Sparse Systems Research Articles

Polyhedral homotopies in Cox coordinates

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety [Formula: see text]. The algorithm lends its name from a construction, described by Cox, of [Formula: see text] as a GIT quotient [Formula: see text] of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space [Formula: see text] of [Formula: see text] and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of [Formula: see text]. It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the [Formula: see text]-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of [Formula: see text]. In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound.

Sparse echo cancellation using variants of least mean fourth and least mean square algorithms

Echo cancellation is the most essential and indispensable component of telephone networks. The impulse responses of most of the networks are sparse in nature; that is, the impulse response has a small percentage of its components with a significant magnitude (large energy), while the rest are zero or small. In these sparse environments, conventional adaptive algorithms like least mean square (LMS) and normalized LMS (NLMS) show substandard and inferior performances. In this paper, the performances of the normalized least mean square (NLMS) algorithm, the normalized least mean fourth (NLMF) and the proportionate normalized least mean fourth (PNLMF) are compared for sparse echo cancellation. The sparseness of both the echo response and the input signal is exploited in this algorithm to achieve improved results at a low computational cost. The PNLMF algorithm showed better results and faster convergence in sparse and non sparse systems, but its results in sparse environments are more impressive. The NLMF algorithm shows good results in sparse environments but not in non-sparse environments. The PNLMS algorithm can be considered superior to the NLMF and NLMS algorithms with respect to the error profile. A modified algorithm, the sparse controlled modified proportionate normalized LMF (SCMPNLMF) algorithm, is proposed, and its performances are compared with the other algorithms.

Sparse analytic systems

Abstract Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal {F}$ of (real or complex) analytic functions, such that $\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$ is countable for every x. We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim $ on $\mathbb {R}$ such that any ‘analytic-anonymous’ attempt to predict the map $x \mapsto [x]_\sim $ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman [2].

Study of curved structures health monitoring with flexible omnidirectional guided-wave transducers

In this paper, structure health monitoring (SHM) of curved plates with improved flexible transducers capable of generating and receiving omnidirectional guided waves is studied. The improved thin and flexible transducer consists of a dual-layer ring coil and a thin magnetostrictive patch is firstly proposed to generate and receive a single-mode and almost non-dispersive guided wave omnidirectionally in curved plates. A finite element simulation is then used to investigate the design of the transducer. The flexible omnidirectional guided-wave (GW) transducers made with flexible printed circuits (FPC) and handwound coils, respectively, are measured and compared by experiment. Finally, a sparse array system based on the proposed flexible omnidirectional GW transducer is developed for the SHM of the curved plate. The influence of the curvature of the plate on the performance of the developed transducer is investigated. A post-processing method based on full matrix capture (FMC) and total focusing method (TFM) with the sparse array of flexible omnidirectional GW transducers is applied to inspect and image the defect in the curved plate. • A new flexible GW transducer can generate and receive a single-mode and almost non-dispersive guided wave omnidirectionally in curved plates. • The O-GW MPT is light in weight, small in size, and flexible in structure. • The experiment results show that the curvature and deformation of the plate almost do not influence the performance of the flexible O-GW transducer. • Combined with the post-processing method, the omnidirectional transducer can apply to inspect and image the defect in the curved plate.

"Testing one or multiple: How beliefs about sparsity affect causal experimentation": Correction to Coenen et al. (2019).

