and training data and capture the system behavior in the prediction phase.

Immersed boundary simulations of flows driven by moving thin membranes

Immersed boundary methods are extensively used for simulations of dynamic solid objects interacting with fluids due to their computational efficiency and modeling flexibility compared to body-fitted grid methods. However, thin geometries, such as shells and membranes, cause a violation of the boundary conditions across the surface for many immersed boundary projection algorithms. Using a one-dimensional analytical derivation and multi-dimensional numerical simulations, this manuscript shows that adjustment of the Poisson matrix itself is require to avoid large velocity, pressure, and force prediction errors when the pressure jump across the interface is substantial and that these errors increase with Reynolds number. A new minimal thickness modification is developed for the Boundary Data Immersion Method (BDIM-σ), which avoids these issues while still enabling the use of efficient projection algorithms for high-speed immersed surface simulations.

Thermodynamically admissible 13-moment equations

Grad's 13-moment equations describe transport in mildly rarefied gases well, but are not properly embedded into nonequilibrium thermodynamics since they are not accompanied by a formulation of the second law. In this work, the Grad-13 equations are embedded into the framework of GENERIC (general equation for the nonequilibrium reversible–irreversible coupling), which demands additional contributions in the equations to guarantee thermodynamic structure. As GENERIC building blocks, we use a Poisson matrix for the basic convection behavior and antisymmetric friction matrices to correct for additional convective transport terms. The ensuing GENERIC-13 equations completely match the Grad-13 equations up to second-order terms in the Knudsen number and fulfill all thermodynamic requirements.

Bayesian Poisson Factorization With Side Information for User Interest Prediction in Hierarchical Edge-Caching Systems

Edge-caching is an effective solution to cope with the unprecedented data traffic growth by storing contents in the vicinity of end-users. In this paper, we formulate a hierarchical caching policy where the end-users and cellular base station (BS) are equipped with limited cache capacity with the objective of minimizing the total data traffic load in the network. The caching policy is a nonlinear combinatorial programming problem and difficult to solve. To tackle the issue, we design a heuristic algorithm as an approximate solution which can be solved efficiently. Moreover, to proactively serve the users, it is of high importance to extract useful information from data requests and predict user interest about contents. In practice, the data often contain <i>implicit feedback</i> from users which is quite noisy and complicates the reliable prediction of user interest. In this regard, we introduce a Bayesian Poisson matrix factorization model which utilizes the available side information about contents to effectively filter out the noise in the data and provide accurate prediction. Subsequently, we design an efficient Markov chain Monte Carlo (MCMC) method to perform the posterior approximation. Finally, a real-world dataset is applied to the proposed proactive caching-prediction scheme and our results show significant improvement over several commonly-used methods. For example, when the BS and the users have caches with storage of 25&#x0025; and 10&#x0025; of the total contents size respectively, our approach yields around 8&#x0025; improvement with respect to the state-of-the-art approach in terms of caching performance.

The discrepancy of random rectangular matrices

AbstractA recent approach to the Beck–Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models: matrices with Bernoulli or Poisson entries. For Poisson matrices, we further characterize the discrepancy for any rectangular aspect ratio. These results give sharp answers to questions of Hoberg and Rothvoß (SODA 2019) and Franks and Saks (Random Structures Algorithms2020). Our main tool is a conditional second moment method combined with Stein's method of exchangeable pairs. While previous approaches are limited to dense matrices, our techniques allow us to work with matrices of all densities. This may be of independent interest for other sparse random constraint satisfaction problems.

Geometric Particle-in-Cell Simulations of the Vlasov--Maxwell System in Curvilinear Coordinates

Numerical schemes that preserve the structure of the kinetic equations can provide stable simulation results over a long time. An electromagnetic particle-in-cell solver for the Vlasov--Maxwell equations that preserves at the discrete level the noncanonical Hamiltonian structure of the Vlasov--Maxwell equations has been presented in [Kraus et al., J. Plasma Phys., 83 (2017)]. While the original formulation has been obtained for Cartesian coordinates, we extend the formulation to curvilinear coordinates in this paper. For the discretization in time, we discuss several (semi-)implicit methods either based on a Hamiltonian splitting or a discrete gradient method combined with an antisymmetric splitting of the Poisson matrix and discuss their conservation properties and computational efficiency.

