Given two points on a soup can or conical cup with lid, we find and classify all paths of minimal length connecting them. When the number of minimal paths is finite, there are at most four on a can and three on a cup. At worst, minimal paths are piece-wise smooth with three components, each of which is a classical geod…
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48 results for “minimizing paths”
In this paper we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of L. Polterovich and M. Schwarz, we study the role of the fixed global extrema in the Floer complex of the generating Hamiltonian. Our…
Characterizes paths minimizing anisotropic lengths in Euclidean space.
problem Finding paths of minimal anisotropic length between points.
method Characterization through geometric connection to anisotropic isoperimetric set.
result Established a connection between minimizing paths and anisotropic isoperimetric geometry.
Algorithm minimizes risk for multiclass classification of stochastic diffusion paths.
problem Multiclass classification of stochastic diffusion paths with distinct drift functions.
method Empirical risk minimization using L2 risk.
result Achieves fast rates of convergence under margin assumption.
We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group . For a compact symplectic manifold of dimension two or four, we show that a path in , generated by an autonomous Hamiltonian and starting at the identity, which induces no non-cons…
In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on . We then discuss conditions which guarantee that su…
Diagonal linear networks converge to lasso regularization path during training.
problem Understanding the regularization behavior of diagonal linear networks.
method Analyzing the training trajectory of diagonal linear networks and comparing it to the lasso regularization path.
result The training trajectory of diagonal linear networks is closely related to the lasso regularization path.
We solve the paradox of score-based methods by minimizing path variance.
problem Score-based methods are path-dependent, leading to inaccurate and unstable estimators.
method Propose MVP Principle to minimize path variance, derive closed-form expression, and use flexible Kumaraswamy Mixture Model.
result Establishes new state-of-the-art results on challenging benchmarks.
We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-R…
New analysis of annealing paths in sampling and estimation.
problem Sampling from complex distributions and estimating normalization constants.
method Extending known results on Bregman divergence to quasi-arithmetic means under monotonic embedding.
result Analogous result for quasi-arithmetic means, highlighting the interplay between means, parametric families, and divergence functionals.
The recently developed bag-of-paths (BoP) framework consists in setting a Gibbs-Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature ) controlling the entropic level of the distribution. This formalism enables…
This paper optimizes paths for generative models using kinetic energy.
problem Improving generative model performance and sample quality.
method Investigating and optimizing Gaussian probability paths with kinetic energy.
result Kinetic optimal Gaussian paths simplify particle trajectories and improve model performance.
Sub-Riemannian geometry connects bike paths to mathematical curves.
problem Understanding bike paths and their mathematical properties.
method Relating sub-Riemannian geometry to bicycle motion and curve shapes.
result Geodesics in sub-Riemannian geometry correspond to specific bike paths.
Algorithm minimizes regret and converges to equilibria in Markov games.
problem Regret minimization and convergence to equilibria in general-sum Markov games under adversarial opponents.
method Decentralized algorithm that uses policy optimization and controls path length to achieve sublinear regret.
result Sublinear regret guarantees for convergence to correlated equilibrium in Markov games.
New method for efficient proximal mapping of 1-path-norm in shallow networks.
problem Efficiently handling the 1-path-norm of shallow neural networks.
method Closed-form proximal operator for efficient computation and upper bound on Lipschitz constant.
result Proximal mapping allows robust training against adversarial perturbations.
The paper develops fair machine learning models using causal path-specific effects.
problem Fairness in machine learning models under causal constraints.
method Lagrange multiplier approach for infinite-dimensional functional estimation, closed-form solutions for constrained optimization.
result Theoretical and flexible semiparametric estimation strategies for fair predictions.
In this paper, we use Floer theory to study the Hofer length functional for paths of Hamiltonian diffeomorphisms which are sufficiently short. In particular, the length minimizing properties of a short Hamiltonian path are related to the properties and number of its periodic orbits.
Introduces q-paths for generalizing geometric annealing paths in machine learning.
problem Limited applicability of existing path methods in machine learning.
method Develops a family of paths derived from a generalized mean, including geometric and arithmetic mixtures.
result Empirical gains in Bayesian inference and generative model evaluation.
CNFs learn on manifolds using PPD, improving likelihood and sample quality.
problem Training CNFs on manifolds efficiently and accurately.
method Minimizing PPD, a novel divergence, to train CNFs on manifolds.
result CNFs trained with PPD achieve state-of-the-art results on manifold benchmarks.
Study on Kähler metrics on ruled surfaces, proving existence and non-existence.
problem Existence and non-existence of Kähler metrics on minimal ruled surfaces.
method Analysis of twisted and coupled constant scalar curvature Kähler metrics.
result Bound for Chen-Cheng invariant on ruled surfaces.
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.
This paper deals with the question of analytic continuation of holonomy germs of holomorphic foliations. We prove that for a quasi-minimal Riccati foliation of the complex projective plane, any holonomy germ of the foliation between complex projective lines can be analytically continued along a generic Brownian path.
Update rules for learning in dynamic time warping spaces are based on optimal warping paths between parameter and input time series. In general, optimal warping paths are not unique resulting in adverse effects in theory and practice. Under the assumption of squared error local costs, we show that no two warping paths …
Implicit Finite-Horizon Approximation and Efficient Optimal Algorithms for Stochastic Shortest Pathcs.LG
New algorithms minimize regret in SSP with optimal sparse updates.
problem Minimizing regret in Stochastic Shortest Path models.
method Implicit finite-horizon approximation for analysis, model-free and model-based algorithms developed.
result Minimax optimal regret for both model-free and model-based algorithms.
Recently, path norm was proposed as a new capacity measure for neural networks with Rectified Linear Unit (ReLU) activation function, which takes the rescaling-invariant property of ReLU into account. It has been shown that the generalization error bound in terms of the path norm explains the empirical generalization b…
Linear time dynamic programming for the exact path of optimal models selected from a finite setcs.LG
New algorithm speeds up path computation for optimal models.
problem Finding the exact path of optimal models from a finite set.
method Dynamic programming approach for linear time computation.
result Dynamic programming achieves linear time for breakpoints computation.
On existence, uniqueness and scalability of adversarial robustness measures for AI classifiersstat.ML
Mathematical conditions and practical computations for adversarial robustness measures are established.
problem Existence, uniqueness, and scalability of adversarial robustness measures for AI classifiers.
method Formulated and proven mathematical conditions for existence, uniqueness, and explicit analytical computation of minimal adversarial paths and distances. Practical computation demonstrated on various AI tools and synthetic benchmarks.
result Explicit mathematical conditions and practical computations for adversarial robustness measures are established.
We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of Itô's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlle…
Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation …
Optimizes diffusion processes for target distributions.
problem Efficiently generating target distributions from point masses.
method Stochastic interpolant framework with conditional expectation drift.
result Optimal diffusion coefficient minimizes path-space KL divergence.
Develops a new solver for path-dependent PDEs using signature kernels.
problem Solving path-dependent PDEs (PPDEs) efficiently and accurately.
method Uses signature kernels to solve PPDEs by approximating the solution with minimal norm in a reproducing kernel Hilbert space.
result Proves the consistency of the numerical scheme, ensuring convergence to PPDE solutions as the number of collocation points increases.
Transformer model improves asset allocation by unifying forecasting and optimization.
problem Separation of forecasting and optimization leads to suboptimal portfolios.
method Signature Informed Transformer using path signatures and specialized attention.
result Direct minimization of Conditional Value at Risk improves performance.
Gradient descent implicitly follows regularization for general losses.
problem The implicit bias of gradient descent methods in machine learning.
method Empirical risk minimization over linear predictors with arbitrary convex, strictly decreasing losses.
result Gradient descent and regularization paths converge to the same direction for non-attained risks.
Algorithm reduces regret in SSP problems with LFA.
problem Finding shortest paths in stochastic environments with linear approximations.
method Uses linear function approximation and stationary policies to minimize regret.
result Achieves sublinear regret under minimal assumptions.
Generative Flow Networks solve shortest path problems in graphs.
problem Finding shortest paths in graphs.
method Generative Flow Networks with flow regularization.
result Training a GFlowNet can solve pathfinding problems in arbitrary graphs.
The geodesic equation for the right invariant -metric (which is a weak Riemannian metric) on each Virasoro-Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular solutions of KdV don't define…
Study shows not all smooth paths are optimal in certain geometric structures.
problem Existence of non-smooth sub-Riemannian minimizing geodesics.
method Constructed a but not length-minimizer example.
result Found a real-analytic sub-Riemannian structure with non-smooth minimizers.
Proposes a new method to learn entire solution paths without discretization.
problem Optimizing a family of problems indexed by hyperparameters.
method Parameterizes the solution path with basis functions and solves a single stochastic optimization problem.
result Uniform error of learned path converges linearly to a constant related to basis expressiveness.
The relaxed maximum entropy problem is concerned with finding a probability distribution on a finite set that minimizes the relative entropy to a given prior distribution, while satisfying relaxed max-norm constraints with respect to a third observed multinomial distribution. We study the entire relaxation path for thi…
Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimi…
Proposes MM-DUST for efficient generalized lasso solution paths.
problem Efficiently solve generalized lasso problems in large-scale and non-linear models.
method Majorization-minimization dual stagewise algorithm incorporating quadratic majorizers and stagewise learning.
result Established the uniform convergence of approximated solution paths.
This study examines biases in flow matching samplers using finite-sample estimation.
problem Biases in flow matching samplers when using finite-sample surrogates.
method Finite-sample plug-in estimation and hierarchy of empirical FM models.
result Exact empirical minimizer and smoothed plug-in regime identified for affine conditional flows.
Improved sampling efficiency for molecular systems using path gradients after Flow Matching.
problem Improving sampling efficiency for complex molecular systems.
method Hybrid approach combining Flow Matching and path gradients.
result Up to a threefold increase in sampling efficiency for molecular systems.
A new slicing method speeds up sliced Wasserstein estimation.
problem Efficiently estimating sliced Wasserstein distance.
method Random-Path Projecting Direction (RPD) for fast sampling.
result RPSW and IWRPSW show favorable performance in training generative models.
Optimal transport with path constraints for distributions of different masses.
problem Comparing distributions with different total masses under path constraints.
method Introduces a model for unbalanced optimal transport with path constraints, proving existence of solutions.
result Existence of solutions to path constrained unbalanced optimal transport for various constraints.
A new method to rescale ReLU neural networks based on path-lifting.
problem Lack of principled ways to leverage rescaling symmetries in ReLU neural networks.
method Introduces a geometrically motivated criterion to rescale neural network parameters, aligning a kernel in the path-lifting space with a chosen reference.
result Proposed method can speed up training and aligns a kernel in the path-lifting space with a chosen reference.
Unconstrained Online Optimization: Dynamic Regret Analysis of Strongly Convex and Smooth Problemscs.LG
Paper analyzes regret bounds for unconstrained online optimization.
problem Minimizing regret in dynamic online learning for strongly convex and smooth functions.
method Preconditioned OGD, Online Optimistic Newton (OON), multiple gradient queries.
result Achieves regret bound with one gradient query per round.
Geodesics in Jet Spacemath.OC
Geodesics in jet space are constructed from polynomials, with some yielding globally minimizing paths.
problem Characterize geodesics in jet space and identify those that are globally minimizing.
method Sub-Riemannian geometry, Hamilton-Jacobi equations, and analysis of period degenerations.
result Some polynomials yield globally minimizing geodesics, with conjectures on the independence of cut time.