Pl\"ucker Coordinates of the best-fit Stiefel Tropical Linear Space to a Mixture of Gaussian Distributions

In this research, we investigate a tropical principal component analysis (PCA) as a best-fit Stiefel tropical linear space to a given sample over the tropical projective torus for its dimensionality reduction and visualization. Especially, we characterize the best-fit Stiefel tropical linear space to a sample generated from a mixture of Gaussian distributions as the variances of the Gaussians go to zero. For a single Gaussian distribution, we show that the sum of residuals in terms of the tropical metric with the max-plus algebra over a given sample to a fitted Stiefel tropical linear space converges to zero by giving an upper bound for its convergence rate. Meanwhile, for a mixtures of Gaussian distribution, we show that the best-fit tropical linear space can be determined uniquely when we send variances to zero. We briefly consider the best-fit topical polynomial as an extension for the mixture of more than two Gaussians over the tropical projective space of dimension three. We show some geometric properties of these tropical linear spaces and polynomials.Comment: To appear in Information Geometr

<Poster Presentation 11>Noise-induced Phenomena in Two Strongly Pulse-coupled Spiking Neuron Models

[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA

Tropical neural networks and its applications to classifying phylogenetic trees

Deep neural networks show great success when input vectors are in an Euclidean space. However, those classical neural networks show a poor performance when inputs are phylogenetic trees, which can be written as vectors in the tropical projective torus. Here we propose tropical embedding to transform a vector in the tropical projective torus to a vector in the Euclidean space via the tropical metric. We introduce a tropical neural network where the first layer is a tropical embedding layer and the following layers are the same as the classical ones. We prove that this neural network with the tropical metric is a universal approximator and we derive a backpropagation rule for deep neural networks. Then we provide TensorFlow 2 codes for implementing a tropical neural network in the same fashion as the classical one, where the weights initialization problem is considered according to the extreme value statistics. We apply our method to empirical data including sequences of hemagglutinin for influenza virus from New York. Finally we show that a tropical neural network can be interpreted as a generalization of a tropical logistic regression

Hit and Run Sampling from Tropically Convex Sets

In this paper we propose Hit and Run (HAR) sampling from a tropically convex set. The key ingredient of HAR sampling from a tropically convex set is sampling uniformly from a tropical line segment over the tropical projective torus, which runs linearly in its computational time complexity. We show that this HAR sampling method samples uniformly from a tropical polytope which is the smallest tropical convex set of finitely many vertices. Finally, we apply this novel method to any given distribution using Metropolis-Hasting filtering over a tropical polytope

The dorsomedial striatum encodes net expected return, critical for energizing performance vigor

Decision making requires an actor to not only steer behavior towards specific goals, but also determine the optimal vigor of performance. Current research and models have largely focused on the former problem of how actions are directed, while overlooking the latter problem of how they are energized. Here, we designed a self-paced decision-making paradigm that showed that rats' performance vigor globally fluctuates with the net value of their options, suggesting that they maintain long-term estimates of the value of their current state. Lesions of the dorsomedial (DMS), and to a lesser degree, in the ventral striatum (VS) impaired such state-dependent modulation of vigor, rendering vigor to depend more exclusively on the outcomes of immediately preceding trials. The lesions, however, spared choice biases. Neuronal recordings showed that the DMS is enriched with net-value-coding neurons. In sum, the DMS encodes one's net expected return, which drives the general motivation to perform

Tropical Fermat-Weber Polytropes

We study the geometry of tropical Fermat-Weber points in terms of the symmetric tropical metric over the tropical projective torus. It is well known that a tropical Fermat-Weber point of a given sample is not unique and in this paper we show that the set of all possible Fermat-Weber points forms a polytrope. Then, we introduce the tropical Fermat-Weber gradient and using them, we show that the tropical Fermat-Weber polytrope is a bounded cell of a tropical hyperplane arrangement given by both min- and max-tropical hyperplanes with apices which are observations in the input data

