Nonzero Weights Research Articles

Modified double brackets and a conjecture of S. Arthamonov

Around 20 years ago, M. Van den Bergh introduced double Poisson brackets as operations on associative algebras inducing Poisson brackets under the representation functor. Weaker versions of these operations, called modified double Poisson brackets, were later introduced by S. Arthamonov in order to induce a Poisson bracket on moduli spaces of representations of the corresponding associative algebras. Moreover, he defined two operations that he conjectured to be modified double Poisson brackets. The first case of this conjecture was recently proved by M. Goncharov and V. Gubarev motivated by the theory of Rota-Baxter operators of nonzero weight. We settle the conjecture by realising the second case as part of a new family of modified double Poisson brackets. These are obtained from mixed double Poisson algebras, a new class of algebraic structures that are introduced and studied in the present work.

Development of Machine Learning Algorithms Using EEG Data to Detect the Presence of Chronic Pain.

Chronic pain impacts more than one in five adults in the United States (US) and the costs associated with the condition amount to hundreds of billions of dollars annually. Despite the tremendous impact of chronic pain in the US and worldwide, the standard of care for diagnosis depends on subjective self-reporting of pain state, with no effective objective assessment procedure available. This study investigated the application of signal processing and machine learning to electroencephalography (EEG) data for the development of classification algorithms capable of differentiating subjects in pain from pain free subjects. In this study, nineteen (19) channels of EEG data were obtained from subjects in an eyes closed resting state, and ultimately data from 186 participants were used for algorithm development, including 35 healthy controls and 151 chronic pain patients. Signal processing was applied to identify noise free segments of EEG data and 6375 quantitative EEG (qEEG) measures were calculated for each subject. Various machine learning methodologies were applied to the data, with Elastic Net chosen as the optimal methodology. The final classifier developed using Elastic Net contained 34 qEEG features with non-zero weights. The classifier was able to differentiate pain versus no pain subjects with an accuracy of 79.6%, sensitivity of 82.2%, and specificity of 66.7%. The features used in the classifier were evaluated and found to align well with contemporary literature regarding changes in neurological characteristics associated with chronic pain.

Inference of human contact networks based on epidemic data

A key challenge in epidemic modeling is the lack of adequate data on population interactions (such as traffic flow or mobility patterns) which result in the spread of infectious diseases. Knowledge of social contact patterns is crucial for public health professionals to devise effective non-pharmaceutical interventions to control epidemics. This paper focuses on inferring social contact rates from reported infection counts during the spread of an infectious disease, addressing the increased difficulty that arises when dealing with “sparse” contact networks where only a small subset of the edges have non-zero weights. Specifically, a new geographically constrained lasso approach for network reconstruction for non-homogeneous mixing Susceptible-Infected-Removed (SIR) disease spread models is presented. The new network reconstruction method can explicitly account for the spatial proximity of network nodes in estimating the disease transmission rates and predicting the future evolution of the epidemic dynamics. Extensive numerical experiments are presented to show the proposed method outperforms existing approaches in terms of accuracy of contact identification under various graph topologies. A case study based on real data from the COVID-19 pandemic is presented to demonstrate the application of the approach for inferring contact structures and a counterfactual scenario analysis to assess effectiveness of containment strategies.

Intermediate-grained kernel elements pruning with structured sparsity

Neural network pruning provides a promising prospect for the deployment of neural networks on embedded or mobile devices with limited resources. Although current structured strategies are unconstrained by specific hardware architecture in the phase of forward inference, the decline in classification accuracy of structured methods is beyond the tolerance at the level of general pruning rate. This inspires us to develop a technique that satisfies high pruning rate with a small decline in accuracy and has the general nature of structured pruning. In this paper, we propose a new pruning method, namely KEP (Kernel Elements Pruning), to compress deep convolutional neural networks by exploring the significance of elements in each kernel plane and removing unimportant elements. In this method, we apply a controllable regularization penalty to constrain unimportant elements by adding a prior knowledge mask and obtain a compact model. In the calculation procedure of forward inference, we introduce a sparse convolution operation which is different from the sliding window to eliminate invalid zero calculations and verify the effectiveness of the operation for further deployment on FPGA. A massive variety of experiments demonstrate the effectiveness of KEP on two datasets: CIFAR-10 and ImageNet. Specially, with few indexes of non-zero weights introduced, KEP has a significant improvement over the latest structured methods in terms of parameter and float-point operation (FLOPs) reduction, and performs well on large datasets.

Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators

Abstract We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang–Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota–Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota–Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota–Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight 0, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.

