scholarly journals Divergence Functions in Dually Flat Spaces and Their Properties

2018 ◽  
Author(s):  
Tomohiro Nishiyama
Keyword(s):  
Probability Distributions ◽  
Flat Space ◽  
Exponential Families ◽  
Geometrical Approach ◽  
Bhattacharyya Distance ◽  
Dual Affine ◽  
The Law ◽  
Law Of Cosines ◽  
Dually Flat Space ◽  
Jensen Shannon Divergence

In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual affine coordinate systems and strictly convex functions called potential and a canonical divergence is naturally introduced as a function of the affine coordinates and potentials. The canonical divergence satisfies a relational expression called triangular relation. This can be regarded as a generalization of the law of cosines in Euclidean space.In this paper, we newly introduce two kinds of divergences. The first divergence is a function of affine coordinates and it is consistent with the Jeffreys divergence for exponential or mixture families. For this divergence, we show that more relational equations and theorems similar to Euclidean space hold in addition to the law of cosines. The second divergences are functions of potentials and they are consistent with the Bhattacharyya distance for exponential families and are consistent with the Jensen-Shannon divergence for mixture families respectively. We derive an inequality between the the first and the second divergences and show that the inequality is a generalization of Lin's inequality.

scholarly journals minicore: Fast scRNA-seq clustering with various distances

2021 ◽  
Author(s):  
Daniel N. Baker ◽  
Nathan Dyjack ◽  
Vladimir Braverman ◽  
Stephanie C. Hicks ◽  
Ben Langmead
Keyword(s):  
Open Source ◽  
Count Data ◽  
Probability Distributions ◽  
Expression Profiles ◽  
Distance Measures ◽  
Bhattacharyya Distance ◽  
Link Type ◽  
Leibler Divergence ◽  
Careful Handling ◽  
Jensen Shannon Divergence

AbstractSingle-cell RNA-sequencing (scRNA-seq) analyses typically begin by clustering a gene-by-cell expression matrix to empirically define groups of cells with similar expression profiles. We describe new methods and a new open source library, minicore, for efficient k-means++ center finding and k-means clustering of scRNA-seq data. Minicore works with sparse count data, as it emerges from typical scRNA-seq experiments, as well as with dense data from after dimensionality reduction. Minicore’s novel vectorized weighted reservoir sampling algorithm allows it to find initial k-means++ centers for a 4-million cell dataset in 1.5 minutes using 20 threads. Minicore can cluster using Euclidean distance, but also supports a wider class of measures like Jensen-Shannon Divergence, Kullback-Leibler Divergence, and the Bhattacharyya distance, which can be directly applied to count data and probability distributions.Further, minicore produces lower-cost centerings more efficiently than scikit-learn for scRNA-seq datasets with millions of cells. With careful handling of priors, minicore implements these distance measures with only minor (<2-fold) speed differences among all distances. We show that a minicore pipeline consisting of k-means++, localsearch++ and minibatch k-means can cluster a 4-million cell dataset in minutes, using less than 10GiB of RAM. This memory-efficiency enables atlas-scale clustering on laptops and other commodity hardware. Finally, we report findings on which distance measures give clusterings that are most consistent with known cell type labels.AvailabilityThe open source library is at https://github.com/dnbaker/minicore. Code used for experiments is at https://github.com/dnbaker/minicore-experiments.

Information geometry of divergence functions

2010 ◽  
Vol 58 (1) ◽  
pp. 183-195 ◽  
Author(s):  
S. Amari ◽  
A. Cichocki
Keyword(s):  
Probability Distributions ◽  
Flat Space ◽  
Information Geometry ◽  
Dual Pair ◽  
Bregman Divergence ◽  
Bregman Divergences ◽  
Leibler Divergence ◽  
Fisher Information Metric ◽  
Information Metric ◽  
Dually Flat Space

Information geometry of divergence functionsMeasures of divergence between two points play a key role in many engineering problems. One such measure is a distance function, but there are many important measures which do not satisfy the properties of the distance. The Bregman divergence, Kullback-Leibler divergence andf-divergence are such measures. In the present article, we study the differential-geometrical structure of a manifold induced by a divergence function. It consists of a Riemannian metric, and a pair of dually coupled affine connections, which are studied in information geometry. The class of Bregman divergences are characterized by a dually flat structure, which is originated from the Legendre duality. A dually flat space admits a generalized Pythagorean theorem. The class off-divergences, defined on a manifold of probability distributions, is characterized by information monotonicity, and the Kullback-Leibler divergence belongs to the intersection of both classes. Thef-divergence always gives the α-geometry, which consists of the Fisher information metric and a dual pair of ±α-connections. The α-divergence is a special class off-divergences. This is unique, sitting at the intersection of thef-divergence and Bregman divergence classes in a manifold of positive measures. The geometry derived from the Tsallisq-entropy and related divergences are also addressed.

scholarly journals On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1568
Author(s):  
Shaul K. Bar-Lev
Keyword(s):  
Power Function ◽  
Exponential Family ◽  
Probability Distributions ◽  
Infinite Divisibility ◽  
Variance Function ◽  
Exponential Families ◽  
Natural Exponential Family ◽  
Natural Exponential Families ◽  
Convolution Properties ◽  
The Relationship

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

Developing and Applying the Law of Cosines

2020 ◽  
pp. 274-288
Author(s):  
Vecihi S. Zambak ◽  
Budi Mulyono
Keyword(s):  
Instructional Technology ◽  
Ninth Grade ◽  
Steam Education ◽  
The Law ◽  
Law Of Cosines ◽  
Ninth Grade Students ◽  
Space Sciences ◽  
The Given ◽  
Support Students

In history, geometry was founded more as a practical endeavor than a theoretical one. Early developments of the branch portray philosophers' attempts to make sense of their surroundings, including the measurement of distances on earth and in space. Such a link between earth and space sciences and geometry motivated us to develop and implement a multidisciplinary lesson focusing on the conceptual understanding of the law of cosines in the context of astronomy. In our content specific STEAM lesson, the authors aimed to facilitate an understanding of the law of cosines in ninth grade students, and then apply the law in a star map task to find approximate distances between stars. The second part of the lesson also included the use of an instructional technology to support students' work with the star map task. In the conclusion, the authors discuss possible ways to improve the quality of their STEAM education efforts for the given context.

Proof without Words: The Law of Cosines

1990 ◽  
Vol 63 (5) ◽  
pp. 342-342 ◽  
Author(s):  
Sidney H. Kung
Keyword(s):  
The Law ◽  
Law Of Cosines

Proof without Words: The Law of Cosines via Ptolemy's Theorem

1992 ◽  
Vol 65 (2) ◽  
pp. 103 ◽  
Author(s):  
Sidney H. Kung
Keyword(s):  
The Law ◽  
Law Of Cosines ◽  
Ptolemy’S Theorem

Note on the Law of Cosines

1951 ◽  
Vol 58 (10) ◽  
pp. 698 ◽  
Author(s):  
S. L. Thompson
Keyword(s):  
The Law ◽  
Law Of Cosines

Transforming the law of cosines for computational purposes

1955 ◽  
Vol 48 (5) ◽  
pp. 308-309
Author(s):  
Benjamin Greenberg
Keyword(s):  
The Law ◽  
Law Of Cosines

The law of cosines has always frustrated those teachers who wish to use it computationally. The transformation developed in this paper makes it more amenable to the use of logarithmic computation.

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