Active subspaces on Riemannian manifolds generalize Euclidean principles.
problem Understanding how scalar-valued quantities change over Riemannian manifolds.
method Generalization of active subspaces from Euclidean to Riemannian spaces using parallel transport.
result The method provides a new way to study scalar-valued quantities on manifolds, differing from extrinsic approaches.
Quantum kernel machines need to use more complex kernels to fully exploit their potential.
problem Current quantum kernels struggle with complex learning tasks due to limited degrees of freedom.
method Propose using operator-valued kernels and C∗-algebraic representations to enhance quantum kernels. result Quantum operator-valued kernels can reveal structural dependencies that scalar-valued kernels miss.
Method uses optimal transport to complete distributional matrices.
problem Matrix completion for distributional data.
method Nearest neighbors in Wasserstein space.
result Method recovers distributions in Wasserstein metric.
Mapper and Ball Mapper tools for complex data analysis.
problem Exploring and visualizing high-dimensional data and scalar functions.
method Combining Mapper and Ball Mapper, adding new features for encoding structure and symmetries.
result A new hybrid algorithm, Mapper on Ball Mapper, for comparing high-dimensional data descriptors.
Extends complex manifold structures to line bundles, revealing new projective manifolds.
problem Generalizing scalar-valued holomorphic structures to line bundles.
method Study of holomorphic p-contact and s-symplectic structures on complex manifolds with line bundles. result Holomorphic p-contact and s-symplectic manifolds can be projective. This work evaluates deep generative models using RD curves, providing a more comprehensive quality assessment.
problem Quantitative evaluation of deep generative models is challenging, especially for implicit models.
method Proposes using rate distortion (RD) curves to evaluate and compare deep generative models, approximating the entire curve with similar computations to log-likelihood estimation.
result Approximating the entire RD curve provides a more comprehensive quality assessment than scalar-valued metrics.
Algorithm optimizes cascaded functions with known structure.
problem Optimizing a function network with known structure.
method GPN-UCB algorithm with upper confidence bounds and theoretical regret bounds.
result Near-optimal cumulative and simple regret bounds.
Study first-order locally convex Lie algebroids in Bastiani calculus.
problem Define and study first-order locally convex Lie algebroids.
method Define sheaves of Lie algebroid forms and morphisms, prove category structure, study representations and cohomology.
result First-order locally convex Lie algebroids form a category and have applications in Lie II theorems.
Develops vector-valued RKBS for neural networks and operators.
problem Understanding function spaces of Rd-valued neural networks and neural operators. method Defines and constructs vector-valued RKBS (vv-RKBS) without restrictive assumptions.
result Establishes Representer Theorem for neural architectures.
This paper optimizes sampling for least-squares approximation.
problem Optimizing sampling for least-squares approximation in arbitrary linear spaces.
method Introducing the Christoffel function to construct near-optimal random sampling strategies.
result The number of samples scales log-linearly in the dimension of the approximation space.
In this paper, we study the possibility of inferring early warning indicators (EWIs) for periods of extreme bitcoin price volatility using features obtained from Bitcoin daily transaction graphs. We infer the low-dimensional representations of transaction graphs in the time period from 2012 to 2017 using Bitcoin blockc…
We address the problem of classifying discrete differential-geometric Poisson brackets (dDGPBs) of any fixed order on target space of dimension 1. It is proved that these Poisson brackets (PBs) are in one-to-one correspondence with the intersection points of certain projective hypersurfaces. In addition, they can be re…
Let M be a complete Riemannian manifold and let Ω∗(M) denote the space of differential forms on M. Let d:Ω∗(M)→Ω∗+1(M) be the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian. We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$ (where the potential $V…
New algorithm for multi-agent reinforcement learning with reduced communication.
problem Cooperative learning among multiple agents with limited communication.
method Randomized multi-agent actor-critic algorithm for directed graphs.
result Algorithm solves problem for strongly connected graphs with reduced communication.
