Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
A model predicts influential nodes in complex networks by considering indirect interactions.
problem Identifying influential nodes in complex networks using indirect interactions.
method Proposes MOGen, a multi-order generative model that considers all indirect influences up to a maximum distance.
result MOGen consistently outperforms network models and path-based approaches in predicting influential nodes.
Deep nets trained with MSE loss exhibit Neural Collapse, collapsing features and classifiers to class means.
problem Understanding Neural Collapse in MSE-trained deep nets.
method Developed a new MSE loss decomposition and introduced the central path concept.
result Exact dynamics of Neural Collapse along the central path can be predicted.
Graph embedding method captures both local and global network structure.
problem Representing and analyzing complex graph networks.
method Spectral embedding based on a generalized graph Laplacian.
result Significant improvement in data analysis tasks.
This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs-Boltzmann distribution is defined over a bag of paths in a network, that is, on a representati…
The paper introduces surface signatures for irregular surfaces and rough surfaces.
problem Characterizing and integrating highly irregular paths and surfaces.
method Introducing surface signatures and proving extension theorems.
result Surface signatures are universal for surface holonomy and rough surfaces.
Unified approach to stochastic control, filtering, and stopping using rough paths.
problem Addressing gaps in classical problems of stochastic control, filtering, and stopping.
method Combining rough path theory with controlled rough paths to provide a pathwise deterministic framework.
result Established rigorous connection between candidate solutions and Hamilton-Jacobi-Bellman equation.
Paper explores rough path theory for frictionless markets, linking NCFL to unbiased rough integrators.
problem Tackles the limits of rough path theory in frictionless markets.
method Investigates the capacity of rough path theory to support No Free Lunch markets.
result Establishes a 'Rough Kreps-Yan' theorem linking NCFL to unbiased rough integrators.
We study the problem of online path learning with non-additive gains, which is a central problem appearing in several applications, including ensemble structured prediction. We present new online algorithms for path learning with non-additive count-based gains for the three settings of full information, semi-bandit and…
PAN uses path integrals for graph convolution and pooling, improving GNN performance.
problem Designing efficient graph convolution and pooling for graph neural networks.
method Path integral based graph convolution and pooling using learnable weights for path lengths.
result PAN achieves state-of-the-art performance on various graph classification/regression tasks.
Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central…
A new path development layer reduces dimensionality for irregular time series.
problem High-dimensional irregular paths in machine learning.
method Finite-dimensional Lie group representations for dimension reduction.
result The development layer outperforms signature features in accuracy and dimensionality.
New SigSwap model for path-dependent financial risk.
problem Managing complex, path-dependent financial risks.
method Geometry-based approach using path-signature and Signature Expected Shortfall.
result Path-dependent risks can be converted into transparent risk factors.
Path regularization improves GFlowNets exploration and generalization.
problem Improving GFlowNets exploration and generalization.
method Path regularization based on optimal transport theory.
result Path regularization enhances GFlowNets to generate more diverse and novel candidates.
Study on when the lower central series stops for various groups, including braid groups.
problem Understanding when the lower central series stops for different groups.
method Various techniques applied to braid groups and related groups.
result Complete computation of the lower central series for most groups studied.
DAGMA learns DAGs faster and more accurately using log-determinant acyclicity.
problem Learning directed acyclic graphs from data efficiently and accurately.
method DAGMA uses M-matrices and log-determinant acyclicity to optimize DAG learning.
result DAGMA achieves faster and more accurate DAG learning compared to existing methods.
Many convex problems in machine learning and computer science share the same form: \begin{align*} \min_{x} \sum_{i} f_i( A_i x + b_i), \end{align*} where fi are convex functions on Rni with constant ni, Ai∈Rni×d, bi∈Rni and ∑ini=n. This probl…
Let G be a connected Lie group, LG its loop group, and PG->G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A=T(PG)/LG of this bundle. Given a symmetric bilinear form on the Lie algebra g and the corresponding central extension of Lg,…
Novel framework synthesizes stochastic trajectories with anticipated structural breaks.
problem Synthesizing forward-looking, time-evolving stochastic trajectories with anticipated structural breaks.
method Anticipatory Neural Jump-Diffusion (ANJD) flow, AVNSG for dynamic spectral whitening.
result The framework effectively captures non-commutative moments and high-order stochastic texture.
Efficiently computes sparse signature coefficients using kernels.
problem Lack of efficient methods for sparse signature coefficients.
method Signature kernels and PDE-based methods.
result Sparse groups of signature coefficients can be isolated effectively.
A new network learns to prioritize messages for efficient multi-robot path planning.
problem Efficient path planning and coordination for large-scale multi-robot systems.
method Message-Aware Graph Attention Network (MAGAT) incorporating attention mechanisms.
result MAGAT achieves performance close to a coupled centralized expert algorithm.
New method uses neural networks to solve complex PDEs from optimal control theory.
problem Solving high-dimensional Hamilton-Jacobi-Bellman PDEs.
method Iterative diffusion optimization techniques, focusing on path measures and divergences.
result Favourable properties of log-variance divergence for Monte Carlo estimators.
A conformal procedure improves CoT reasoning by aggregating reasoning paths and calibrating abstention rules.
problem Aggregation uncertainty in chain-of-thought reasoning makes correct answers less reliable.
method Introduces a conformal procedure for CoT reasoning that uses weighted score aggregation and abstention rules.
result Achieves higher selective accuracy with abstention, reducing confident-error rate.
SigMA uses signatures and attention to estimate parameters in fBm-driven SDEs.
problem Estimating parameters in SDEs driven by fBm is challenging due to non-Markovian and semimartingale issues.
method SigMA integrates path signatures with multi-head self-attention, using convolutional and MLP layers.
result SigMA outperforms other methods in accuracy, robustness, and model compactness.
The paper develops statistical inference for gradient flows in optimization.
problem Uncertainty quantification along the entire optimization path.
method Uniform central limit theorem and algorithm-aware covariance estimator.
result Asymptotically valid confidence intervals for target parameter.
A new framework uses stochastic optimal control to estimate rare events more accurately.
problem Estimating rare events like chemical reactions in biomolecules is computationally challenging.
method The approach casts committor estimation as a stochastic optimal control problem, developing direct and off-policy Value Matching losses.
result The framework yields more accurate committor estimates, reaction rates, and equilibrium constants.
Motivated by considerations of euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this end, we study analytic paths of metrics that induce isospectral Laplace-Beltram…
Develops a lifting theory for exponential maps in semi-Riemannian geometry.
problem Overcoming singularities in exponential maps to prove geodesic connectivity.
method Lifting theory for semi-Riemannian manifolds with path-continuation property.
result General path-lifting theorem extending globally under certain conditions.
The Constant Elasticity of Variance (CEV) model significantly outperforms the Black-Scholes (BS) model in forecasting both prices and options. Furthermore, the CEV model has a marked advantage in capturing basic empirical regularities such as: heteroscedasticity, the leverage effect, and the volatility smile. In fact, …
In this note we propose a method based on artificial neural network to study the transition between states governed by stochastic processes. In particular, we aim for numerical schemes for the committor function, the central object of transition path theory, which satisfies a high-dimensional Fokker-Planck equation. By…
In this work, we propose an algorithm to price American options by directly solving the dual minimization problem introduced by Rogers. Our approach relies on approximating the set of uniformly square integrable martingales by a finite dimensional Wiener chaos expansion. Then, we use a sample average approximation tech…
Consider the problem of a central bank that wants to manage the exchange rate between its domestic currency and a foreign one. The central bank can purchase and sell the foreign currency, and each intervention on the exchange market leads to a proportional cost whose instantaneous marginal value depends on the current …
Develops methods for estimating constrained function-valued parameters in infinite-dimensional models.
problem Estimating function-valued parameters with structural constraints in complex models.
method Characterizes constrained solutions as minimizers of penalized population risk, using a Lagrange-type formulation and path through unconstrained space.
result Proposes estimators that achieve optimal risk and constraint satisfaction, applicable across various statistical learning approaches.
