Negative Entries Research Articles

Taming numerical imprecision by adapting the KL divergence to negative probabilities

The Kullback–Leibler (KL) divergence is frequently used in data science. For discrete distributions on large state spaces, approximations of probability vectors may result in a few small negative entries, rendering the KL divergence undefined. We address this problem by introducing a parameterized family of substitute divergence measures, the shifted KL (sKL) divergence measures. Our approach is generic and does not increase the computational overhead. We show that the sKL divergence shares important theoretical properties with the KL divergence and discuss how its shift parameters should be chosen. If Gaussian noise is added to a probability vector, we prove that the average sKL divergence converges to the KL divergence for small enough noise. We also show that our method solves the problem of negative entries in an application from computational oncology, the optimization of Mutual Hazard Networks for cancer progression using tensor-train approximations.

Benders decomposition for the discrete ordered median problem

Ordered median optimization has been proven to be a powerful tool to generalize many well-known problems from the literature. In Location Theory, the Discrete Ordered Median Problem (DOMP) is a facility location problem where clients are first ranked according to their allocation cost to the nearest open facility, and then these costs are multiplied by a suitable weight vector λ. That way, DOMP generalizes many well-known discrete location problems including p-median, p-center or centdian. In this article, we also allow negative entries of λ, allowing us to derive models for better addressing equity and fairness in facility location, for modeling obnoxious facility location problems or for including other client preference models. We present new mixed integer programming models for DOMP along with algorithmic enhancements for solving the DOMP to optimality using mixed integer programming techniques. Specifically, starting from state-of-the-art formulations from the literature, we present several Benders decomposition reformulations applied to them. Using these approaches, new state-of-the-art results have been obtained for different ordered weighting vectors.

Some points of view on Grothendieck's inequalities

Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps may be used to show that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constantkGC.

A characterization of sets realizable by compensation in the SNIEP

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability.

Optimal Transport and Seismic Rays

We present a theoretical framework that links Fermat’s principle of least time to optimal transport theory via a cost function that enforces local transport. The proposed cost function captures the physical constraints inherent in wave propagation; when paired with specific mass distributions, it yields shortest paths in the considered media through the optimal transport plans. In the discrete setting, our formulation results in physically significant optimal couplings, whose off-diagonal entries identify shortest paths in both directed and undirected graphs. For undirected graphs with positive edge weights, commonly used to parameterize seismic media, our method provides solutions to the Eikonal equation consistent with those from the Dijkstra algorithm. For directed negative-weight graphs, corresponding to transportation cost matrices with negative entries, our approach aligns with the Bellman–Ford algorithm but offers considerable computational advantages. We also highlight potential research directions. These include the use of sparse cost matrices to reduce the number of unknowns and constraints in the considered transportation problem, and solving specific classes of optimal transport problems through the Dijkstra algorithm to enhance computational efficiency.

Multi-view clustering with Laplacian rank constraint based on symmetric and nonnegative low-rank representation

Multi-view spectral clustering has attracted more and more attention due to its promising advantages in data clustering, and many related methods have been proposed, among which the method based on low-rank representation have become increasingly popular. However, the similarity matrices learned by the existing methods inevitably contain some negative entries and are generally nonsymmetric. In order to guarantee weight consistency for each pair of data points and highly correlated data points to be together, we propose a new multi-view spectral clustering model based on the symmetric and nonnegative low-rank representation. This model can learn the view-specific consistent similarity matrix and also capture the local structure and discriminative information among data by imposing symmetric and nonnegative constraints on the representation matrix. By employing the compatible and complementary information from multi-view data points, it can also learn the consensus similarity matrix for all views and perform the spectral clustering simultaneously. By the augmented Lagrangian multiplier method, an alternating iterative optimization algorithm is designed for solving this model and its computational complexity and convergence is analyzed. Comprehensive experiments on multiple multi-view data sets also verify the effectiveness and superiority of the proposed approach.

Finding phylogeny-aware and biologically meaningful averages of metagenomic samples: L2UniFrac.

