Related papers: Geographic Space as Manifolds
Spatially embedded networks are shaped by a combination of purely topological (space-independent) and space-dependent formation rules. While it is quite easy to artificially generate networks where the relative importance of these two…
Advances in reinforcement learning have led to its successful application in complex tasks with continuous state and action spaces. Despite these advances in practice, most theoretical work pertains to finite state and action spaces. We…
Generative adversarial networks (GANs) have emerged as a powerful unsupervised method to model the statistical patterns of real-world data sets, such as natural images. These networks are trained to map random inputs in their latent space…
Neural models learn representations of high-dimensional data on low-dimensional manifolds. Multiple factors, including stochasticities in the training process, model architectures, and additional inductive biases, may induce different…
The scattering transform is a multilayered, wavelet-based transform initially introduced as a model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance…
A finite-dimensional linear representation of a group or an algebra may be regarded as a map into a space of matrices, endowing abstract elements with coordinates, and encoding algebraic operations as matrix products. With this in mind, we…
Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…
Deep generative models are tremendously successful in learning low-dimensional latent representations that well-describe the data. These representations, however, tend to much distort relationships between points, i.e. pairwise distances…
Mathematical descriptions of dynamical systems are deeply rooted in topological spaces defined by non-Euclidean geometry. This paper proposes leveraging structure-rich geometric spaces for machine learning to achieve structural…
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…
We consider the formulation of entropic gravity in two spacetime dimensions. The usual gravitational force law is derived even in the absence of area, as normally required by the holographic principle. A special feature of this perspective…
Circular and non-flat data distributions are prevalent across diverse domains of data science, yet their specific geometric structures often remain underutilized in machine learning frameworks. A principled approach to accounting for the…
The Random Geometric Graph (RGG) is a random graph model for network data with an underlying spatial representation. Geometry endows RGGs with a rich dependence structure and often leads to desirable properties of real-world networks such…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…
Simplified representations of macromolecules help in rationalising and understanding the outcome of atomistic simulations, and serve to the construction of effective, coarse-grained models. The number and distribution of coarse-grained…
Motivated by the empirical analysis of the air transportation system, we define a network model that includes geographical attributes along with topological and weight (traffic) properties. The introduction of geographical attributes is…
Focus of this study is to explore some aspects of mathematical foundations for using complex manifolds as a model for space-time. More specifically, certain equations of motions have been derived as a Projective geodesic on a real manifold…
Next-token predictors often appear to develop internal representations of the latent world and its rules. The probabilistic nature of these models suggests a deep connection between the structure of the world and the geometry of probability…
The internal representations learned by language models consistently exhibit striking geometric structure: calendar months organize into a circle, historical years form a smooth one-dimensional manifold, and cities' latitudes and longitudes…
Learning useful representations is a key ingredient to the success of modern machine learning. Currently, representation learning mostly relies on embedding data into Euclidean space. However, recent work has shown that data in some domains…