Derivative formulas on measure spaces of Riemannian manifolds are characterized.
problem Characterizing derivatives in measure spaces on Riemannian manifolds.
method Introducing and characterizing derivatives in measure spaces for functions on the space of finite measures over a Riemannian manifold.
result Derivatives in measure spaces for functions on Riemannian manifolds are linked and calculated.
We review the nature of some well-known phenomena such as volatility smiles, convexity adjustments and parallel derivative markets. We propose that the market is incomplete and postulate the existence of intrinsic risks in every contingent claim as a basis for understanding these phenomena. In a continuous time framewo…
Study shows stable graphs in Heisenberg group are essentially planes.
problem Characterizing stable graphs in the Heisenberg group.
method Analyzes Sobolev intrinsic graphs in the Heisenberg group with sub-Riemannian area stability.
result Stable graphs are cosets of two-dimensional subgroups.
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
Revisits VIC method to correct intrinsic reward bias in stochastic environments.
problem Intrinsic reward bias in VIC leading to suboptimal solutions.
method Proposes two methods based on transitional probability model and Gaussian mixture model to correct bias.
result Achieves maximal empowerment through corrected intrinsic reward.
We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the $\SUn(n)$-structure. Furthermore, we apply the obtained results …
We study the intrinsic torsion of almost quaternion-Hermitian manifolds via the exterior algebra. In particular, we show how it is determined by particular three-forms formed from simple combinations of the exterior derivatives of the local Kaehler forms. This gives a practical method to compute the intrinsic torsion a…
Graphs and their complements are intrinsically knotted.
problem Characterizing maximal linklessly embeddable graphs and their complements.
method Analyzing maximal linklessly embeddable graphs, deriving connected domination numbers, and proving intrinsic knotting properties.
result Complements of maximal linklessly embeddable graphs of order 12 and 15 are intrinsically knotted.
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
New method estimates intrinsic dimensionality using angles, not distances.
problem Estimating local intrinsic dimensionality accurately.
method Introduces a new estimator using the distribution of angles between neighbor points.
result New estimator behaves similarly but complementarily to existing measures of intrinsic dimensionality.
Formulas for curvature measures of sublevel sets on manifolds derived from function and derivatives.
problem Calculating intrinsic volumes of sublevel sets on Riemannian manifolds.
method Established formulas involving integrals of functionals of function and its derivatives.
result Formulas for intrinsic volumes of sublevel sets on Riemannian manifolds.
Study provides bounds for estimating intrinsic dimension using Gaussian kernels.
problem Estimating intrinsic dimension from data.
method Finite-sample concentration and anti-concentration bounds for Gaussian kernel sums.
result Explicit dependence on sample size, bandwidth, and geometric parameters.
We propose a new method for estimating the intrinsic dimension of a dataset by applying the principle of regularized maximum likelihood to the distances between close neighbors. We propose a regularization scheme which is motivated by divergence minimization principles. We derive the estimator by a Poisson process appr…
Following the approach of Bryant we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing of the intrinsic torsion.
Study recovers Riemannian quantities from noisy data densities.
problem Recovering geometric structure from noisy data on submanifolds.
method Derive uniform small-noise expansions of noisy density and its derivatives; construct estimators for tangent spaces, intrinsic dimension, and second fundamental form.
result Fundamental Riemannian quantities identifiable from density derivatives.
Monetary risk measures are usually interpreted as the smallest amount of external capital that must be added to a financial position to make it acceptable. We propose a new concept: intrinsic risk measures and argue that this approach provides a direct path from unacceptable positions towards the acceptance set. Intrin…
A new method classifies almost contact metric manifolds using intrinsic endomorphisms.
problem Difficult classification of almost contact metric manifolds into 2^12 classes.
method Introducing intrinsic endomorphisms S and h, providing a flowchart based on algebraic conditions.
result A more natural classification scheme with fewer classes, including H-parallel manifolds. Analyzes intrinsic time in financial markets, linking it to physical time.
problem Understanding the intrinsic nature of time in financial data.
method Presented an analytic relationship linking intrinsic and physical time, using empirical scaling laws.
result A novel empirical scaling law relating intrinsic time variability to overshoots.