Metagenomic samples have high spatiotemporal variability. Hence, it is useful to summarize and characterize the microbial makeup of a given environment in a way that is biologically reasonable and interpretable. The UniFrac metric has been a robust and widely used metric for measuring the variability between metagenomic samples. We propose that the characterization of metagenomic environments can be improved by finding the average, a.k.a. the barycenter, among the samples with respect to the UniFrac distance. However, it is possible that such a UniFrac-average includes negative entries, making it no longer a valid representation of a metagenomic community. To overcome this intrinsic issue, we propose a special version of the UniFrac metric, termed L2UniFrac, which inherits the phylogenetic nature of the traditional UniFrac and with respect to which one can easily compute the average, producing biologically meaningful environment-specific "representative samples." We demonstrate the usefulness of such representative samples as well as the extended usage of L2UniFrac in efficient clustering of metagenomic samples, and provide mathematical characterizations and proofs to the desired properties of L2UniFrac. A prototype implementation is provided at https://github.com/KoslickiLab/L2-UniFrac.git. All figures, data, and analysis can be reproduced at https://github.com/KoslickiLab/L2-UniFrac-Paper.

On Matrices with Only One Non-SDD Row

The class of H-matrices, also known as Generalized Diagonally Dominant (GDD) matrices, plays an important role in many areas of applied linear algebra, as well as in a wide range of applications, such as in dynamical analysis of complex networks that arise in ecology, epidemiology, infectology, neurology, engineering, economy, opinion dynamics, and many other fields. To conclude that the particular dynamical system is (locally) stable, it is sufficient to prove that the corresponding (Jacobian) matrix is an H-matrix with negative diagonal entries. In practice, however, it is very difficult to determine whether a matrix is a non-singular H-matrix or not, so it is valuable to investigate subclasses of H-matrices which are defined by relatively simple and practical criteria. Many subclasses of H-matrices have recently been discussed in detail demonstrating the many benefits they can provide, though one particular subclass has not been fully exploited until now. The aim of this paper is to attract attention to this class and discuss its relation with other more investigated classes, while showing its main advantage, based on its simplicity and elegance. This new approach, which we are presenting in this paper, will be compared with the existing ones, in three possible areas of applications, spectrum localization; maximum norm estimation of the inverse matrix in the point, as well as the block case; and error estimation for LCP problems. The main conclusion is that the importance of our approach grows with the matrix dimension.

Using deep cnn architectures and tensorflow for bee identification on images

Problem statement. Bees are important pollinators for various plant species and are essential in maintaining our planet's biodiversity. With the decline in the number of bees worldwide, monitoring the bee hive state has become increasingly important. Pur-pose. Individual bee identification is an important task that, if performed well, will help to track the bee hive population and its health state in general. Related work. With the development of deep learning and computer vision technologies, several studies were done on their application in detecting insects like wasps, spiders, or fruit flies. Materials and methods. Photographs of bee hive frames with bees on them are being used as a da-ta source for model training. These photographs were manually annotated using Remo annotation software. This study uses five widely used deep learning architectures (Fast-er R-CNN ResNet152, CenterNet ResNet50, SSD ResNet50, Faster R-CNN Inception ResNet V2, SSD MobileNet V2) for the purpose of bee detection on test images. These models were trained on the same dataset and evaluated on the same bee image set. Re-sults and discussion. According to the confusion matrix, SSD MobileNet V2 architecture showed the best detection performance with 80% detected bees on test files, but it also had the falsest negative entries. On the other hand, Faster R-CNN ResNet152 model showed the best results in accuracy and evaluation speed, but successfully identified on-ly 61% of the bees. Conclusions. The experiment showed that state-of-the-art CNN ar-chitecture SSD MobileNet V2 is a better performer with detecting 80% and 95% accura-cy on bee images test set.

Lattice Paths and Negatively Indexed Weight-Dependent Binomial Coefficients

In 1992, Loeb (Adv Math, 91:64–74, 1992) considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub (Ann Comb, 23:725–748, 2019) showed that these results can be extended to the q-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining the arithmetic properties of the q-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser (Sém Lothar Combin, 81:24, 2020), to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. We also examine important special cases of the weight-dependent binomial coefficients, including ordinary, q- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions.

