This paper improves the efficiency of generative models by optimizing the straightness of Rectified Flow.
problem Improving the efficiency of generative models by reducing discretization error.
method Introducing a novel Piecewise Straightness parameter, γ2,T, to optimize the straightness of Rectified Flow.
result Minimizing curvature in Rectified Flow models leads to high-fidelity, one-step sampling.
A piecewise flat Finsler metric on a triangulated surface M is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…
SURF simplifies distribution estimation with simple, robust, and fast algorithms.
problem Efficient and accurate distribution estimation in statistics and machine learning.
method Piecewise polynomial approximation using empirical probability interpolation and divide-and-conquer merging.
result Surpassing state-of-the-art algorithms in efficiency and accuracy, SURF estimates distributions robustly and quickly.
We present two families of knots which have straight number higher than crossing number. In the case of the second family, we have computed the straight number explicitly. We also give a general theorem about alternating knots that states adding an even number of crossings to a twist region will not change whether the …
Jablan and Radović originally defined two invariants called the Meander number and OGC number of knots for certain classes of knots. We generalize these definitions to all knots and name the straight number and contained straight number of a knot, respectively, and prove they are well defined. We answer two questions a…
Kirigami-inspired math reveals shortest paths and ultimate shapes of cut paper.
problem Geodesics and isometric immersions in paper with cuts.
method Constructive proof of geodesics and rectification of polygonal geodesics.
result Polygonal geodesics can be rectified into a straight line by flat-folding.
Training activation quantized neural networks involves minimizing a piecewise constant function whose gradient vanishes almost everywhere, which is undesirable for the standard back-propagation or chain rule. An empirical way around this issue is to use a straight-through estimator (STE) (Bengio et al., 2013) in the ba…
The study characterizes straight-line flows in dynamic measure transport.
problem Tackles the challenge of designing flows that are easy to integrate.
method Characterizes straight-line flows using a PDE and Reynolds tensor.
result Characterizes affine-in-time interpolants and necessary conditions for flow geometry.
The paper triangulates Heisenberg groups with horizontal and straight simplexes.
problem Triangulating Heisenberg groups with specific regularity properties.
method Constructing triangulations with horizontal and straight simplexes on a polyhedral structure and extending to the whole Heisenberg group.
result Explicit examples of grid and triangulations provided.
A new method learns straight trajectories in one step for optimal flow matching.
problem Learning flows with straight trajectories for fast inference.
method Optimal Flow Matching (OFM) approach using convex functions for vector fields.
result Recovering straight OT displacements in just one FM step for quadratic transport.
Study on straight-line flows for generative modeling with theoretical obstructions.
problem Existence and obstructions of straight-line flows in generative modeling.
method Characterizations of straight-line flows through PDEs involving conditional statistics of stochastic processes.
result Sharp dichotomy in the existence of straight-line flows for targets with well-separated modes.
We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally …
Simpler algorithms for morphing planar and toroidal graphs.
problem Constructing smooth transitions between isomorphic drawings of planar and toroidal graphs.
method Barycentric interpolation and scaling strategy.
result Simplified and more natural morphs with improved computational efficiency.
Correct method found for drawing precise envelope of straight lines.
problem Widespread method fails to represent the precise shape of envelope.
method Recently discovered correct method for straight line families in the plane.
result Correct method precisely represents the envelope of straight lines.
Study of straight-line flows on a unique infinite surface.
problem Understanding straight-line flows on a specific infinite surface.
method Geometric description and characterization of periodic and drift orbits; use of rigid symmetries and Veech group.
result Complete characterization of periodic directions and proof of density of periodic and ergodic directions.
New method for estimating gradients in stochastic binary networks.
problem Challenges in training neural networks with binary activations and weights.
method Combines sampling and analytic approximation steps to estimate gradients accurately.
result Significantly reduced variance at the cost of small bias, leading to practical tradeoffs.
We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.
Smooth compactness theorem for elasticae, except straight segments.
problem Compactness of elasticae space.
method Smooth compactness theorem proof.
result Smooth stability results for minimizers.
The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled surfaces. The finite energy case yields a limit of images, under a suitable topolo…
To make deep neural networks feasible in resource-constrained environments (such as mobile devices), it is beneficial to quantize models by using low-precision weights. One common technique for quantizing neural networks is the straight-through gradient method, which enables back-propagation through the quantization ma…
Reintroduces straight-through estimators for binary neural networks.
problem Training neural networks with binary weights and activations is challenging due to gradient issues and discrete weight optimization.
method Derives ST methods as estimators in the SBN model, analyzes properties and estimation accuracy, explains latent weights and mirror descent method.
result Reintroduces ST methods as sound approximations and provides clearer application and improvements.
In this paper, we investigate the ruled surfaces generated by a straight line according to rotation minimizing frame (RMF). Using this frame of a straight line, we obtained the necessary and sufficient conditions when the ruled surface is developable. Also, we give some new results and theorems related to be the asympt…
The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincid…
Paper improves Gumbel-Softmax estimator variance reduction.
problem Challenges in gradient estimation for models with discrete latent variables.
method Rao-Blackwellization applied to straight-through Gumbel-Softmax estimator.
result Reduces mean squared error and variance of Gumbel-Softmax estimator.
