The Fréchet mean is a fundamental notion of central tendency defined as a minimizer of a sum of squared distances in a general metric space. In this paper, we study Fréchet means in tropical geometry—a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry—by formulating and solving the associated tropical quadratic optimization problem. We give a geometric characterization of the collection of all tropical Fréchet means as a bounded set that is simultaneously tropically and classically convex, hence a polytrope. We establish the existence of positivity certificates for maxima of finitely many quadratic polynomials in R [ x 1 , … , x n ] whose homogeneous quadratic components are sums of squares, which provides a symbolic framework for exact optimization. Using this structure, we develop algorithms for computing tropical Fréchet means and the associated Fréchet mean polytrope. We further describe a combinatorial type decomposition of the objective function induced by braid arrangements, yielding a piecewise quadratic representation and a fully symbolic method for exact computation.

The Fréchet mean is a key measure of central tendency as a barycenter for a given set of points in a general metric space. It is computed by solving an optimization problem and is a fundamental quantity in statistics. In this paper, we study Fréchet means in tropical geometry—a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry that has gained prominence in applications. A key property of Fréchet means is that uniqueness is generally not guaranteed, which is true in tropical settings. In solving the tropical Fréchet mean optimization problem, we obtain a geometric characterization of the collection of all Fréchet means in a general tropical space as a tropically and classically convex polytope. Furthermore, we prove that a certificate of positivity for finitely many quadratic polynomials in \(\mathbb {R}[x_1,\ldots ,x_n]\) always exists, given that their quadratic homogeneous components are sums of squares. We propose an algorithm to symbolically compute the Fréchet mean polytope based on our exact quadratic optimization result and study its complexity.

Abstract We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given by the tree’s branching configuration and leaf labeling. We use the tropical line segment as a framework to define notions of variance as well as invariance of tree topologies: we provide a combinatorial search theorem that describes all tree topologies occurring along a tropical line segment, as well as a setting under which tree topologies do not change along a tropical line segment. Our study is motivated by comparison to the moduli space endowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann (referred to as BHV space); we consider the tropical geometric setting as an alternative framework to BHV space for sets of phylogenetic trees. We give an algorithm to compute tropical line segments which is lower in computational complexity than the fastest method currently available for BHV geodesics and show that its trajectory behaves more subtly: while the BHV geodesic traverses the origin for vastly different tree topologies, the tropical line segment bypasses it.

Electroencephalogram (EEG) is a non-invasive collection method for brain signals. It has broad prospects in brain-computer interface (BCI) applications. Recent advances have shown the effectiveness of the widely used convolutional neural network (CNN) in EEG decoding. However, some studies reveal that a slight disturbance to the inputs, e.g., data translation, can change CNN's outputs. Such instability is dangerous for EEG-based BCI applications because signals in practice are different from training data. In this study, we propose a multi-scale activity transition network (MSATNet) to alleviate the influence of the translation problem in convolution-based models. MSATNet provides an activity state pyramid consisting of multi-scale recurrent neural networks to capture the relationship between brain activities, which is a translation-invariant feature. In the experiment, Kullback-Leibler divergence is applied to measure the degree of translation. The comprehensive results demonstrate that our method surpasses the AUC of 0.0080, 0.0254, 0.0393 in 1, 5, and 10 KL divergence compared to competitors with various convolution structures.

Parkinson’s disease is the second most common neurodegenerative disorder, commonly affecting elderly people over the age of 65. As the cardinal manifestation, hypomimia, referred to as impairments in normal facial expressions, stays covert. Even some experienced doctors may miss these subtle changes, especially in a mild stage of this disease. The existing methods for hypomimia recognition are mainly dominated by statistical variable-based methods with the help of traditional machine learning algorithms. Despite the success of recognizing hypomimia, they show a limited accuracy and lack the capability of performing semantic analysis. Therefore, developing a computer-aided diagnostic method for semantically recognizing hypomimia is appealing. In this article, we propose a Semantic Feature based Hypomimia Recognition network , named SFHR-NET , to recognize hypomimia based on facial videos. First, a Semantic Feature Classifier (SF-C) is proposed to adaptively adjust feature maps salient to hypomimia, which leads the encoder and classifier to focus more on areas of hypomimia-interest. In SF-C, the progressive confidence strategy (PCS) ensures more reliable semantic features. Then, a two-stream framework is introduced to fuse the spatial data stream and temporal optical stream, which allows the encoder to semantically and progressively characterize the rigid process of hypomimia. Finally, to improve the interpretability of the model, Gradient-weighted Class Activation Mapping (Grad-CAM) is integrated to generate attention maps that cast our engineered features into hypomimia-interest regions. These highlighted regions provide visual explanations for decisions of our network. Experimental results based on real-world data demonstrate the effectiveness of our method in detecting hypomimia.

Parkinson’s disease is a progressive nervous system disorder afflicting millions of patients. Among its motor symptoms, bradykinesia is one of the cardinal manifestations. Experienced doctors are required for the clinical diagnosis of bradykinesia, but sometimes they also miss subtle changes, especially in early stages of such disease. Therefore, developing auxiliary diagnostic methods that can automatically detect bradykinesia has received more and more attention. In this article, we employ a two-stage framework for bradykinesia recognition based on the video of patient movement. First, convolution neural networks are trained to localize keypoints in each video frame. These time-varying coordinates form motion trajectories that represent the whole movement. From the trajectory, we then propose novel measurements, namely stability , completeness , and self-similarity , to quantify different motor behaviors. We also propose a periodic motion model called PMNet . An encoder--decoder structure is applied to learn a low dimensional representation of a motion process. The compressed motion process and quantified motor behaviors are combined as inputs to a fully-connected neural network. Different from the traditional means, our solution extends the application scenario outside the hospital and can be easily transplanted to conduct similar tasks. A commonly used clinical assessment is served as a case study. Experimental results based on real-world data validate the effectiveness of our approach for bradykinesia recognition.