New upper bounds for the dominant eigenvalue of a matrix with Perron–Frobenius property

In this paper, we derive some upper bounds for the dominant eigenvalue of a matrix with some negative entries, which possess the Perron–Frobenius property. Numerical examples are given to illustrate the effectiveness of our new upper bounds.

Distributions of cherries and pitchforks for the Ford model

Distributional properties of tree shape statistics under random phylogenetic tree models play an important role in investigating the evolutionary forces underlying the observed phylogenies. In this paper, we study two subtree counting statistics, the number of cherries and that of pitchforks for the Ford model, the alpha model introduced by Daniel Ford. It is a one-parameter family of random phylogenetic tree models which includes the proportional to distinguishable arrangement (PDA) and the Yule models, two tree models commonly used in phylogenetics. Based on a non-uniform version of the extended Pólya urn models in which negative entries are permitted for their replacement matrices, we obtain the strong law of large numbers and the central limit theorem for the joint distribution of these two statistics for the Ford model. Furthermore, we derive a recursive formula for computing the exact joint distribution of these two statistics. This leads to exact formulas for their means and higher order asymptotic expansions of their second moments, which allows us to identify a critical parameter value for the correlation between these two statistics. That is, when the number of tree leaves is sufficiently large, they are negatively correlated for 0≤α≤1/2 and positively correlated for 1/2<α<1.

On numerical ranges of matrices with Perron–Frobenius properties and some negative entries

In this paper, we give a characterization of when numerical radius is a sharp point of numerical range of a real matrix with Perron-Frobenius properties and some negative entries. It extends the sharp point property of numerical ranges of nonnegative matrices previously established in the literature. A closely related problem of characterizing when 0 is a sharp point of numerical range of a Laplacian matrix for a signed digraph is also studied.

Multivariate Conway-Maxwell-Poisson Distribution: Sarmanov Method and Doubly Intractable Bayesian Inference

ABSTRACT In this article, a multivariate count distribution with Conway-Maxwell (COM)-Poisson marginals is proposed. To do this, we develop a modification of the Sarmanov method for constructing multivariate distributions. Our multivariate COM-Poisson (MultCOMP) model has desirable features such as (i) it admits a flexible covariance matrix allowing for both negative and positive nondiagonal entries; (ii) it overcomes the limitation of the existing bivariate COM-Poisson distributions in the literature that do not have COM-Poisson marginals; (iii) it allows for the analysis of multivariate counts and is not just limited to bivariate counts. Inferential challenges are presented by the likelihood specification as it depends on a number of intractable normalizing constants involving the model parameters. These obstacles motivate us to propose Bayesian inferential approaches where the resulting doubly intractable posterior is handled with via the noisy exchange algorithm or the Grouped Independence Metropolis–Hastings algorithm. Numerical experiments based on simulations are presented to illustrate the proposed Bayesian approach. We demonstrate the potential of the MultCOMP model through a real data application on the numbers of goals scored by the home and away teams in the English Premier League from 2018 to 2021. Here, our interest is to assess the effect of a lack of crowds during the COVID-19 pandemic on the well-known home team advantage. A MultCOMP model fit shows that there is evidence of a decreased number of goals scored by the home team, not accompanied by a reduced score from the opponent. Hence, our analysis suggests a smaller home team advantage in the absence of crowds, which agrees with the opinion of several football experts. Supplementary materials for this article are available online.

"I Let Depression and Anxiety Drown Me…": Identifying Factors Associated With Resilience Based on Journaling Using Machine Learning and Thematic Analysis.

Over the years, there has been a global increase in the use of technology to deliver interventions for health and wellness, such as improving people's mental health and resilience. An example of such technology is the Q-Life app which aims to improve people's resilience to stress and adverse life events through various coping mechanisms, including journaling. Using a combination of sentiment analysis and thematic analysis methods, this paper presents the results of analyzing 6023 journal entries from 755 users. We uncover both positive and negative factors that are associated with resilience. First, we apply two lexicon-based and eight machine learning (ML) techniques to classify journal entries into positive or negative sentiment polarity, and then compare the performance of these classifiers to determine the best performing classifier overall. Our results show that Support Vector Machine (SVM) is the best classifier overall, outperforming other ML classifiers and lexicon-based classifiers with a high F1-score of 89.7%. Second, we conduct thematic analysis of negative and positive journal entries to identify themes representing factors associated with resilience either negatively or positively, and to determine various coping mechanisms. Our findings reveal 14 negative themes such as stress, worry, loneliness, lack of motivation, sickness, relationship issues, as well as depression and anxiety. Also, 13 positive themes emerged including self-efficacy, gratitude, socialization, progression, relaxation, and physical activity. Seven (7) coping mechanisms are also identified including time management, quality sleep, and mindfulness. Finally, we reflect on our findings and suggest technological interventions that address the negative factors to promote resilience.