We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…
The study explores isometric deformations of surfaces of translation.
problem Determine the ways surfaces of translation bend isometrically.
method Analyzes existence conditions and provides closed-form expressions for infinitesimal and finite bendings of surfaces of translation.
result Surfaces of translation admit various infinitesimal and finite bendings, including purely torsional and torsion-free.
New method describes entanglement of straight lines in 3D space.
problem Tackles the geometry and topology of configurations of straight lines.
method Introduces direction matrices and a discrete motion principle.
result Shows n-crosses as links of pairwise connected unknots.
Proves cup product homomorphism for bounded cohomology on negatively curved manifolds.
problem Understanding cup product behavior in bounded cohomology.
method Analyzes map Ψ* associating closed forms to bounded cohomology classes via integration.
result Proves Ψ* preserves cup product in sufficiently high degrees.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
Classifies branched Willmore spheres using conformal Gauss maps.
problem Classifying branched Willmore spheres.
method Analyzing the asymptotic expansion of the conformal Gauss map at branched points.
result Full classification of branched Willmore spheres.
Neural networks can represent complex piecewise functions efficiently.
problem Representing continuous piecewise affine functions with neural networks.
method Two hidden layers with ReLU activation, O(p) neurons for p pieces. result CPA functions can be represented by a neural network with linear size.
Theorem proves integrability for piecewise-smooth distributions.
problem Integrability of piecewise-smooth distributions.
method Generalizations of Frobenius integrability theorem.
result Sufficient criteria for complete integrability with bi-Lipschitz coordinates.
New insights into quantized neural networks reveal learning dynamics and generalization errors.
problem Understanding the impact of quantization hyperparameters on learning dynamics in high-dimensional models.
method Theoretical analysis and fixed-point analysis of STE dynamics in quantized models.
result STE training in quantized models converges to a plateau followed by a sharp drop in generalization error, influenced by quantization range.
In this paper, on the first, we prove Δr=2H where Δ is the Laplacian operator, r=(r1,r2,r3) the position vector field and H is the mean curvature vector field of a surface S in the 3-dimensional Heisenberg group H3. In the second, we classify the ruled surfaces by straight…
Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
The study identifies unique fluid flow patterns.
problem Understanding incompressible fluid flows with straight streamlines.
method Local differential geometry of line congruences to integrate Euler equations.
result Only specific fluid flows are possible with straight streamlines.
New concept of effective isometries for compliant shells.
problem Inadequate classification of isometric deformations for compliant shells.
method Introduce effective isometric deformations defined by first-order isometry in a small scale separation parameter.
result Effective isometries are solutions to a quasilinear second-order PDE.
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.
A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …
Study geometrically characterizes piecewise circular curves with decreasing curvature.
problem Characterizing piecewise circular curves with decreasing curvature.
method Introducing moduli spaces and relating them to Legendrian polygons.
result Proves the moduli space contains a connected component homeomorphic to the Fock-Goncharov space of positive flags.
The center of a quotient group of piecewise linear homeomorphisms is trivial.
problem Understanding the structure of a specific group of homeomorphisms.
method Analyzing a quotient of piecewise linear homeomorphisms of the real line.
result The center of the quotient group is trivial.
New heat trace coefficients reveal curvature effects in polygonal domains.
problem Understanding heat trace behavior in polygonal domains with curved corners.
method Local heat trace expansion through order t1/2, analyzing both Dirichlet and Neumann boundary conditions. result Sharp sign law for the Dirichlet angular factor of the first corner-curvature heat invariant.
GraN-GAN normalizes gradients for better GAN performance.
problem Improving image generation in GANs with piecewise linear discriminators.
method Piecewise Gradient Normalization (GraN) for input-dependent normalization.
result Significant performance gains in image generation across various datasets.
The paper extends a variance gamma model to quadratic functions, reducing arbitrage and computational costs.
problem Creating an arbitrage-free interpolation for option pricing models.
method Generalizing the local variance gamma model to a piecewise quadratic local variance function.
result The quadratic model results in an arbitrage-free interpolation of class C3, reducing knots and computational cost.
Study knot diagrams on a sphere without vertical lines, focusing on minimal crossings.
problem Understanding minimal crossings of knot diagrams on a punctured sphere.
method Mathematical model of string figures using knot diagrams on xyz-space with missing vertical lines, analyzing minimal crossings under Reidemeister moves. result Minimal number of crossings of knot diagrams on a punctured sphere.
The aim of this paper is to investigate properties preserved and co-preserved by coarsely n-to-1 functions, in particular by the quotient maps X→X/∼ induced by a finite group G acting by isometries on a metric space X. The coarse properties we are mainly interested in are related to asymptotic dimension a…
This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of piecewise linear spaces we define and discuss (normalized) Ricci flows. …
We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …