We extend rectified flow to infinite-dimensional Hilbert space.
problem Extending rectified flow to infinite-dimensional spaces.
method Established a rigorous functional formulation using the superposition principle for continuity equations.
result Demonstrated superior performance compared to existing models.
The paper explores rectified flows and their relation to optimal transport.
problem Understanding the connection between rectified flows and optimal transport.
method Investigates invariance properties, explicit constructions, and analysis of rectified flows in various settings.
result Rectified flows, when gradient constrained, do not generally solve optimal transport problems.
Rectified flows achieve optimal sample complexity for generating data.
problem Generating high-quality data samples efficiently.
method Rectified flows constrain transport trajectories to be linear, enabling efficient sampling.
result Achieve sample complexity of ildeO(ε−2), matching optimal rate for mean estimation. The rectified flow method is analyzed for its statistical properties.
problem Theoretical support for rectified flow methods is lacking.
method Empirical analysis of rectified flow's statistical properties using regression and density estimation.
result Convergence rates for rectified flow estimators are faster than for nonparametric regression and density estimation.
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
We prove that monotonicity of density and energy inequality imply the rectifiability of the singular sets for Yang-Mills flow.
The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
problem Proving strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
method Establishing a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption of geometry near the base point being close to a generalized cylinder. result Proves strong uniqueness of generalized cylindrical tangent flows and shows that the subset of points with rectifiable Sqck(N) is horizontally parabolic. Study proves existence of a specific type of flow in geometry.
problem Existence of canonical multi-phase free boundary Brakke flows.
method Global-in-time existence established using Brakke flow and uniform density ratio assumption.
result Existence of the flow with no positive mass on the free boundary for some short time.
Language Rectified Flow improves diffusion language generation by simplifying complex steps.
problem Complexity in diffusion language models limits their implementation in NLP applications.
method Reformulates probabilistic flow models to learn neural ODE models for efficient domain transfer.
result Consistently outperforms baselines on fine-grained control tasks and text editing.
Brakke flow support is parabolically rectifiable
problem Support of Brakke flow is parabolically rectifiable
method Developed approach to Brakke flow as space-time-Grassmann measure
result Standard convergence of Brakke flows is equivalent to space-time-Grassmann Radon measures
ReDi improves few-step generation for discrete data models.
problem Slow sampling speeds in discrete flow-based models.
method Rectified Discrete Flow (ReDi) reduces factorization error by rectifying coupling.
result Empirically, ReDi reduces Conditional Total Correlation and enables few-step generation.
The paper studies singular sets in Ricci flow limits, proving rectifiability and curvature bounds.
problem Understanding singular sets in Ricci flow limits.
method Stratification of singular sets, analysis of tangent flows, and geometric measure theory.
result Parabolic rectifiability of singular sets in certain dimensions and uniform curvature bounds.
The abstract proves the existence and regularity of Brakke flows starting from a given set.
problem Existence and regularity of Brakke flows starting from a given set.
method Proves the existence and regularity of Brakke flows using a closed countably 1-rectifiable set in R^2.
result For almost all time, the flow locally consists of a finite number of embedded curves of class W^{2,2} whose endpoints meet at junctions with angles of 0, 60, or 120 degrees.
Study evaluates Tree-Ring Watermarking in rectified flow-based models, revealing detection and separability limitations.
problem Detecting and separating Tree-Ring Watermarks in rectified flow-based models.
method Extensive experimentation comparing SD 2.1 and FLUX.1-dev models with various text guidance configurations and augmentation attacks.
result Inversion limitations affect watermark recovery and statistical separation.
Paper proposes efficient image inversion and editing using rectified stochastic differential equations.
problem Inversion and editing of real images using generative models.
method Proposes RF inversion using dynamic optimal control and a linear quadratic regulator, extending to stochastic sampler for Flux.
result Allows state-of-the-art performance in zero-shot inversion and editing, outperforming prior works.
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 new method for optimal transport using neural ODEs that preserves marginal constraints.
problem Optimal transport between two continuous distributions with specific cost functions.
method Iterative construction of neural ODEs to minimize transport cost while preserving marginal constraints.
result Monotonic interior approach that decreases transport cost efficiently.
Paper explores how Rectified Flow adapts to low-dimensional data.
problem Improving sampling efficiency in low-dimensional data.
method Investigates Rectified Flow's adaptation to low-dimensional support and introduces a stochastic version.
result Shows improved sampling efficiency with O(k/ε) complexity. In this article, we show that, for any compact 3-manifold, there is a C1 volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the exam…
This paper proves a general Uhlenbeck compactness theorem for sequences of solutions of Yang-Mills flow on Riemannian manifolds of dimension n≥4, including rectifiability of the singular set at finite or infinite time.
Paper proposes a novel method to reduce mutual information for missing data imputation.
problem Missing data imputation in datasets with missingness patterns.
method Iterative minimization of KL divergence between imputed data and missingness mask, using rectified flow training objective.
result The proposed method achieves superior imputation performance on synthetic and real-world datasets.
