The paper introduces new Finsler metrics and associated weighted quasi-metrics.
problem Geometric properties of Finsler metrics and quasi-metric spaces.
method Construction of weighted quasi-metrics associated with Finsler metrics.
result Investigation of geometric properties of weighted quasi-metric spaces.
Study of classification in asymmetric quasi-metric spaces.
problem Classification in asymmetric quasi-metric spaces.
method Sample compression and nearest neighbor algorithm.
result Algorithm has favorable statistical properties.
New RL approach speeds up training across tasks.
problem Training reinforcement learning models quickly and efficiently.
method Combines planning quasi-metric and task-specific aimers.
result Achieves multiple-fold speed-up on bit-flip and robotic arm tasks.
We investigate the relation between weighted quasi-metric Spaces and Finsler Spaces. We show that the induced metric of a Randers space with reversible geodesics is a weighted quasi-metric space.
Characterizes isometries between non-reversible Finsler manifolds.
problem Understanding isometries in non-reversible Finsler manifolds.
method Generalization of Myers-Nakai Theorem for Riemannian manifolds, modification of function spaces to accommodate asymmetric structure.
result Functional characterization of isometries between non-reversible Finsler manifolds.
Disk Embeddings tackle embedding DAGs with exponential growth.
problem Embedding DAGs with exponentially increasing ancestors and descendants.
method Disk Embeddings framework for quasi-metric spaces, including Hyperbolic Disk Embeddings.
result Disk Embeddings outperform existing methods in complex DAGs.
For any geodesic current we associated a quasi-metric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.
We develop the basics of a theory of almost isometries for spaces endowed with a quasi-metric. The case of non-reversible Finsler (more specifically, Randers) metrics is of particular interest, and it is studied in more detail. The main motivation arises from General Relativity, and more specifically in spacetimes endo…
Neural Spacetimes learn DAGs by embedding nodes in a spacetime manifold.
problem Learning representations of weighted directed acyclic graphs (DAGs).
method Trainable deep learning-based geometries (Neural Spacetimes) that encode both edge weights and causality.
result Universal embedding theorem for DAGs with sub-cubic parameters and low distortion.
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
Enhances financial market reinforcement learning with new state generation.
problem Improving reinforcement learning models for financial markets.
method Lipschitz extensions and artificial state generation.
result Improved investment strategy through enriched state space.