Reports an error in "Testing one or multiple: How beliefs about sparsity affect causal experimentation" by Anna Coenen, Azzurra Ruggeri, Neil R. Bramley and Todd M. Gureckis (Journal of Experimental Psychology: Learning, Memory, and Cognition, 2019[Nov], Vol 45[11], 1923-1941). In the article, there were errors in Equations 2 through 5. In Equation 2, SE(H ls, o) should have been SE(Hls, o = j). In Equation 3, the Sum should have been from i = 1 to m, and the log should not have been italicized. In Equation 4, the denominator of the fraction on the right hand side should have been\∑ {j=1} m P(o lCj ) P(Cj ). In Equation 5, the Sum should have been from i = 1 to m, and the log should not have been italicized. The online version of this article has been corrected. (The following abstract of the original article appeared in record 2019-27247-001.) What is the best way of discovering the underlying structure of a causal system composed of multiple variables? One prominent idea is that learners should manipulate each candidate variable in isolation to avoid confounds (sometimes known as the control of variables [CV] strategy). We demonstrate that CV is not always the most efficient method for learning. Using an optimal actor model, which aims to minimize the average number of tests, we show that when a causal system is sparse (i.e., when the outcome of interest has few or even just one actual cause among the candidate variables), it is more efficient to test multiple variables at once. Across a series of behavioral experiments, we then show that people are sensitive to causal sparsity and adapt their strategies accordingly. When interacting with a dense causal system (high proportion of actual causes among candidate variables), they use a CV strategy, changing one variable at a time. When interacting with a sparse causal system, they are more likely to test multiple variables at once. However, we also find that people sometimes use a CV strategy even when a system is sparse. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Exploring Sparse Visual Odometry Acceleration With High-Level Synthesis

Visual Odometry (VO) systems are widely used to determine the position and orientation of a robot or camera in an unknown environment. They are deployed on resource-constrained platforms, such as drones and Virtual Reality (VR) or Augmented Reality (AR) headsets. VO systems harnessing modern System-on-Chip (SoCs) with integrated Field Programmable Gate Array (FPGA) have the potential to improve the overall systems performance. This paper explores the FPGA acceleration of sparse VO kernels using High-level Synthesis (HLS) as this kind of VO system has been designed to use with low-power SoCs. We show that both computational and data transfer overheads between the processing cores of the CPU of the SoC and the accelerators on the FPGA need to be optimized to obtain better end-to-end performance. This is a result of the additional data movement incurred when using an FPGA accelerator and also because of the sparse computational nature with predictable or random memory access patterns of the kernels involved. However, state-of-the-art HLS tools are not yet able to perform the required optimizations automatically because they usually assume that the kernels to be accelerated have dense computational patterns with regular memory access. In this paper we propose three, potentially generic, methods to reduce the data transfer between the CPU and the customised hardware kernels on the FPGA; these methods are: (a) approximation based on domain-specific knowledge, (b) image compression, and (c) the use of on-the-fly computation. We present a case study of the use of these methods on SVO, a state-of-the-art sparse VO system with a semi-direct front-end. We demonstrate that our proposed methods can reduce data transfer overhead to achieve better end-to-end performance and that they can be applied not only when using standard Xilinx HLS tools but also with other state-of-the-art HLS tools, such as HeteroFlow. Compared to the baseline performance of the original SVO software on an Arm CPU, our proposed methods assist the HLS and HeteroFlow designs to achieve a speedup of 2.4x and 2.14x, respectively, without noticeable accuracy loss. The HLS and HeteroFlow designs also achieve a 1.85x and 1.89x, respectively, improvement in energy efficiency on the SoC system used. Compared to the SVO software baseline running on the Intel Xeon CPU, our proposed methods assist the HLS and HeteroFlow designs to achieve 8.2x and 8.3x improvement in energy efficiency, respectively.

Personalized Computational Causal Modeling of the Alzheimer Disease Biomarker Cascade.