A semigroup of annuli integrates a central extension of vector fields on S^1.
problem No Lie group exists for complexified vector fields on S^1.
method Introduced an enlargement of the semigroup of annuli and proved it integrates a central extension of vector fields.
result Every partially thin annulus is the time-ordered exponential of a path in the cone of inward pointing complexified vector fields.
Study finds non-IID data causes FL performance issues.
problem Reduced performance in federated learning due to non-IID data.
method Investigated from IID to non-IID settings, categorized methods into two strategies.
result Inconsistencies in client loss landscapes are the primary cause of performance degradation.
Given a limited number of entries from the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, recovery of the low-rank and sparse components is a fundamental task subsuming compressed sensing, matrix completion, and principal components pursuit. This paper devel…
This paper extends rack and quandle covering theory using higher categorical Galois theory.
problem Developing a higher covering theory of racks and quandles.
method Applying techniques from higher categorical Galois theory to extend and clarify the foundations of rack and quandle coverings.
result Identification of meaningful higher-dimensional centrality conditions defining higher coverings of racks and quandles.
In this paper we study the tensor powers of the standard representation of the quantum super-algebra Uq(sl(2∣1), focusing on the rings of its algebra endomorphisms, called centraliser algebras and denoted by LGn. Their dimensions were conjectured by I. Marin and E. Wagner \cite{MW}. We prove this conjecture, desc…
This paper analyzes Local SGD for federated learning, achieving both statistical and communication efficiency.
problem Statistical estimation and inference in federated learning with decentralized data.
method Local SGD, a multi-round estimation procedure using intermittent communication.
result Local SGD achieves both statistical efficiency and communication efficiency.
A new method uses deep learning to efficiently sample rare transitions for estimating committor functions.
problem Efficiently sampling rare transitions to estimate committor functions in high-dimensional problems.
method DASTR (Deep Adaptive Sampling on Transition Paths) method using deep generative models.
result Significantly improved accuracy in approximating committor functions through efficient sampling.
Since the early days of digital communication, Hidden Markov Models (HMMs) have now been routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. An HMM (Xi,Yi)i≥1 assumes observations X1,X2,... to be conditionally independent given an "explanotary" Markov proc…
Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM (Xi,Yi)i≥1, observations X1,X2,... are assumed to be conditionally independent given an ``explanator…
This paper analyzes the multi-armed bandit model using path-integral methods.
problem Understanding the stochastic dynamics and optimal strategies in multi-armed bandit problems.
method Path-integral analysis of statistical physics.
result Emergence of multimodal regret distribution with large regrets from exploitation of sub-optimal arms.
New method uses resurgent analysis to determine growth rate of quantum field theory coefficients.
problem Determining the growth rate of quantum field theory coefficients.
method Resurgence analysis on the Stokes line, leading to transseries decomposition and continued across natural boundary.
result Essential exponent of growth has Cardy-like interpretation as effective central charge.
Stochastic kernel based dimensionality reduction approaches have become popular in the last decade. The central component of many of these methods is a symmetric kernel that quantifies the vicinity between pairs of data points and a kernel-induced Markov chain on the data. Typically, the Markov chain is fully specified…
Let Σ be a closed surface, G a compact Lie group, with Lie algebra g, ξ:P→Σ a principal G-bundle, let N(ξ) denote the moduli space of central Yang-Mills connections on ξ, for suitably chosen additional data, and let $\roman{Rep}_ξ(Γ,G)$ be the space of representations of the universal central ex…
In this paper we address the question of the existence of a model for the string 2-group as a strict Lie-2-group using the free loop group LSpin (or more generally LG for compact simple simply-connected Lie groups G). Baez-Crans-Stevenson-Schreiber constructed a model for the string 2-group using a based loop gro…
New technique identifies lead-lag relationships in FX market during pandemic.
problem Identifying lead-lag relationships in financial markets, especially during crises.
method Dynamic Programming technique for finding optimal lead-lag path, using a loose metric.
result The proposed technique gives the best results in identifying statistically significant paths and closest forecasts.