Introduces an unobservable intrinsic electricity price to link storage theory with risk premium.
problem Connecting storage theory with risk premium in electricity markets.
method Introduces an unobservable intrinsic electricity price and derives prices for various contracts.
result Finds an overall negative risk premium in empirical analysis.
Intrinsic formulation of noncommutative geometry for quantum gravity.
problem Formalizing noncommutative differential geometry for quantum gravity.
method Geometric definitions and proofs of noncommutative Ricci curvatures and Bianchi identities.
result Quantum fluctuations and curvatures of (pseudo-) Riemannian metrics are renormalizable.
Formula derived for Spin(7)-structures on 8D manifolds.
problem Deriving the intrinsic torsion for Spin(7)-structures.
method Explicit formula derivation using Levi-Civita and minimal Spin(7)-connections.
result Formula relating intrinsic torsion to Lee form and codifferential components.
Study on curves in Riemannian surfaces, focusing on total intrinsic curvature.
problem Understanding the total intrinsic curvature of irregular curves in Riemannian surfaces.
method Weak notion of parallel transport, bounded variation of angle, energy functional analysis.
result Total intrinsic curvature of irregular curves matches an energy functional.
Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.
problem Stochastic dynamics of rigid inclusions on curved surfaces.
method Cartan's method of moving frames, Hamiltonian equations, intrinsic Langevin equations, Fokker-Planck equation.
result Extracted overdamped equations for accurate simulations of diffusion processes.
The paper derives inequalities for Riemannian submersions and applies them to specific space forms.
problem Deriving inequalities for Riemannian submersions.
method Introducing and deriving inequalities for vertical, horizontal, and mixed distributions of Riemannian submersions.
result Established relationships between intrinsic and extrinsic invariants of Riemannian submersions.
New estimators for intrinsic dimension and Wasserstein distance improve OT accuracy.
problem Intrinsic dimension estimation and Wasserstein distance estimation in large-scale OT.
method Introduces novel estimators for intrinsic dimension and Wasserstein distance.
result Simple, tuning-free estimator of OT and fast intrinsic dimension estimator.
Study shows DNNs perform well with low intrinsic data dimensions.
problem Understanding DNN performance with high-dimensional data.
method Derived bounds for approximation and generalization errors, developed novel proof technique.
result Convergence rates of DNN errors are independent of high dimensionality but dependent on intrinsic low dimensionality.
New RL approach uses future state and action visitation measures for better exploration.
problem Improving exploration in reinforcement learning.
method Intrinsic reward based on future state and action visitation measures, using contraction operators.
result Policies achieve good state-action space coverage and high performance.
We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor field defining a parallelism structure on M. Also we give a novel and intrinsic pr…
In this contribution we present an intrinsic description of time-variant Port Hamiltonian systems as they appear in modeling and control theory. This formulation is based on the splitting of the state bundle and the use of appropriate covariant derivatives, which guarantees that the structure of the equations is invari…
The intrinsic entropy model accurately estimates stock market volatility.
problem Accurately estimating historical volatility of stock market indices.
method Incorporates traded volumes alongside OHLC prices in daily data.
result Intrinsic entropy model delivers reliable estimates with lower coefficient of variation.
We consider an agent's uncertainty about its environment and the problem of generalizing this uncertainty across observations. Specifically, we focus on the problem of exploration in non-tabular reinforcement learning. Drawing inspiration from the intrinsic motivation literature, we use density models to measure uncert…
In many sequential decision making tasks, it is challenging to design reward functions that help an RL agent efficiently learn behavior that is considered good by the agent designer. A number of different formulations of the reward-design problem, or close variants thereof, have been proposed in the literature. In this…
Researchers compute differential invariants for Carrollian spacetimes.
problem Understanding the geometry and symmetries of Carrollian spacetimes.
method Derived from the geometry of the screen bundle, computed differential invariants using jet-spaces and Spencer cohomology.
result Specified how to generate the entire algebra of differential invariants for generic Carrollian structures, focusing on dimension 3.
