arXiv research

The volume of the quantum mechanical state space over $n$-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endo…

ZoomRL learns efficient strategies for large state-action spaces using a metric.

Study noncommutative Sobolev inequalities using quantum state metrics.

We study online reinforcement learning for finite-horizon deterministic control systems with {\it arbitrary} state and action spaces. Suppose that the transition dynamics and reward function is unknown, but the state and action space is endowed with a metric that characterizes the proximity between different states and…

We find a Sasaki-Einstein metric from a CFT state in AdS5.

Extended Vaisman theorem to compact spaces with singularities.

State-space systems generate probabilistic dependencies between inputs and outputs.

Kernel-UCBVI algorithm balances exploration and exploitation in metric state-action spaces.

Model-free Reinforcement Learning (RL) algorithms such as Q-learning [Watkins, Dayan 92] have been widely used in practice and can achieve human level performance in applications such as video games [Mnih et al. 15]. Recently, equipped with the idea of optimism in the face of uncertainty, Q-learning algorithms [Jin, Al…

KeRNS tackles non-stationary reinforcement learning in metric spaces.

Fixed point assertions in digital topology are often incorrect or poorly stated.

The theory of monotone Riemannian metrics on the state space of a quantum system was established by Denes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant - monotone - metric can be interpreted as an average statistical uncertainty. The present paper contributes to this su…

Develops risk measures on Lipschitz spaces for financial positions.

The study examines a semi-symmetric metric connection in perfect fluid space-time and phantom barriers.

A new metric based on hitting probabilities for directed graphs and Markov chains.

The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we d…

Study shows reinforcement learning algorithm's performance depends on metric space's size.

Proves compactness for timed-metric spaces using new distance and maps.

Generalizes embedding formalism for CFTs on curved backgrounds.

Efficient algorithm for reinforcement learning in large state-action spaces with adaptive discretization.

A new method estimates multi-dimensional value distributions using Hilbert space embeddings.

I consider compact metric spaces which admit intrinsic isometries to Euclidean d-space. The main result roughly states that the class of these spaces coincides with class of inverse limits of Euclidean d-polyhedra.

Metric learning makes it plausible to learn distances for complex distributions of data from labeled data. However, to date, most metric learning methods are based on a single Mahalanobis metric, which cannot handle heterogeneous data well. Those that learn multiple metrics throughout the space have demonstrated superi…

A new method for Gaussian filtering using gradient flows and Wasserstein metrics.

The main result of this article states that the (K;N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner…

Abstract notes on hyperbolic surfaces and Teichmüller spaces.

Geometric approach to quantum thermodynamics models state spaces and processes.

In this paper we study isometry-invariant Finsler metrics on inner product spaces over $\mathbb{R}$ or $\mathbb{C}$, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most g…

Paper introduces a new framework to improve sample efficiency in POMDPs learning.

Entangled bisimulation improves policy learning from visual input.

New method glues Lorentzian spaces, preserving curvature bounds.

Most of parameters used to describe states and dynamics of financial market depend on proportions of the appropriate variables rather than on their actual values. Therefore, projective geometry seems to be the correct language to describe the theater of financial activities. We suppose that the object of interest of ag…

On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that if non-empty the space of metrics with inver…

Link condition for simplicial complexes to be CUB spaces.

This paper investigates the notion of learning user and item representations in non-Euclidean space. Specifically, we study the connection between metric learning in hyperbolic space and collaborative filtering by exploring Mobius gyrovector spaces where the formalism of the spaces could be utilized to generalize the m…

We consider the problem of metric learning for multi-view data and present a novel method for learning within-view as well as between-view metrics in vector-valued kernel spaces, as a way to capture multi-modal structure of the data. We formulate two convex optimization problems to jointly learn the metric and the clas…

Incorrect fixed point assertions in digital topology are discussed.

We consider a quasi-metric topological structure for the construction of a new reinforcement learning model in the framework of financial markets. It is based on a Lipschitz type extension of reward functions defined in metric spaces. Specifically, the McShane and Whitney extensions are considered for a reward function…

Differentiable optimization bridges arbitrary metrics to tree metrics.

New bounds for low-regularity Riemannian metrics defined via distributional curvature.

A nonnegative number d_infinity, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open …

New RL policy for unbounded state space with stability guarantee.

Study on minimal disks in metric spaces, focusing on branch set structure.

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists closed $n$-dimensional Riemannian manifolds $M$ with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that $π_{1}(\mathcal{T}^{<0}(M))$ is non-trivial. $\mathcal{T}^{<0}(M)$ denotes the Teichmüller space…

Study finds all hyper-Kähler 4-manifolds with specific symmetries and structures.

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an ap…

A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any Riemannian metric on $S^2$ with curvature $K>-1$ is induced on a unique convex surface in $H^3$. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another …

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…