Tropical Geometric Tools for Machine Learning: the TML package

In the last decade, developments in tropical geometry have provided a number of uses directly applicable to problems in statistical learning. The TML package is the first R package which contains a comprehensive set of tools and methods used for basic computations related to tropical convexity, visualization of tropically convex sets, as well as supervised and unsupervised learning models using the tropical metric under the max-plus algebra over the tropical projective torus. Primarily, the TML package employs a Hit and Run Markov chain Monte Carlo sampler in conjunction with the tropical metric as its main tool for statistical inference. In addition to basic computation and various applications of the tropical HAR sampler, we also focus on several supervised and unsupervised methods incorporated in the TML package including tropical principal component analysis, tropical logistic regression and tropical kernel density estimation

Tropical Support Vector Machines: Evaluations and Extension to Function Spaces

Support Vector Machines (SVMs) are one of the most popular supervised learning models to classify using a hyperplane in an Euclidean space. Similar to SVMs, tropical SVMs classify data points using a tropical hyperplane under the tropical metric with the max-plus algebra. In this paper, first we show generalization error bounds of tropical SVMs over the tropical projective space. While the generalization error bounds attained via VC dimensions in a distribution-free manner still depend on the dimension, we also show theoretically by extreme value statistics that the tropical SVMs for classifying data points from two Gaussian distributions as well as empirical data sets of different neuron types are fairly robust against the curse of dimensionality. Extreme value statistics also underlie the anomalous scaling behaviors of the tropical distance between random vectors with additional noise dimensions. Finally, we define tropical SVMs over a function space with the tropical metric and discuss the Gaussian function space as an example

Mircea Eliade and Japanese Studies of Folklore and Ethnology

ルーマニア出身の宗教学者ミルチャ・エリアーデは、小説家や神話学者としても知られているが、ルーマニアやバルカンのフォークロアに関心を持った民俗学者であったことは、故国を除き、よく認識されていないので、エリアーデの民俗学研究の重要性を明らかにしたい。　エリアーデは、フォークロア研究は文化の様式やシンボルの解明に資すると認識したが、その背景には、3 年間のインド滞在経験や両大戦間期のルーマニア人の宗教性とアイデンティティーをめぐる知識人の論争があった。エリアーデのフォークロア研究は、建築をめぐる人柱伝説である「マノーレ親方伝説」のバラッドおよび羊飼いの謀殺と死を結婚に擬する儀礼をめぐる口承叙事詩「ミオリッツァ」の研究に集約される。　「マノーレ親方伝説」研究では、伝説の宗教的神話的意味を究明し、人柱となった妻の「犠牲としての死」と建築現場から飛び降りた親方の「非業の死」を、宇宙創造神話における巨人の創造のための犠牲としての死を反復したものであり、「創造性ある死」であるとの解釈を提示した。また、「ミオリッツァ」研究では、羊飼いの死は、死を前にした諦念を表しているとの伝統的解釈を避け、叙事詩に歌われる「神秘的結婚」は、自分の運命を変えたいとする羊飼いの意思を表しており、強大な周りの民族の侵入の恐怖に晒されたルーマニア人は、羊飼いの運命を自己の運命に重ね合わせていると解釈している。　日本の民俗学、民族学との関係については、呪術的植物である「マンドレーク」の伝説研究と、日本の霊魂観に関心を示した著作『永遠回帰の神話』を取り上げる。　エリアーデは、「マンドレーク伝説」研究で、植物学、民俗学者南方熊楠が、1880 年代に英国の雑誌『ネイチャー』において、欧州のマンドレークに関する民俗が、近東、アラブ世界を通じて中国にまで伝播したことを最初に論証したとその先駆的研究を高く評価した。　『永遠回帰の神話』では、民族学者岡正雄の論文『古日本の文化層』を間接的ではあるが参照し、日本の男性秘密結社、来訪者、新年の儀礼につき論じ、特に「タマ」等の日本人の霊魂観に強い関心を示しているが、岡論文の論拠の一つとなった柳田國男や折口信夫の業績には直接触れておらず、その研究は時代的制約を蒙っていたものと思われる