Sharp inequality for p-harmonic maps with new optimal constant.
problem Deriving the sharp vectorial Kato inequality for p-harmonic mappings. method Analyzing the inequality for p-harmonic mappings and comparing with scalar valued cases. result Established the optimal constant for p-harmonic maps and enhanced the range of p values for regularity. Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.
problem Limited understanding of self-normalized concentration for vector-valued processes outside sub-Gaussian frameworks.
method Developed concentration inequalities for self-normalized processes with light tails (e.g., Bennett, Bernstein bounds) for vector-valued data.
result Provided new insights and bounds for self-normalized processes with non-sub-Gaussian distributions.
In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalar-valued functions, are considered. In part…
We consider the problem of learning a vector-valued function f in an online learning setting. The function f is assumed to lie in a reproducing Hilbert space of operator-valued kernels. We describe two online algorithms for learning f while taking into account the output structure. A first contribution is an algorithm,…
Study finds conserved quantities for two types of curves on conformal sphere.
problem Identifying conserved quantities for specific types of curves on a conformal sphere.
method Used parallel tractor and Lagrangian formalism to compute conserved quantities.
result Found relation between conserved quantities of two curve types.
Ecker's and Huisken's quantities agree for ancient mean curvature flows.
problem Understanding the finiteness of integral quantities for ancient mean curvature flows.
method Comparison of Ecker's and Huisken's integral quantities.
result Finiteness of Ecker's integral quantity implies finiteness of entropy at infinity.
New sample complexity bounds for linear predictors and neural networks, focusing on initialization.
problem Understanding sample complexity for vector-valued linear predictors and neural networks, especially under initialization-dependent conditions.
method Size-independent bounds on Frobenius norm distance from a fixed reference matrix, applying to vector-valued predictors and neural networks.
result Established new sample complexity bounds for feed-forward neural networks, resolving open questions and introducing a new learnable problem.
We formulate a private learning model to study an intrinsic tradeoff between privacy and query complexity in sequential learning. Our model involves a learner who aims to determine a scalar value, v∗, by sequentially querying an external database and receiving binary responses. In the meantime, an adversary observes…
Enhances SVGD with matrix-valued kernels for faster inference.
problem Efficient approximate inference in complex probability landscapes.
method Integrates geometric information through matrix-valued kernels in SVGD.
result Significant improvement in real-world Bayesian inference tasks.
Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of infinite-dimensional operator-valued kernels which are suitable for extending func…
The paper extends Noether's theorem to contact systems, finding dissipated quantities instead of conserved ones.
problem Noether's theorem for contact systems does not produce conserved quantities.
method Classification of infinitesimal symmetries in contact Lagrangian systems, leading to dissipated quantities.
result Infinitesimal symmetries in contact dynamics lead to dissipated quantities rather than conserved ones.
In this paper, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman's reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni and Topping. This es…
Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved…
Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.
problem Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.
method Follow the strategy developed in Miao.
result Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.
The paper introduces a method to incorporate expert opinion on observable quantities into statistical models.
problem Tackling the challenge of integrating expert knowledge on observable quantities into statistical models.
method The approach involves updating a prior belief using a loss function that reflects expert opinion on observable quantities.
result The method allows for a flexible specification of expert opinion and is straightforward to implement.
Efficient inference for adaptive data with directional stability condition.
problem Efficient inference on scalar targets after adaptive data collection.
method Introduces directional stability, a weaker condition than i.i.d. data, and shows asymptotic normality and efficiency of estimators.
result Estimators remain asymptotically normal and semiparametrically efficient under directional stability.
The Cheap Gradient Principle (Griewank 2008) --- the computational cost of computing the gradient of a scalar-valued function is nearly the same (often within a factor of 5) as that of simply computing the function itself --- is of central importance in optimization; it allows us to quickly obtain (high dimensional) …
We investigate invariants of compact hyperk{ä}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifo…
The study finds resonance points in polarised curves with polynomial conserved quantities.
problem Finding resonance points in polarised curves with polynomial conserved quantities.
method Using the non-orthogonality assumption on the conserved quantity, the study deduces the existence of resonance points.
result Every finite type polarised curve in the conformal 2-sphere with a polynomial conserved quantity admits a resonance point.
The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper a new theory of Frobenius-algebra valued integrable s…
This paper shows connections between two complex mathematical theories are equivalent.
problem Establishing equivalence between two complex mathematical theories.
method Using geometric quantisation and conformal field theory, the paper establishes equivalence between the Hitchin connection and the Knizhnik-Zamolodchikov connection.
result The Hitchin and Knizhnik-Zamolodchikov connections are projectively equivalent in genus zero.
The paper bounds solutions to complex optimization problems with uncertain data.
problem Distributionally robust optimization problems with multivariate uncertainty sets.
method Conditions and bounds derived for multivariate and univariate Wasserstein distances, Bregman-Wasserstein divergences, and signed Choquet integrals.
result Computable lower and upper bounds for DRO problems, derived from scalar-valued aggregation functions and Wasserstein distances.
Derives monotonic quantities for p-harmonic functions on manifolds.
problem Understanding p-harmonic functions on manifolds with nonnegative scalar curvature. method Derives local and global monotonic quantities associated with p-harmonic functions. result Establishes inequalities relating mass, capacity, and Willmore functional.
We study n-dimensional area-minimizing currents T in Rn+1, with boundary ∂T satisfying two properties: ∂T is locally a finite sum of (n−1)-dimensional C1,α orientable submanifolds which only meet tangentially and with same orientation, for some α∈(0,1]; ∂T has…
This work connects symmetries and conserved quantities in machine learning.
problem Improving machine learning models by learning conserved quantities.
method Using Noether's theorem, learn symmetries and conserved quantities directly from data.
result Correctly identifies conserved quantities and improves model performance.
Study almost rigidity of super Ricci flow with non-negative Muller quantity.
problem Almost rigidity properties of super Ricci flow with non-negative Muller quantity.
method Almost splitting and quantitative stratification theorems established by Bamler for Ricci flow.
result Obtained almost constancy for a certain integral quantity concerning scalar curvature at an almost self-similar point.
We construct a class of monotonic quantities along the normalized Ricci flow on closed n-dimensional manifolds.
New definitions of conserved quantities at null infinity resolve ambiguities in general relativity.
problem Ambiguities in defining conserved quantities like angular momentum at null infinity.
method New definitions based on Chen-Wang-Yau quasilocal conserved quantities and optimal isometric embedding theory.
result These new definitions are free of supertranslation ambiguity and limit to classical Bondi mass.
Model for hedging price and quantity risks in electricity markets.
problem Hedging risks for energy retailers in a regulated electricity market.
method Closed-form solution for optimal portfolio using financial instruments based on price and weather indexes.
result Closed-form solution for mean-var model in discrete setting without distributional assumptions.
New quantity helps map homotopy classes in complex spaces.
problem Understanding homotopic classes of maps between complex spaces.
method Identified a new monotone quantity in mean curvature flows of maps between Riemannian manifolds.
result Sharp criteria for homotopic classes of maps between complex projective spaces and spheres.
In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild met- rics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively cor- respond to the closed and co-closed, or polar and…
New geometric quantities help classify manifolds and relate to entropy.
problem Classifying Riemannian manifolds using geometric quantities.
method Introducing and analyzing asymptotic geometric quantities like p-capacity, eigenvalues, and Maz'ya constant.
result Geometric quantities coincide with entropy in specific conditions, characterizing manifolds.
Proves properties of neural network basins of attraction and their expressiveness.
problem Characterize the properties of basins of attraction in neural networks.
method Analyzes width-bounded neural networks, proving properties of basins of attraction.
result Boundedness and path-connectedness of basins of attraction under certain conditions.
New inequalities for convex curves with multiple geometric factors.
problem Establishing inequalities for convex curves with multiple geometric factors.
method Parametric isoperimetric-type inequalities for closed convex curves with parameter conditions and equality conditions.
result Derived new inequalities and improved versions of existing inequalities.