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.
Develops a new method for solving sparse non-negative least squares problems.
problem Sparse non-negative least squares problem.
method Bayesian evidence maximization framework using Rectified Gaussian Scale Mixture (R-GSM) prior and Expectation-Maximization (EM) algorithm.
result Proposed R-SBL method outperforms existing solvers in signal and support recovery.
New rectified flow method improves image generation and converges to optimal transport.
problem Improving computational and statistical guarantees of rectified flow for image generation.
method Introducing c-rectified flow, which projects velocity fields onto a gradient class while preserving marginals.
result Iterative c-rectified flow always converges to the optimal transport coupling under suitable assumptions.
Paper introduces a new GGM variant for efficient inference and unsupervised learning.
problem Limited modeling abilities of traditional Gaussian graphical models.
method Introduces a novel variant of Gaussian graphical models with truncated normal distributions and bipartite structure.
result Efficient inference and unsupervised learning capabilities demonstrated.
New model learns deep, sparse hierarchies of features from images.
problem Learning deep, sparse hierarchies of features from natural images.
method Structured variational auto-encoder with rectified Gaussian units.
result Joint training of deep models with many layers of latent variables.
Improves AI-prior reliability for Bayesian inference.
problem Error propagation from predictive models into posterior inference.
method Rectified AI-informed prior elicitation framework.
result Significant reduction in bias and improvement in predictive performance.
Estimates parameters of a rectified Gaussian distribution using ReLU networks.
problem Estimating parameters of a rectified Gaussian distribution from i.i.d. samples.
method Simple algorithm using O(1/ε2) samples and O(d2/ε2) time. result Estimates distribution up to ε in total variation distance. New insights into neural network kernels for rectified MLPs.
problem Understanding the behavior of weight distributions in rectified MLPs.
method Deriving equivalent kernels for MLPs with ReLU or Leaky ReLU activations under various weight distributions.
result Kernels corresponding to layers with certain weight distributions are asymptotically universal and well approximated by Gaussian kernels.
Paper tackles constrained bandit problems with a new learning framework.
problem Optimizing a black-box reward function subject to a black-box constraint function over a continuous space.
method Rectified Pessimistic-Optimistic Learning (RPOL) framework, incorporating optimistic and pessimistic GP bandit learning.
result RPOL achieves sublinear regret and minimal cumulative constraint violation.
Convex relaxations improve CNNs with fixed weights.
problem Improving CNNs with fixed weights.
method Convex relaxations for CNNs with fixed weights using second order cone programs.
result The relaxation recovers the global minimum under a planted model assumption.
Paper proposes copula-based models for analyzing multivariate zero-inflated continuous data.
problem Challenges in analyzing multivariate zero-inflated continuous data with mixed discreteness and continuity.
method Proposes two copula-based density estimation models and rectified Gaussian copula.
result Demonstrates superior performance compared to conventional methods.
Random neural networks with ReLU activations are non-Gaussian processes.
problem Understanding the behavior of neural networks with random initialization and rectified linear units.
method Proving these networks are non-Gaussian processes and deriving their properties.
result These networks can converge to non-Gaussian processes under certain conditions.
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.
A software library for constructing and learning probabilistic models is presented. The library offers a set of building blocks from which a large variety of static and dynamic models can be built. These include hierarchical models for variances of other variables and many nonlinear models. The underlying variational B…
Generative ConvNet derived from ConvNet for image generation.
problem Creating realistic natural image patterns from ConvNet.
method Assuming a ConvNet for classification and a Gaussian white noise reference, deriving a piecewise Gaussian generative model with auto-encoders and ReLU non-linearities.
result Maximum likelihood learning synthesizes realistic natural images.
Rectifying submanifolds are characterized by their tangential position vector field component.
problem Characterizing rectifying submanifolds in Euclidean spaces.
method Introducing rectifying submanifolds and proving their properties.
result Rectifying submanifolds are identified by a specific tangential vector field property.
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.
Study of flat ribbons constructed along curves in 3D space.
problem Determine the conditions for a ruled structure to form a flat ribbon.
method Investigate the ruled structure of flat ribbons and calculate energy bounds.
result There exists a well-defined flat ribbon only up to an initial condition.
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.
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. New curves generalize helix and rectifying curves.
problem Generalizing helix and rectifying curves.
method Introducing f-rectifying curves with f-position vector in rectifying plane.
result Classification and characterization of f-rectifying curves.