New method for long-term sampling of complex dynamics on curved spaces.
problem Sampling ergodic dynamics on Riemannian manifolds efficiently over long periods.
method Intrinsic geometric operations for sampling invariant measure without embeddings.
result Outperforms previous methods in long-term sampling efficiency.
Study on rolling Stiefel manifolds with specific metrics.
problem Intrinsic and extrinsic rolling of Stiefel manifolds with α-metrics. method Investigation of intrinsic rolling of normal naturally reductive homogeneous spaces, derivation of ODEs for rolling, and explicit solutions.
result Explicit solutions for intrinsic and extrinsic rolling of Stiefel manifolds.
We compute persistent homology using an intrinsic metric derived from density.
problem Estimating topological features from high-dimensional data.
method Density-based metric learning for persistent homology.
result Persistent homology converges to intrinsic manifold metric.
New risk measures for financial networks avoid external capital, reducing systemic risk.
problem Systemic risk in financial networks is underestimated by traditional methods.
method Developed set-valued, intrinsic risk measures for financial networks.
result Systemic intrinsic risk measures are more stable and avoid reliance on external capital.
We consider a regular distribution D in a Riemannian manifold (M,g). The Levi-Civita connection on (M,g) together with the orthogonal projection allow to endow the space of sections of D with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant deri…
Unified geometric framework for Brownian motion on various manifolds.
problem Modeling Brownian motion on complex Riemannian manifolds.
method Constructing stochastic differential equations with noise and drift terms aligned with Laplace-Beltrami operators.
result Geometrically transparent and mathematically consistent foundation for diffusion processes.
A new reinforcement learning method uses mutual information to encourage agents to control their environment.
problem Learning from internal drives instead of external rewards.
method Formulate an intrinsic objective as mutual information between goal states and controllable states, derive a surrogate objective for efficient optimization.
result Demonstrated the efficacy of the approach in robotic tasks.
DPA autoencoders learn data distribution and intrinsic dimensionality with guarantees.
problem Learning data distribution and intrinsic dimensionality in unsupervised learning.
method Combines distributionally correct reconstruction with principal-component-like interpretability.
result Exact theoretical guarantees on disentangling factors of variation and intrinsic dimensionality.
We derive intrinsic curvature and radius estimates for compact disks embedded in R3 with nonzero constant mean curvature and apply these estimates to study the global geometry of complete surfaces embedded in R3 with nonzero constant mean curvature.
We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. …
We define the notion of characteristic classes for supermanifolds endowed with a homological vector field Q. These take values in the cohomology of the Lie derivative operator LQ acting on arbitrary tensor fields. We formulate a classification theorem for intrinsic characteristic classes and give their explicit de…
AutoDIME automates design of multi-agent environments for RL.
problem Designing multi-agent environments for reinforcement learning is challenging.
method Developed intrinsic teacher rewards for multi-agent settings and evaluated them in various tasks.
result Value disagreement was found to be most consistent and effective across tasks.
The paper explores how neural networks generalize differently from natural and medical images.
problem Discrepancies in generalization error between natural and medical images.
method Established and empirically validated a generalization scaling law with respect to intrinsic dataset properties.
result Higher intrinsic 'label sharpness' of medical images leads to higher adversarial vulnerability.
In anlogy with the work of R. Bryant on the Ricci tensor of a G2-structure, we study the intrinsic torsion of an SU(2)-structure on a 5-dimensional manifold deriving an explicit expression for the Ricci and the scalar curvature in terms of torsion forms and its derivative. As a consequence of this formula we prove…
Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.
problem Understanding intrinsic Hopf-Lax semigroup and its relation to intrinsic slope.
method Introduces and proves the link between intrinsic Hopf-Lax semigroup and intrinsic slope.
result Intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.