Mathematical models of complex diseases, such as Alzheimer's disease, have the potential to play a significant role in personalized medicine. Specifically, models can be personalized by fitting parameters with individual data for the purpose of discovering primary underlying disease drivers, predicting natural history, and assessing the effects of theoretical interventions. Previous work in causal/mechanistic modeling of Alzheimer's Disease progression has modeled the disease at the cellular level and on a short time scale, such as minutes to hours. No previous studies have addressed mechanistic modeling on a personalized level using clinically validated biomarkers in individual subjects. This study aimed to investigate the feasibility of personalizing a causal model of Alzheimer's Disease progression using longitudinal biomarker data. We chose the Alzheimer Disease Biomarker Cascade model, a widely-referenced hypothetical model of Alzheimer's Disease based on the amyloid cascade hypothesis, which we had previously implemented mathematically as a mechanistic model. We used available longitudinal demographic and serial biomarker data in over 800 subjects across the cognitive spectrum from the Alzheimer's Disease Neuroimaging Initiative. The data included participants that were cognitively normal, had mild cognitive impairment, or were diagnosed with dementia (probable Alzheimer's Disease). The model consisted of a sparse system of differential equations involving four measurable biomarkers based on cerebrospinal fluid proteins, imaging, and cognitive testing data. Personalization of the Alzheimer Disease Biomarker Cascade model with individual serial biomarker data yielded fourteen personalized parameters in each subject reflecting physiologically meaningful characteristics. These included growth rates, latency values, and carrying capacities of the various biomarkers, most of which demonstrated significant differences across clinical diagnostic groups. The model fits to training data across the entire cohort had a root mean squared error (RMSE) of 0.09 (SD 0.081) on a variable scale between zero and one, and were robust, with over 90% of subjects showing an RMSE of < 0.2. Similarly, in a subset of subjects with data on all four biomarkers in at least one test set, performance was high on the test sets, with a mean RMSE of 0.15 (SD 0.117), with 80% of subjects demonstrating an RMSE < 0.2 in the estimation of future biomarker points. Cluster analysis of parameters revealed two distinct endophenotypic groups, with distinct biomarker profiles and disease trajectories. Results support the feasibility of personalizing mechanistic models based on individual biomarker trajectories and suggest that this approach may be useful for reclassifying subjects on the Alzheimer's clinical spectrum. This computational modeling approach is not limited to the Alzheimer Disease Biomarker Cascade hypothesis, and can be applied to any mechanistic hypothesis of disease progression in the Alzheimer's field that can be monitored with biomarkers. Thus, it offers a computational platform to compare and validate various disease hypotheses, personalize individual biomarker trajectories and predict individual response to theoretical prevention and therapeutic intervention strategies.

Efficient Linearization of Explicit Multilinear Systems using Normalized Decomposed Tensors

Multilinear systems allow multiplications of states, inputs, and states with inputs, in all possible combinations. Recently, a new normalized decomposed tensor format of explicit multilinear models was introduced. This paper presents a linearization method for the normalized canonical polyadic decomposed tensor format of explicit multilinear models. The proposed method computes the Jacobian matrix to obtain the linear system evaluated at the equilibrium point. An adaption for large-scale sparse systems is outlined. Performance and computational time are evaluated for different number of states and sparsity structures. The results suggest computational advantages of the explicit multilinear format compared to the non-normalized one. The adaptation to large-scale sparse systems shows clear computational advantage.

Generalized Soft-Root-Sign Based Robust Sparsity-Aware Adaptive Filters

Robust adaptive filters utilizing hyperbolic cosine and correntropy functions have been successfully employed in non-Gaussian noisy environments. However, these filters suffer from high steady-state misalignment due to significant weight update in the presences of outliers. In addition, several practical systems exhibit sparse characteristics, which is not taken into account by these filters. In this paper, a generalized soft-root-sign (GSRS) function is proposed and the corresponding GSRS adaptive filter is designed. The proposed GSRS provides negligible weight update in the occurrence of large outliers and thereby results in lower steady-state misalignment. To further improve modelling performance for sparse systems and to achieve robustness, sparsity-aware GSRS algorithms are also developed in this paper. The bound on learning rate and the computational complexity of proposed algorithm is also investigated. Simulation studies confirmed the improved convergence characteristics achieved by the proposed algorithms over existing algorithms.

Generalized Modified Blake–Zisserman Robust Sparse Adaptive Filters

In the past years, the generalized maximum correntropy criterion (GMCC) has been widely used in adaptive filters to provide robust behavior under non-Gaussian/impulsive noise environments. However, GMCC-based adaptive filters are affected by high steady-state misalignment. In order to enhance the robustness under non-Gaussian noise environments and reduce steady-state misalignment, a generalized modified Blake–Zisserman (GMBZ) robust loss function is introduced in this correspondence. Furthermore, a GMBZ adaptive filter (GMBZ-AF) has been developed that provides improved convergence performance over other existing algorithms. The proposed learning scheme has a computational complexity very similar to that of the GMCC-based adaptive filtering method. In order to further exploit the sparse nature of the system for identifying sparse systems and simultaneously provide robust convergence, two new robust sparse adaptive filters: 1) zero attracting GMBZ-AF (ZA-GMBZ-AF) and 2) reweighted ZA-GMBZ-AF (RZA-GMBZ-AF) have also been proposed. To further enhance the filter convergence performance, a new robust and sparsity-aware loss function called generalized modified dual Blake–Zisserman (GMDBZ) is also introduced in this correspondence and the corresponding GMDBZ adaptive filter (GMDBZ-AF) has been developed.

Sparsity-Promoting Affine Projection Algorithm With Periodically-Updated Gain Matrix and Its Performance Analysis

Sparse system identification is often encountered in applications such as network and acoustic echo cancellation. This work applies the sparsity promoting method to the affine projection algorithm (APA) to develop a sparsity-promoting APA (SAPA). To reduce its computational complexity, the gain matrix of SAPA is periodically updated, which leads to a periodically updated gain matrix based SAPA (PSAPA). In addition, the steady-state and tracking performance of PSAPA is analyzed using an improved weighted energy conservation method, which takes account of the correlation between the adaptive filter weight vector and the noise vector. The results of performance analysis are also suited to other proportionate APAs (PAPAs). Simulation results verify the advantages of the proposed algorithms and show that the theoretical expressions on the steady-state and tracking performance can predict the stochastic behaviors well.

Использование метода декомпозиции области для распараллеливания моделирования течения вязкой несжимаемой среды методом LS-STAG и дополнительного предобуславливания

As a rule, the main part of the computational costs in the numerical solution of problems in continuum mechanics consists in the solving of large sparse systems of linear algebraic equations. For this reason, efficient parallelization of this particular procedure can significantly speed up the simulation. To solve this problem, two main approaches can be used. The simplest approach consists in parallelizing of matrix-vector operations in a usual iterative solver. It requires several synchronization points and exchanges of coefficients at each iteration of the solver, which does not significantly speed up the simulation process as a whole. Therefore, domain decomposition methods are preferable. These methods involve dividing the computational domain into subdomains, constructing and solving separate problems in them, as well as some procedure to coordinate the solution between subdomains to ensure global convergence. Subdomains can overlap, as in the Schwartz method used in OpenFOAM, or they can be separated by interface sections, on which their own interface task is built, as in the Schur complement method. The latter method is used in this research to construct a parallel algorithm for viscous incompressible flow simulation by using the immersed boundary method LS-STAG with cut-cells and level-set functions. The resulting matrix of the interface system has a block tridiagonal structure. To speed up prototyping, OpenMP parallel programming technology is used in the software implementation of the developed algorithm, so computational experiments are carried out only on systems with shared memory, in particular on individual nodes of the educational and experimental cluster of the Applied Mathematics Department, Bauman Moscow State Technical University. To verify and evaluate the effectiveness of the developed parallel algorithm, a well-studied test problem about simulation of two-dimensional flow around a stationary circular airfoil is considered. Computations on a sequence of meshes with different numbers of subdomains show that the parallel algorithm allows one to obtain the same numerical solution as the original algorithm of the LS-STAG method, and the computed values of the Strouhal number and drag coefficient are in good agreement with the experimental and computational data known in the literature. Experiments demonstrate that the developed algorithm with domain decomposition allows to accelerate simulation even in sequential mode by reducing the number of solver iterations, i.e. the domain decomposition method acts as an additional preconditioner. Due to this property, the acceleration is superlinear when simulating in parallel mode with developed algorithm. This effect persists up to a certain number of subdomains, which depends on the size of the problem.

