Zero-one reaction networks play a crucial role in cell signaling. Determining the equivalence of reaction networks is a fundamental computational problem in the field of chemical reaction networks. In this work, we develop an efficient method for determining the equivalence of zero-one networks. The efficiency comes from several criteria for determining the equivalence of the steady-state ideals arising from zero-one networks, which helps for cutting down the expenses on computing Gröbner bases. Experiments show that our method can successfully classify over three million networks according to their equivalence in a reasonable time.

We develop a probabilistic algorithm for computing elimination ideals of likelihood equations. We show experimentally that it is far more efficient than directly computing Groebner bases or the interpolation method for medium to large size models. Furthermore, we deduce discriminants of the elimination ideals, which play a central role in real root classification. In particular, we can compute the discriminant of one Jukes-Cantor model in phylogenetics (with size 8 GB text file).

We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by Lemmens and Seidel; namely, we use linear algebra and combinatorial arguments to bound the number of vectors within an equiangular set which have inner products of certain signs with a negative clique. After projection and rescaling, such sets are also certain spherical two-distance sets, and semidefinite programming techniques may be used to bound the size of the set. Applying our method, we prove new relative bounds for the angle $\arccos(1/5)$. Experiments show that our relative bounds for all possible angles are considerably smaller than the known semidefinite programming bounds for a range of larger dimensions. Our computational results also show an explicit bound on the size of a set of equiangular lines in $\mathbb{R}^r$ regardless of angle, which is strictly less than the well-known Gerzon's bound if r+2 is not a square of an odd number.

Abstract In recent years, deep learning techniques have been applied to the diagnosis of pulmonary nodules. In order to improve the pulmonary nodule diagnostic performance effectively, we propose a novel pulmonary nodule diagnosis method using dual‐modal deep supervised autoencoder based on extreme learning machine for which discriminative features are automatically learnt from the input data. The network is fed with nodule images in pairs obtained from computed tomography and positron emission tomography respectively. For each pair image, the high‐level discriminative features of nodules in computed tomography and positron emission tomography are extracted from stacked supervised autoencoder layers. The outputs of the proposed architecture are combined using an ideal fusion method to get the final classification. In the experiments, 5‐fold cross‐validation method is used to validate the proposed method on 1,600 pulmonary nodule images and our method reaches high‐classification sensitivities of 91.75% at 1.58 false positives per scan. Meanwhile, compared with other deep learning diagnosis methods, our method achieves better discriminative results and is highly suited to be used for pulmonary nodule diagnosis.

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The ${CAT}(0)$ metric of Billera--Holmes--Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen--Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric exhibit properties that are desirable for geometric statistics, such as geodesics of small depth.

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.