Combines observational and controlled data to learn dose-response curves.
problem Learning causal effects from limited controlled interventions and large observational datasets.
method Hierarchical Gaussian process prior and nonparametric affine transform.
result Function composition from different data sources speeds up learning.
Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
Algorithm finds optimal affine transformation to minimize overall distortion.
problem Minimizing distortion in affine transformations.
method Riemannian geometry approach to define and minimize distortion.
result Mean distorting transformation found for minimizing overall distortion.
Explains affine rough models and their transform formulas.
problem Rough volatility models in financial markets.
method Explains affine structure and transforms formulas.
result Shows exponential-affine transform formulas for rough models.
In this paper, we show that a compact affine manifold endowed with an Affine Anosov transformation is finitely covered by a complete affine nilmanifold.
The paper develops methods to generate invariant quantities in Metric-Affine Geometry.
problem Developing methods to generate invariant quantities in Metric-Affine Geometry.
method The paper introduces a theorem to generate invariant quantities under transformations of the affine connection, proving invariance conditions.
result Theorem establishing conditions for invariance of functionals under transformations of the affine connection.
Characterizes flat affine connections on manifolds.
problem Understanding flat affine connections on manifolds.
method Characterization through natural affine representation of diffeomorphisms.
result Group of affine transformations acts on R^n with open orbit when dimension > n.
New method detects symmetries beyond affine transformations.
problem Current methods limit symmetry detection to affine transformations.
method Framework for discovering continuous symmetry beyond affine transformations.
result Method is competitive for large sample sizes and superior for small sample sizes.
Transformers can efficiently approximate nonparametric regression with minimal parameters and sequences.
problem Efficiently approximating nonparametric regression functions with transformers.
method Kernel-weighted polynomial basis and gradient descent.
result Achieves minimax optimal rate of convergence with fewer parameters and sequences.
This research studies affine invariance in continuous-domain convolutional neural networks.
problem Recognizing patterns and features under affine transformations in continuous domains.
method Introduces a new criterion for assessing affine invariance, embeds images into the affine Lie group, and analyzes convolution over this group.
result Extends the scope of geometrical transformations that deep-learning pipelines can handle.
Improves MCMC performance with adaptive affine transformations.
problem Improving the performance of Markov Chain Monte Carlo samplers.
method Adaptive learning of bijective affine transformations during sampling.
result Adaptive affine transformations improve the quality of samples at low computational cost.
The paper generalizes fractals to higher dimensions using affine transformations.
problem Generalizing fractals to higher dimensions.
method Using affine transformations and characterizations of affinely-equivalent Sierpinski carpet.
result Menger sponge and Sierpinski simplex in 4-dimensional space can be drawn out clearly.
The paper explores flat affine and symplectic structures on Lie groups.
problem Exploring flat affine and symplectic structures on Lie groups.
method Left invariant affine structures, immersion of Lie groups, Koszul's method, Lagrangian bi-foliation.
result Flat left invariant affine symplectic connections and their associated affine symplectomorphisms.
This paper describes integral affine structures on compact 3-manifolds.
problem Understanding integral affine structures on compact 3-manifolds.
method Analyzing complete integral affine structures on compact 3-manifolds up to finite-sheeted coverings.
result A complete list of integral affine structures on the three-dimensional torus and compact three-dimensional nilmanifolds was obtained.
Standard Transformers approximate Hölder functions and achieve optimal nonparametric regression rate.
problem Approximating Hölder functions and achieving optimal nonparametric regression rate with Transformers.
method Using the size tuple and dimension vector metrics, the paper characterizes Transformer structures and derives upper bounds for their Lipschitz constant and memorization capacity.
result Standard Transformers achieve the minimax optimal rate in nonparametric regression for Hölder target functions.
Proves nonemptyness of domains for specific group actions.
problem Nonemptyness of domains of proper discontinuity for Anosov groups of affine Lorentzian transformations.
method Proof of nonemptyness of domains of proper discontinuity.
result Proves nonemptyness of domains for Anosov groups of affine Lorentzian transformations.
Study shows complex affine transformations index is at most 2 for Kähler manifolds.
problem Understanding transformations on Kähler manifolds.
method Analyzing groups of transformations on Kähler manifolds.
result Establishes a stronger version of Yano-Obata conjecture for complete Kähler manifolds.
Transforms classical connections using pushforwards and gauge transformations.
problem Transforming classical connections in categorical settings.
method Constructing pushforwards and applying gauge transformations to decorated path spaces.
result Combines traditional gauge transformation with affine translation.
Extend classical theory of affine processes to path-dependent setting
problem Path-dependent affine processes
method Introduce path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform
result Define path-dependent affine processes through their exponential-affine Fourier--Laplace transform and establish a characterization theorem
FineMorphs models smooth transformations for multivariate regression.
problem Efficiently modeling complex transformations for multivariate regression.
method Optimal control of affine and diffeomorphic transformations using smooth vector fields.
result FineMorphs can reduce dimensionality and adapt to large datasets.
We show that conformal transformations on the generalized Minkowski space Rp,q map hyperboloids and affine hyperplanes into hyperboloids and affine hyperplanes. We also show that this action on hyperboloids and affine hyperplanes is transitive when p or q is 0, and that this action has exactly three…
This paper classifies reversible and strongly reversible elements in affine groups.
problem Classifying reversible and strongly reversible elements in affine groups.
method Identifying affine transformations and using conjugacy by involutions.
result Classification of reversible and strongly reversible elements in affine groups.
This paper uses linear rational splines for invertible modeling, offering a simpler inverse and similar costs.
problem Creating expressive invertible models with tractable Jacobian determinants.
method Replacing affine transformations with linear rational splines in coupling layers.
result Linear rational splines offer a simpler inverse and similar costs for inference and generation.