Dual parameters optimization [formula omitted]-LMS for estimating underwater acoustic channel with uncertain sparsity

The sparse norm constraint (l0,l1,l2 and lp) least mean square algorithm (LMS) is established technique for modeling sparse systems. However, when applied in target systems with uncertain sparsity, such as temporal-spatial-varying sparse underwater acoustic (UWA) channel, the parameters tuning (the step-size and parameter p) of lp-LMS faces significant challenges. In this paper, with the purpose to simplify the complicated dual-parameter selection problem via gradient strategy, a dual parameters optimization lp-LMS (DPO-lp-LMS) algorithm is derived by iteratively adjusting the step-size and the parameters p in parallel along the descent gradient. Convergence analysis of the proposed algorithm is given. A numerical simulation under varying sparsity systems exhibits that the proposed algorithm outperforms the lp-LMS algorithms driven by the existing optimization approaches in convergence speed and steady-state error. Meanwhile, a field shallow water experiment of UWA communication demonstrated that the proposed algorithm achieves superior performance under the framework of direct adaptation turbo equalization (DA-TEQ).

Solution of Large Sparse System of Linear Equations over GF(2) on a Multi Node Multi GPU Platform

We provide an efficient multi-node, multi-GPU implementation of the Block Wiedemann Algorithm (BWA)to find the solution of a large sparse system of linear equations over GF(2). One of the important applications ofsolving such systems arises in most integer factorization algorithms like Number Field Sieve. In this paper, wedescribe how hybrid parallelization can be adapted to speed up the most time-consuming sequence generation stage of BWA. This stage involves generating a sequence of matrix-matrix products and matrix transpose-matrix products where the matrices are very large, highly sparse, and have entries over GF(2). We describe a GPU-accelerated parallel method for the computation of these matrix-matrix products using techniques like row-wise parallel distribution of the first matrix over multi-node multi-GPU platform using MPI and CUDA and word-wise XORing of rows of the second matrix. We also describe the hybrid parallelization of matrix transpose-matrix product computation, where we divide both the matrices row-wise into equal-sized blocks using MPI. Then after a GPU-accelerated matrix transpose-matrix product generation, we combine all those blocks using MPI_BXOR operation in MPI_Reduce to obtain the result. The performance of hybrid parallelization of the sequence generation step on a hybrid cluster using multiple GPUs has been compared with parallelization on only multiple MPI processors. We have used this hybrid parallel sequence generation tool for the benchmarking of an HPC cluster. Detailed timings of the complete solution of number field sieve matrices of RSA-130, RSA-140, and RSA-170 are also compared in this paper using up to 4 NVidia V100 GPUs of a DGX station. We got a speedup of 2.8 after parallelization on 4 V100 GPUs compared to that over 1 GPU.

Piston Sensing for Golay-6 Sparse Aperture System with Double-Defocused Sharpness Metrics via ResNet-34

In pursuit of high imaging quality, optical sparse aperture systems must correct piston errors quickly within a small range. In this paper, we modified the existing deep-learning piston detection method for the Golay-6 array, by using a more powerful single convolutional neural network based on ResNet-34 for feature extraction; another fully connected layer was added, on the basis of this network, to obtain the best results. The Double-defocused Sharpness Metric (DSM) was selected first, as a feature vector to enhance the model performance; the average RMSE of the five sub-apertures for valid detection in our study was only 0.015λ (9 nm). This modified method has higher detecting precision, and requires fewer training datasets with less training time. Compared to the conventional approach, this technique is more suitable for the piston sensing of complex configurations.

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.

Improved NMR-data-compliant protein structure modeling captures context-dependent variations and expands the scope of functional inference.

Nuclear magnetic resonance (NMR) spectroscopy can reveal conformational states of a protein in physiological conditions. However, sparsely available NMR data for a protein with large degrees of freedom can introduce structural artifacts in the built models. Currently used state-of-the-art methods deriving protein structure and conformation from NMR deploy molecular dynamics (MD) coupled with simulated annealing for building models. We provide an alternate graph-based modeling approach, where we first build substructures from NMR-derived distance-geometry constraints combined in one shot to form the core structure. The remaining molecule with inadequate data is modeled using a hybrid approach respecting the observed distance-geometry constraints. One-shot structure building is rarely undertaken for large and sparse data systems, but our data-driven bottom-up approach makes this uniquely feasible by suitable partitioning of the problem. A detailed comparison of select models with state-of-art methods reveals differences in the secondary structure regions wherein the correctness of our models is confirmed by NMR data. Benchmarking of 106 protein-folds covering 38-282 length structures shows minimal experimental-constraint violations while conforming to other structure quality parameters such as the proper folding, steric clash, and torsion angle violation based on Ramachandran plot criteria. Comparative MD studies using select protein models from a state-of-art method and ours under identical experimental parameters reveal distinct conformational dynamics that could be attributed to protein structure-function. Our work is thus useful in building enhanced NMR-evidence-based models that encapsulate the contextual secondary and tertiary structure variations present during the experimentation and expand the scope of functional inference.

Evaluating tools for capture-recapture model selection to estimate the size of hidden populations: it works in practice, but does it work in theory?

Capture-recapture methods estimate the size of hidden populations by leveraging the proportion of overlap of the population on independent lists. Log-linear modeling relaxes the assumption of list independence, but best model selection criteria remain uncertain. Incorrect model selection can deliver incorrect and even implausible size estimates. We used simulations to model when capture-recapture methods deliver biased or unbiased estimates and compare model selection criteria. Simulations included five scenarios for list dependence among three incomplete lists of population of interest. We compared metrics of log-linear model selection, accuracy, and precision. We also compared log-linear model performance to the decomposable graph approach (a Bayesian model average), the sparse multiple systems estimation (SparseMSE) approach that accounts for zero or low cell counts, and the Sample Coverage approach. Log-linear models selected by Akaike's information criterion (AIC) calculated accurate population size estimates in all scenarios except for those with sparse or zero cell counts. In these scenarios, the decomposable graph and the Sample Coverage models produced more accurate size estimates. Conventional capture-recapture model selection fails with sparse cell counts. This naïve approach to model selection should be replaced with the implementation of multiple different models in order triangulate the truth in real-world applications.

A Polyhedral Homotopy Algorithm for Real Zeros

We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients satisfying certain concavity conditions. It operates entirely over the real numbers and tracks the optimal number of solution paths. In more technical terms; we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba.

A highly parallel fully implicit domain decomposition method for the simulation of the hemodynamics of a patient-specific artery at the full-body scale

The numerical simulation of blood flows in the human body with a certain level of clinical accuracy is important for the understanding of the human physiology. The success of the modeling relies on a robust numerical method with the corresponding software that can handle the complex geometry, the complex fluid flows and run efficiently on a supercomputer. In this work, we introduce a highly parallel domain decomposition method to solve the three-dimensional incompressible Navier-Stokes equations on a patient-specific artery at the full-body scale from neck to feet with 222 outlets and a minimum diameter around 1.0 mm. A locally refined, unstructured mesh is used to resolve the complex fluid flow. Moreover, a two-level method is introduced to determine the model parameters in the Windkessel outlet boundary condition to guarantee clinically correct flow distributions to 14 major regions. A fully implicit Newton-Krylov-Schwarz method is used to solve the nonlinear algebraic system at each time step and numerical experiments show that the proposed method is robust with respect to the complex geometry, the graph-based partition of the complex mesh, the ill-conditioned sparse systems with locally dense blocks, and different model parameters and is scalable with up to 15,360 processor cores. With the proposed method, one simulation of the blood flow in a full-body arterial network can be obtained in about 8 hours per cardiac cycle, which enables its potential use in a wide range of clinical scenarios.