On the Entire Paranormed Triple Sequence Spaces of Defined by Binomial Poisson Matrix

In this paper the entire triple sequence space are the generalization of the classical Maddox's paranormed sequence space have been introduced and investigated some topological properties of entire triple sequence space of binomial Poisson matrix of and

Energy-conserving time propagation for a structure-preserving particle-in-cell Vlasov–Maxwell solver

This paper discusses energy-conserving time-discretizations for finite element particle-in-cell discretizations of the Vlasov–Maxwell system. A geometric spatially discrete system can be obtained using a standard particle-in-cell discretization of the particle distribution and compatible finite element spaces for the fields to discretize the Poisson bracket of the Vlasov–Maxwell model (see Kraus et al. (2017) [1]). In this paper, we derive energy-conserving time-discretizations based on the discrete gradient method applied to an antisymmetric splitting of the Poisson matrix. Firstly, we propose a semi-implicit method based on a splitting that yields constant Poisson matrices in each substep. Moreover, we devise an alternative discrete gradient that yields a time discretization that can additionally conserve Gauss' law. Finally, we explain how substepping for fast species dynamics can be incorporated.

Answer Keyword Generation for Community Question Answering by Multiaspect Gamma–Poisson Matrix Completion

Community question answering (CQA) recommends appropriate answers to existing and new questions. Such answer recommendation is challenging since CQA data are often sparse and decentralized, and lacks sufficient information to generate suitable answers to existing questions. Matching answers to new questions is more challenging in modeling Q/A sparsity, generating answers to cold-start/novel questions, and integrating metadata about Q/A into models, etc. This article addresses these issues by a novel statistical model to automatically generate answer keywords in CQA with multiaspect Gamma-Poisson matrix completion (MAGIC). MAGIC is the first trial in CQA to model multiple aspects of Q/A sentence information in CQA by involving Q/A metadata, Q/A sparsity, and both lexical and semantic Q/A information in a hierarchical Gamma-Poisson model. MAGIC can efficiently generate answer keywords for both existing and new questions against nonnegative matrix factorization (MF), probability MF, and relevant Poisson factorization models w.r.t. recommending appropriate and informative answer keywords.

Degenerate Laplacian describing topologically constrained diffusion: helicity constraint as an alternative to ellipticity

We study a second-order non-elliptic partial differential operator that generalizes the classical Laplacian with a degenerate coefficient matrix. The theory is motivated by the statistical mechanics of topologically constrained particles. In the context of diffusion models, the coefficient matrix is given by , with a generalized Poisson matrix that dictates particle dynamics. When is the symplectic matrix of canonical Hamiltonian systems, the operator reduces to the classical Laplacian. However, topological constraints make the matrix degenerate. The ellipticity is broken in the direction parallel to the nullity of . We call such a diffusion operator an orthogonal Laplacian and denote it by . When the nullity foliates the space, reduces into a classical Laplacian on the leaf that is orthogonal to the null space. However, when the nullity of is not integrable (which is the case when has helicity), is akin to elliptic operators having some useful properties of convex analysis. We show that defines an inner product of a Sobolev-like Hilbert space, and is a coercive functional in the L2 topology. Constructing a trace theorem, we discuss a boundary value problem (the orthogonal Poisson equation) for the orthogonal Laplacian, and obtain a unique weak solution by application of Riesz’s representation theorem.

Nambu–Poisson gauge theory

We generalize noncommutative gauge theory using Nambu–Poisson structures to obtain a new type of gauge theory with higher brackets and gauge fields. The approach is based on covariant coordinates and higher versions of the Seiberg–Witten map. We construct a covariant Nambu–Poisson gauge theory action, give its first order expansion in the Nambu–Poisson tensor and relate it to a Nambu–Poisson matrix model.

A systematic derivation of Green's and Poisson's matrices for boundary-value problems in elastostatic half-spaces

Green's and Poisson's matrices in an n-dimensional half-space are computed for the system of linear isotropic elasticity and four types of boundary conditions attacked by Boussinesq and Cerruti by means of potential-theoretic methods and reconsidered by Sneddon (1993). This involves the solution of convolution equations and the computation of various convolution products by means of distribution theory. The question of uniqueness is also addressed.

On a partial integral which can be derived from poisson matrix

Consider a surface which is a common level of some functions. Suppose that this surface is invariant under a Hamiltonian system. The question is if a partial integral can be derived explicitly from the Poisson matrix of these functions. In some cases such an integral is equal to the determinant of the matrix. This paper establishes a necessary and sufficient condition for this to hold true. The partial integral that results is not trivial if the induced Poisson structure is non-degenerate at one point at least. Therefore, the invariant surface must be even-dimensional.

Об одном частном интеграле, который можно извлечь из матрицы Пуассона

Об одном частном интеграле, который можно извлечь из матрицы Пуассона

Boussinesq–Cerruti problems for the hexagonally symmetric system of elasticity in n–dimensional half-space

The Poisson matrices of the analoga to the Boussinesq–Cerruti boundary value problems for the operator of transversely isotropic elastostatics in n–dimensional half-space are computed by Fourier transformation and given in explicit form.