Automorphisms of Kodaira surfaces are shown to be affine transformations.
problem Characterizing automorphisms of Kodaira surfaces.
method Analyzing lifts to the universal cover and conditions on affine transformations.
result Precise description of Kodaira surfaces' automorphism groups.
This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. Th…
An (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space R3 with transition maps in the affine transformation group Aff(R3). Equivalently an affine 3-manifold is a 3-manifold with a flat torsion-free affine connection. We show that a closed affine 3-mani…
The study examines lower and upper bounds of Wasserstein distances for affine transformations of random vectors.
problem Understanding Wasserstein distances for affine transformations of random vectors.
method Lower and upper bounds for affine transformations of random vectors in Rn are derived using Bures metric and compositions of affine maps. result Concrete lower bounds and upper bounds for affine transformations are derived and applied to various distributions.
Classification of cubics (that is, third order planar curves in the R2 up to certain transformations is interested since Newton, and treated by several authors. We classify cubics up to affine transformations, in seven class, and give a complete set of representatives of the these classes. This result is complete an…
Algorithm learns affine transformations robustly from corrupted samples.
problem Learning affine transformations from corrupted samples.
method New geometric certificate and iterative improvement method.
result Total variation distance of O(ε) between learned and original distributions. Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cas…
This project improves model robustness to affine transformations.
problem Vulnerability of models to affine transformations.
method Evolution strategies for finding worst affine transforms.
result Effective robust models against non-parametric adversarial perturbations.
Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption i…
Constructs proper affine actions for right-angled Coxeter groups.
problem Proper affine actions for right-angled Coxeter groups.
method Constructs proper actions of right-angled Coxeter groups on O(p,q+1) and its Lie algebra by affine transformations.
result Any virtually special group admits proper affine actions on some R^n.
Defines CAMC discrete nets and their properties.
problem Understanding CAMC discrete nets and their properties.
method Defining CAMC discrete nets and proving properties.
result Properties of CAMC discrete nets are equivalent to properties of compatible interpolating quadrics.
Paper proves a new geometric result about curvature and transformations.
problem Index of affine transformations in projective transformations.
method Projectively invariant objects and differential geometry.
result Index of affine transformations is at most two on complete Riemannian manifolds of nonconstant curvature.
Characterizes affine vector fields on Finsler manifolds with rigidity results.
problem Understanding affine vector fields on Finsler manifolds.
method Utilizing the Jacobi type equation and spray characterization, proving rigidity theorems.
result Rigidity theorems for affine vector fields on Finsler manifolds with non-positive total Ricci curvature.
Clarifies relation for solving control-affine Schrödinger bridge problems.
problem Solving control-affine Schrödinger bridge problems via Hopf-Cole transform.
method Applies Hopf-Cole transform to conditions of optimality, resulting in nonlinear PDEs.
result Generic control-affine Schrödinger bridge requires further algorithmic development.
Study finds solutions for degenerate affine curve shortening flow.
problem Analyzing degenerate affine curve shortening flow.
method Solved equations for affine self-similar solutions.
result New special solutions discovered for affine curve shortening flow.
This paper offers a simple method for Bayesian regression with unknown transformations.
problem Joint inference of unknown transformations and model parameters in Bayesian regression is computationally inefficient and cumbersome.
method The paper introduces a Bayesian nonparametric model via the Bayesian bootstrap to directly target the posterior distribution of the transformation.
result The approach delivers joint posterior consistency and efficient Monte Carlo inference for the transformation and all parameters.
The paper studies third-order PDEs invariant under affine transformations and connects them to the Fubini-Pick invariant.
problem Investigating third-order PDEs invariant under affine transformations.
method Using a general method introduced in [D.V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno: A general method to construct invariant PDEs on homogeneous manifolds].
result Derives third-order PDEs from the Fubini-Pick invariant.
Yau's Affine Normal Descent optimizes smooth unconstrained problems with geometrically adapted directions.
problem Optimizing smooth unconstrained problems with geometrically adapted directions.
method Yau's Affine Normal Descent (YAND) uses the equi-affine normal of level-set hypersurfaces as search directions.
result YAND converges globally under standard smoothness assumptions and locally quadratically near nondegenerate minimizers.
New method calculates geometric Brownian motion with affine drift and its integral.
problem Calculating the distribution of geometric Brownian motion with affine drift and its integral.
method Laplace transform approach and Heun differential equation.
result Joint distribution of geometric Brownian motion with affine drift and its integral can be determined.
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence questio…
Transformers learn new tasks from few examples via optimal approximation.
problem Learning new tasks from limited examples using large language models.
method Developed approximation and generalization error bounds for transformers trained on nonparametric regression tasks.
result Transformers achieve minimax optimal estimation risk in context.
Transforms conditional density estimation into a nonparametric regression problem.
problem Conditional density estimation in high dimensions.
method Introduces auxiliary samples to transform into nonparametric regression.
result Estimator converges to true conditional density in data limit.
Affine 3-manifolds with centralizing holonomy are complete.
problem Closed flat affine manifolds with parallel volume are not always complete.
method Showed the Markus conjecture holds for 3-manifolds with centralizing holonomy.
result Holonomy centralizing an affine transformation preserves volume completeness.
This work extends variance reduction for path-dependent derivatives to affine stochastic volatility models.
problem Pricing path-dependent derivatives in affine stochastic volatility models.
method Prove large deviations principle, apply Esscher transform, use Varadhan's lemma.
result Numerical efficiency demonstrated on Heston model with and without jumps.
There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dime…