Differential Ephaptic Networks

Ephaptic coupling is the phenomenon where the various axons in an axon tract contribute to the evolution of the voltage variable on each axon. In this paper, we relax the central assumption of electroneutrality used by Reutskiy et al. That assumption is based on charge conservation laws. However, we present data from the literature in support of this relaxation. Thus we are able to justify the presence of negative entries in the geometric W matrix. These negative entries then impact the action potential conduction profiles of the axons in the tract. These “signed” and coupled tracts can be envisioned as bearers of neural information which is akin to that borne by a synapse.

Polynomials that preserve nonnegative matrices

In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London ((1978/1979) [8]) posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If Pn denotes the set of all polynomials that preserve all n-by-n nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to Pn. However, it is known that Pn contains polynomials with negative entries. In this work, novel results for Pn with respect to the coefficients of the polynomials belonging to Pn. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed.

Data-driven discovery of multiscale chemical reactions governed by the law of mass action

In this paper, we propose a data-driven method to discover multiscale chemical reactions governed by the law of mass action. First, we use a single matrix to represent the stoichiometric coefficients for both the reactants and products in a system without catalysis reactions. The negative entries in the matrix denote the stoichiometric coefficients for the reactants and the positive ones for the products. Second, we find that the conventional optimization methods usually get stuck in the local minima and could not find the true solution in learning the multiscale chemical reactions. To overcome this difficulty, we propose a partial-parameters-freezing (PPF) technique to progressively determine the network parameters by using the fact that the stoichiometric coefficients are integers. With such a technique, the dimension of the searching space is gradually reduced in the training process and the global minima can be eventually obtained. Several numerical experiments including the classical Michaelis–Menten kinetics, the hydrogen oxidation reactions and the simplified GRI-3.0 mechanism verify the good performance of our algorithm in learning the multiscale chemical reactions. The code is available at https://github.com/JuntaoHuang/multiscale-chemical-reaction.

On asymptotic joint distributions of cherries and pitchforks for random phylogenetic trees

Tree shape statistics provide valuable quantitative insights into evolutionary mechanisms underpinning phylogenetic trees, a commonly used graph representation of evolutionary relationships among taxonomic units ranging from viruses to species. We study two subtree counting statistics, the number of cherries and the number of pitchforks, for random phylogenetic trees generated by two widely used null tree models: the proportional to distinguishable arrangements (PDA) and the Yule-Harding-Kingman (YHK) models. By developing limit theorems for a version of extended Pólya urn models in which negative entries are permitted for their replacement matrices, we deduce the strong laws of large numbers and the central limit theorems for the joint distributions of these two counting statistics for the PDA and the YHK models. Our results indicate that the limiting behaviour of these two statistics, when appropriately scaled using the number of leaves in the underlying trees, is independent of the initial tree used in the tree generating process.

Maximal-Sum submatrix search using a hybrid contraint programming/linear programming approach

A Maximal-Sum Submatrix (MSS) maximizes the sum of the entries corresponding to the Cartesian product of a subset of rows and columns from an original matrix (with positive and negative entries). Despite being NP-hard, this recently introduced problem was already proven to be useful for practical data-mining applications. It was used for identifying bi-clusters in gene expression data or to extract a submatrix that is then visualized in a circular plot. The state-of-the-art results for MSS are obtained using an advanced Constraint Programing approach that combines a custom filtering algorithm with a Large Neighborhood Search. We improve the state-of-the-art approach by introducing new upper bounds based on linear and mixed-integer programming formulations, along with dedicated pruning algorithms. We experiment on both synthetic and real-life data, and show that our approach outperforms the previous methods.