The study evaluates different rectified activation functions in CNNs and finds RReLU superior.
problem Improving performance of CNNs using rectified activation functions.
method Compared standard ReLU, Leaky ReLU, PReLU, and randomized RReLU on image classification tasks.
result RReLU outperforms other rectified activation functions, achieving 75.68% accuracy on CIFAR-100.
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. Uniform rectifiability proven for sets with Poincaré inequalities.
problem Uniform rectifiability of sets with Poincaré inequalities.
method Weak (1,d)-Poincaré inequality and surface measure. result Uniform rectifiability achieved for sets supporting such inequalities.
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…
Paper transforms deep rectifier networks into shallow ones for analysis.
problem Understanding the complexity of deep neural networks.
method Transformation of deep rectifier networks into shallow ones.
result Shallow networks can represent deep networks with fewer functions.
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.
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. 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. 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.
The study characterizes rectifying curves in n-dimensional space.
problem Understanding rectifying curves in arbitrary dimensions.
method Characterization through various conditions and constructions.
result Different ways to characterize rectifying curves in n-dimensional Euclidean space.
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.
The paper characterizes special curves and generalizes rectifying-type curves in n-dimensional space.
problem Characterizing and generalizing special curves in higher dimensions.
method Characterization through Rotation minimizing frame (RMF) and generalization of rectifying-type curves.
result Rectifying-type curves are generalized in n-dimensional space.
Rectifies zero loci of certain spinors in 4D manifolds.
problem Understanding the structure of zero loci of Z/2 harmonic spinors. method Proves rectifiability and Minkowski bounds for the zero locus.
result Zero loci are 2-rectifiable and have locally finite Minkowski content.
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…
Proposes a new normalization technique to address issues with Batch Normalization.
problem Inaccurate batch statistics and inability to use with batch-size 1.
method Normalization Propagation: uses data-independent parametric estimates of mean and standard deviation.
result Computational efficiency and faster convergence compared to BN.
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.
In Heisenberg groups, rectifiability is studied for subsets using C1,α-regular surfaces.
problem Understanding rectifiability of subsets in Heisenberg groups.
method Introducing a new notion of rectifiability and proving conditions for rectifiability using tangent paraboloids.
result A sufficient condition for C1,α-rectifiability of low-codimensional subsets in Heisenberg groups is the existence of suitable approximate tangent paraboloids. The paper characterizes rectifying curves on smooth surfaces using isometries and Darboux frames.
problem Characterizing rectifying curves on smooth surfaces under isometries.
method Using Darboux frames and isometries to investigate rectifying curves.
result Find deviations of rectifying curves under isometries and analyze their properties.
Estimates set dimension for rectifiable sets using Sobolev mappings.
problem Estimating the dimension of rectifiable sets.
method Weak exterior differentiation and low rank property for Sobolev mappings.
result Validates Gromov's dimension comparison estimate for rectifiable sets.
Rectifiable varifolds with bounded curvature can be covered by smooth surfaces.
problem Understanding the structure of rectifiable varifolds with bounded curvature.
method Using curvature of arbitrary closed sets and viscosity solutions of PDEs.
result The support of rectifiable varifolds can be covered by smooth submanifolds.
New geometric definition for subsets of Euclidean space, proving second-order rectifiability.
problem Defining and proving second-order rectifiability for closed subsets of Euclidean space.
method Defining strata based on linear independence of touching directions and proving rectifiability using a new criterion.
result The m-th stratum of a closed subset of an n-dimensional Euclidean space is second-order rectifiable of dimension m. The paper proves smoothness of stationary varifolds.
problem Understanding the smoothness of stationary varifolds.
method Analyzing m-dimensional integer rectifiable varifolds in open sets. result The support of stationary varifolds is C∞ rectifiable. PHP connects to ReLU neural networks for scalable Bayesian inference.
problem Scalability and Bayesian inference in two-layer ReLU neural networks.
method PHP with Gaussian prior, decomposition propositions, annealed sequential Monte Carlo.
result PHP provides an alternative scalable representation for two-layer ReLU neural networks.
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. Study real rectifiable currents, generalize King's theorem, simplify proof, relate to Hodge conjecture.
problem Characterize currents defined by positive real holomorphic chains.
method Use Siu's semicontinuity theorem to simplify King's proof.
result Sufficient condition for the Hodge conjecture.
The paper characterizes timelike rectifying curves in De Sitter 3-space.
problem Characterizing timelike rectifying curves in De Sitter 3-space.
method Defining timelike rectifying curves and conical surfaces, providing characterizations and results.
result Characterizations and results of timelike rectifying curves in De Sitter 3-space.