We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature f…
FlowChef steers RFMs to efficiently guide image generation tasks.
problem Efficiently guiding image generation tasks with RFMs.
method Developed a theoretical and empirical understanding of RFMs' vector field dynamics, proposing FlowChef for gradient-free navigation.
result FlowChef significantly outperforms baselines in performance, memory, and time requirements.
Optimal self-distillation improves generative models' velocity risk and mode recovery.
problem Improving generative models' velocity risk and mode recovery.
method Proved optimal self-distillation for rectified flow via linear probing, derived mixing coefficient, and provided validation tuning.
result Optimal self-distillation improves velocity risk and mode recovery.
Until recently, Ricci flow was viewed almost exclusively as a way of deforming Riemannian metrics of bounded curvature. Unfortunately, the bounded curvature hypothesis is unnatural for many applications, but is hard to drop because so many new phenomena can occur in the general case. This article surveys some of the th…
Flow of curves with curvature and forcing vector field exists.
problem Existence of a curve flow with curvature and forcing.
method Proved existence through Brakke motion law.
result Non-trivial flow of curves exists through singularities.
CCVFM uses coreset to improve generative models by refining residual flows.
problem Generating multimodal distributions from scratch is challenging.
method Augments hierarchical rectified flow with a data-informed source distribution using a coreset.
result CCVFM achieves competitive few-step generation without a learned noise-to-data map.
Proves existence of multi-phase flows from arbitrary initial data.
problem Non-uniqueness issue in Brakke flows.
method Global existence proof for multi-phase mean curvature flow.
result Validates explicit identity for evolving grain volumes.
Study shows non-uniqueness of Brakke flow near flat singular points.
problem Exploring instability of minimal surfaces at flat singular points.
method Analyzes the behavior of stationary varifolds and their blow-ups.
result Proves existence of non-constant Brakke flow near flat singular points.
We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the 1-skeleton. The theorem is proved by applying a sort of volume-increasing flow to any h…
Suppose that Γ0⊂Rn+1 is a closed countably n-rectifiable set whose complement Rn+1∖Γ0 consists of more than one connected component. Assume that the n-dimensional Hausdorff measure of Γ0 is finite or grows at most exponentially near infinity. Under these assumptions, we…
Study rectifying curves in 3D multiplicative Euclidean space.
problem Investigate rectifying curves in a non-Newtonian geometry setting.
method Apply multiplicative differential-geometric concepts to rectifying curves.
result Classify multiplicative rectifying curves using spherical curves.
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
A space curve in a Euclidean 3-space E3 is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n…
Study characterizes k-rectifiable sets in homogeneous groups.
problem Characterizing k-rectifiable sets in arbitrary homogeneous groups. method Proves characterizations using (k,G)-approximate tangent groups. result Existence of (k,G)-approximate tangent groups implies k-rectifiability. Recalls and refines the concept of algebraically rectifiable curves.
problem Classical notion of algebraically rectifiable plane curves.
method Provides new criteria, relates to quadratic differentials, and generalizes to higher order differentials.
result Generalization and new criteria for algebraic rectifiability.
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into F-connected complexes. We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and su…
In this paper we get a version of mean value inequality for generalized self-expander type submanifolds in Euclidean space. As the application, we prove that if mean curvature flow M(t) on the self-expander in Euclidean space subconverges to an n-rectifiable varifold T in weak sense for t goes to the singular t…
New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
Rectifies singular set of harmonic maps into complex.
problem Regularity of harmonic maps into complex manifolds.
method Proves (m−2)-rectifiability of singular set. result Singular set is (m−2)-rectifiable. The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. In this paper we investigate the performance of different types of rectified activation functions in convolutional neural network: standard rectified linear unit (ReLU), leaky rectified linear unit (Leaky ReLU), parametric rectified linear unit (PReLU) and a new randomized leaky rectified linear units (RReLU). We evalu…
Neural networks can approximate rectifiable measures with small error.
problem Approximating complex rectifiable measures using neural networks.
method Using ReLU neural networks to approximate (countably) m-rectifiable measures as push-forwards of the Lebesgue measure.
result The approximation error in terms of Wasserstein distance can be made arbitrarily small.
We show that C^1 hypersurfaces in the Heisenberg group are countably N-rectifiable. As a corollary, this shows that all C^1_H graphs over the xy-plane are countable N-rectifiable, showing the equivalence of this notion of rectifiability with that of Franchi, Serra Cassano and Serapioni for such surfaces.
New method uses neural maps to efficiently sample lattice QCD distributions.
problem Challenges in sampling Boltzmann distributions of lattice field theories.
method Sparse triangular transport maps exploiting conditional independence structure of lattice graphs.
result Sparse triangular maps achieve efficient sampling with linear time complexity in lattice size.
The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and n_2 of the curve. In this study, we have obtained some characterizations of semi-real spatial quaternionic rectifyin…
Develops weak formulation for spacelike flows in pseudo-Euclidean space.
problem Weak formulation of spacelike mean curvature flow in pseudo-Euclidean space.
method Based on spacelike integer rectifiable varifolds and pseudo-Euclidean first variation.
result Existence and compactness of spacelike Brakke flows with fixed boundary.