New upper bounds on the number of non-zero weights of constacyclic codes

For any simple-root constacyclic code C over a finite field Fq, as far as we know, the group G generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group Aut(C) of C. In this paper, by calculating the number of G-orbits of C﹨{0}, we give an explicit upper bound on the number of non-zero weights of C and present a necessary and sufficient condition for C to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in Zhang and Cao (2024) [26]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code C belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of C by substituting G with a larger subgroup of Aut(C). The results derived in this paper generalize the main results in Chen et al. (2024) [9].

Improved Upper Bounds on the Number of Non-Zero Weights of Cyclic Codes

Improved Upper Bounds on the Number of Non-Zero Weights of Cyclic Codes

Linear Codes Constructed from Two Weakly Regular Plateaued Functions with Index (p - 1)/2.

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting p≡1(mod4), we constructed an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index (p-1)/2. Their weight distributions were completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal.

Comparison of weighting algorithms to mitigate respiratory motion in free-breathing neonatal pulmonary radial UTE-MRI

Background. Thoracoabdominal MRI is limited by respiratory motion, especially in populations who cannot perform breath-holds. One approach for reducing motion blurring in radially-acquired MRI is respiratory gating. Straightforward ‘hard-gating’ uses only data from a specified respiratory window and suffers from reduced SNR. Proposed ‘soft-gating’ reconstructions may improve scan efficiency but reduce motion correction by incorporating data with nonzero weight acquired outside the specified window. However, previous studies report conflicting benefits, and importantly the choice of soft-gated weighting algorithm and effect on image quality has not previously been explored. The purpose of this study is to map how variable soft-gated weighting functions and parameters affect signal and motion blurring in respiratory-gated reconstructions of radial lung MRI, using neonates as a model population. Methods. Ten neonatal inpatients with respiratory abnormalities were imaged using a 1.5 T neonatal-sized scanner and 3D radial ultrashort echo-time (UTE) sequence. Images were reconstructed using ungated, hard-gated, and several soft-gating weighting algorithms (exponential, sigmoid, inverse, and linear weighting decay outside the period of interest), with %Nproj representing the relative amount of data included. The apparent SNR (aSNR) and motion blurring (measured by the maximum derivative of image intensity at the diaphragm, MDD) were compared between reconstructions. Results. Soft-gating functions produced higher aSNR and lower MDD than hard-gated images using equivalent %Nproj, as expected. aSNR was not identical between different gating schemes for given %Nproj. While aSNR was approximately linear with %Nproj for each algorithm, MDD performance diverged between functions as %Nproj decreased. Algorithm performance was relatively consistent between subjects, except in images with high noise. Conclusion. The algorithm selection for soft-gating has a notable effect on image quality of respiratory-gated MRI; the timing of included data across the respiratory phase, and not simply the amount of data, plays an important role in aSNR. The specific soft-gating function and parameters should be considered for a given imaging application’s requirements of signal and sharpness.

Non-Projective Two-Weight Codes.

It has been known since the 1970's that the difference of the non-zero weights of a projective Fq-linear two-weight code has to be a power of the characteristic of the underlying field. Here, we study non-projective two-weight codes and, e.g., show the same result under mild extra conditions. For small dimensions we give exhaustive enumerations of the feasible parameters in the binary case.

The R-matrix presentation for the rational form of a quantized enveloping algebra

Let Uq(g) denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra g. Let λ be a nonzero dominant integral weight of g, and let V be the corresponding type 1 finite-dimensional irreducible representation of Uq(g). Starting from this data, the R-matrix formalism for quantum groups outputs a Hopf algebra URλ(g) defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley–Serre type presentation which can be recovered from that of Uq(g) by adding a single invertible quantum Cartan element. We simultaneously establish that URλ(g) can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of V. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining URλ(g), with respect to a natural grading by the root lattice of g compatible with the weight space decomposition of End(V).

Koszul duality, minimal model and L∞-structure for differential algebras with weight

A differential algebra with weight is an abstraction of both the derivation (weight zero) and the forward and backward difference operators (weight ±1). In 2010 Loday established the Koszul duality for the operad of differential algebras of weight zero. He did not treat the case of nonzero weight, noting that new techniques are needed since the operad is no longer quadratic. This paper continues Loday's work and establishes the Koszul duality in the case of nonzero weight. In the process, the minimal model and the Koszul dual homotopy cooperad of the operad governing differential algebras with weight are determined. As a consequence, a notion of homotopy differential algebras with weight is obtained and the deformation complex as well as its L∞-algebra structure for differential algebras with weight are deduced.

Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X ( T ) . For a nonzero p-restricted dominant weight λ ∈ X ( T ) , let V be the associated irreducible kG-module. We define ν G ( V ) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G ( V ) for G of type A l and dim ( V ) ≤ l 3 2 , and for G of type B l , C l , or D l and dim ( V ) ≤ 4 l 3 . Moreover, we give the exact value of ν G ( V ) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 .

New dimension-independent upper bounds on linear insdel codes

To determine the insdel distances of linear codes is a very challenging problem. The half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li is a basic upper bound on the insertion-deletion error-correcting capabilities of linear codes. In this paper we give several new coordinate ordering-free and coordinate ordering-depending upper bounds for the insdel distances of linear codes. These upper bounds do not depend on dimensions of linear codes and only depend on the formations of codewords. It is shown that for many natural well-known linear codes including binary simplex codes, Kasami codes, many linear binary codes with few non-zero weights and some algebraic geometry codes, the new coordinate ordering-free upper bounds on their insdel distances are strictly smaller than the half-Singleton bound and the direct upper bound. On the other hand for many linear binary codes, the ordering-depending upper bound on their insdel distance is 2. We also give ordering-depending upper bounds on the insdel distances of some linear ternary codes with few non-zero weights and some binary and ternary algebraic geometry codes.

Robust Particle Filter Design With an Application to Power System State Estimation

This paper develops statistically robust particle filters that exhibit high statistical efficiency and good robustness to outliers. Three types of outliers, namely, observation outliers, structural outliers and innovation outliers, are considered. Two robust particle filters are developed and their performance are assessed. Firstly, a weight reconstruction particle filter is proposed that makes use of the projection statistics to derive non-zero particle weights in response to observation outliers. Secondly, a fitting-based particle filter is developed that generalizes the former filter to the case of structural and innovation outliers. This filter approximates the next estimated values using both the projection statistics and the linear fitting technique. The designed filters are applied to the state estimation of decentralized machines' internal state estimation and local bus frequency estimation in power systems. Simulation results performed on a four-machine two-area power system and the 39-bus <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">New England</i> power system demonstrate the excellent performance of the proposed robust particle filters in dealing with various types of outliers.

Variance-Guided Structured Sparsity in Deep Neural Networks

The success of deep neural networks, especially convolutional neural networks in various applications, has greatly been possible by the presence of an enormous number of learnable parameters. These parameters increase the learning capacity of the model, but at the same time, it also significantly increases the computational and memory costs. This severely hinders the scalability of these models to limited resource environments, like IoT devices. The majority of the network weights are known to be redundant and can be removed from the network. This paper introduces a regularization scheme, which is the combination of structured sparsity regularization and variance regularization. It simultaneously helps to obtain computationally sparse models by making the majority of parameter groups zero and increasing the variance of non-zero groups to compensate for the accuracy drop. We use sparse group lasso, group sparsity variant of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> (lasso) regularization, which removes redundant connections and unnecessary neurons from the network. For variance regularization, the KL divergence of the current parameter distribution and the target distribution is minimized, which aims to have the concentration of weights towards zero and a high variance of non-zero weights (skewed distribution). To check the effectiveness of the proposed regularizer, the experiments are performed on various benchmark datasets and it is noticed that variance regularization helps to reduce the accuracy drop caused by sparsity regularization. On MNIST, the trainable parameters are reduced from 331984 (baseline model) to 57327 and managed to obtain better accuracy than the baseline (99.6%). Also, on Fashion-MNIST, Cifar-10 and ImageNet the proposed scheme achieved state-of-the-art sparsity with almost no drop in accuracy.

Investigating Thrombosis Risk Factors in Sickle Cell Disease Utilizing Archetypal Analysis

Background: Thrombosis in sickle cell disease (SCD) affects both the arterial and venous vasculature with devastating consequences. Individuals with SCD have an increased risk of venous thromboembolism (VTE). The risks are multifactorial with prior studies identifying both VTE-specific and SCD-specific risk factors. Machine learning (ML) offers the potential to detect non-linear relationships in complex multifactorial diseases and has successfully predicted outcomes in deeply phenotyped patients. We investigated whether archetypal analysis (AA), a method of ML would identify clinical and laboratory factors associated with arteriovenous thrombosis in patients with SCD. Methods: We retrospectively analyzed a prospective longitudinal cohort of patients with SCD (NCT00011648) studied between 2000-2017. A prior history of VTE and/or stroke at the time of study entry as captured by the electronic medical record was used to define index “cases”. SCD patients without a history of thrombosis were “controls”. The demographic and clinical characteristics of cases and controls were compared using Wilcoxon rank-sum test for continuous and Fisher's exact test for categorical variables. Unsupervised AA was used to identify patterns of clinical and laboratory test characteristics within the entire cohort of patients including cases and controls. AA identifies patterns are on the convex hull that encloses all patients in the cohort in the high-dimensional feature space. Once specific archetypes were identified, the frequency of each archetype was determined within cases and controls. In addition, we describe the age and laboratory characteristics of the archetypes that were significantly more prevalent in cases as compared to controls. Results: The cohort consisted of 633 out of 862 patients that had their whole exome sequenced. A combined total of 160 arterial and venous thrombosis events was detected in 148/633 patients (24%) with half (n=79) attributable to arterial stroke and half (n=81) attributable to VTE [Table 1]. A small proportion of patients had both arterial and venous thrombosis (n=12). The median age (31 years, p=0.94) and sex were similar between the cases (n=148) and controls (n=485). The distribution of genotypes was similar between the two groups (p=0.09). A significantly higher proportion of patient with thrombosis had a prior history of blood transfusion (p&amp;lt;0.0001) and history of insertion of a central venous access device (p=&amp;lt;0.0001) [Table 1]. History of medical management with hydroxyurea was similar among the cases and controls (p=0.70). Compared to controls, the cases had elevated laboratory parameters including AST levels (p=0.02), and serum ferritin (p&amp;lt;0.0001), while HbS percentages were significantly lower (p&amp;lt;0.0001). Based on the minimum error in matrix factorization, a total of 15 archetypes were identified in the cohort [Figure 1]. Each patient was decomposed as a weighted combination of the 15 archetypes. The contributing archetypes for each patient were those archetypes that have non-zero weights. Among those 15 archetypes, archetypes 9, 14 and 15 were significantly more enriched among the cases compared to controls (p&amp;lt;0.02) [Figure 1]. We observed elevated serum ferritin levels and low levels of HbS across the three archetypes, which are consistent with the previously identified risk factors [Figure 1]. Conclusions: Machine-identified archetypes are objective and reproducible feature representations. Archetype analysis-based machine learning appears to hold promise to identify both traditional VTE and SCD-specific risks for clinical thrombosis. Future studies will utilize archetypal analysis to investigate associations between clinical, laboratory and genetic markers for thrombosis and assess their reproducibility using validation cohorts.

A tight upper bound on the number of non-zero weights of a constacyclic code

For a simple-root λ-constacyclic code C over Fq, let 〈ρ〉 and 〈ρ,M〉 be the subgroups of the automorphism group of C generated by the cyclic shift ρ, and by the cyclic shift ρ and the scalar multiplication group M, respectively. Let NG(C⁎) be the number of orbits of a subgroup G of the automorphism group of C acting on C⁎=C﹨{0}. In this paper, we establish explicit formulas for N〈ρ〉(C⁎) and N〈ρ,M〉(C⁎). Consequently, we derive a upper bound on the number of non-zero weights of C. We present some irreducible and reducible λ-constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee N〈ρ〉(C⁎)=N〈ρ,M〉(C⁎) is presented.

Optimal Hardy‐weights for elliptic operators with mixed boundary conditions

AbstractWe construct families of optimal Hardy‐weights for a subcritical linear second‐order elliptic operator with degenerate mixed boundary conditions. By an optimal Hardy‐weight for a subcritical operator we mean a nonzero nonnegative weight function W such that is critical, and null‐critical with respect to W. Our results rely on a recently developed criticality theory for positive solutions of the corresponding mixed boundary value problem.

Sparse Bayesian Learning With Weakly Informative Hyperprior and Extended Predictive Information Criterion.

This article considers the regression problem with sparse Bayesian learning (SBL) when the number of weights P is larger than the data size N, i.e., P》 N. The situation induces overfitting and makes regression tasks, such as prediction and basis selection, challenging. We show a strategy to address this problem. Our strategy consists of two steps. The first is to apply an inverse gamma hyperprior with a shape parameter close to zero over the noise precision of automatic relevance determination (ARD) prior. This hyperprior is associated with the concept of a weakly informative prior in terms of enhancing sparsity. The model sparsity can be controlled by adjusting a scale parameter of inverse gamma hyperprior, leading to the prevention of overfitting. The second is to select an optimal scale parameter. We develop an extended predictive information criterion (EPIC) for optimal selection. We investigate the strategy through relevance vector machine (RVM) with a multiple-kernel scheme dealing with highly nonlinear data, including smooth and less smooth regions. This setting is one form of the regression task with SBL in the P》 N situation. As an empirical evaluation, regression analyses on four artificial datasets and eight real datasets are performed. We see that the overfitting is prevented, while predictive performance may be not drastically superior to comparative methods. Our methods allow us to select a small number of nonzero weights while keeping the model sparse. Thus, the methods are expected to be useful for basis and variable selection.