Related papers: The Information-Geometric Perspective of Compositi…
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…
The fundamental equations of various disciplines often seem to share the same basic structure. Natural selection increases information in the same way that Bayesian updating increases information. Thermodynamics and the forms of common…
Supervised dimensionality reduction maps labeled data into a low-dimensional feature space while preserving class discriminability. A common approach is to maximize a statistical measure of dissimilarity between classes in the feature…
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
It is shown that for any ensemble, whether classical or quantum, continuous or discrete, there is only one measure of the "volume" of the ensemble that is compatible with several basic geometric postulates. This volume measure is thus a…
Principal Component Analysis (PCA) is a workhorse of modern data science. While PCA assumes the data conforms to Euclidean geometry, for specific data types, such as hierarchical and cyclic data structures, other spaces are more…
Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact…
A central issue of the science of complex systems is the quantitative characterization of complexity. In the present work we address this issue by resorting to information geometry. Actually we propose a constructive way to associate to a -…
Obtaining meaningful quantitative descriptions of the statistical dependence within multivariate systems is a difficult open problem. Recently, the Partial Information Decomposition (PID) was proposed to decompose mutual information (MI)…
We formulate the Riemannian calculus of the probability set embedded with $L^2$-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold)…
In recent years, the unified theory of information and thermodynamics has been intensively discussed in the context of stochastic thermodynamics. The unified theory reveals that information theory would be useful to understand…
What determines whether a molecular property prediction model organizes its representations so that geometric and compositional information can be cleanly separated? We introduce Compositional Probe Decomposition (CPD), which linearly…
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…
The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in…
Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections…
We construct an information-geometric structure for chemical thermodynamics, applicable to a wide range of chemical reaction systems including non-ideal and open systems. For this purpose, we explicitly construct dual affine coordinate…
Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and…
We call for a theory of the particle-scale structure of materials that is based on the general notion of information rather than its special case of symmetry. An inherent limitation to the symmetry-based understanding of structure is…
Information theory is an outstanding framework to measure uncertainty, dependence and relevance in data and systems. It has several desirable properties for real world applications: it naturally deals with multivariate data, it can handle…
A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g.,…