New method computes affine normal directions efficiently for sparse polynomials.
problem Computing affine normal directions is computationally expensive in high dimensions.
method Reduces third-order tensor contraction to matrix-free formulation using log-determinant gradient.
result Scalable implementations with near-linear scaling in dimension and sparsity.
New samplers improve MCMC efficiency in high dimensions.
problem Efficient sampling in high-dimensional problems.
method Affine invariant ensemble samplers, including derivative-free and derivative-based HMC.
result Affine invariant ensemble HMC outperforms standard HMC in high dimensions.
The Green's function of a complex plane domain is analyzed with affine scaling.
problem Quantitative properties of the Green's function in multiply connected domains.
method Affine scaling of the domain to study boundary behavior.
result Quantitative boundary behavior of the Green's function and related objects.
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.
The paper matches features in images using centro-affine invariants and heat flow.
problem Feature matching in images with invariant algorithms.
method Developed an invariant algorithm using centro-affine invariants and heat flow.
result The algorithm compares favorably with existing feature matching methods.
A new method for clustering high-dimensional data into subspaces efficiently and accurately.
problem Inaccurate clustering due to poor intra-subspace similarity in existing methods.
method Iterative Maximum Correlation (IMC) for affinity matrix learning and Piecewise Correlation Estimation (PCE) for densification.
result SDSC framework improves clustering accuracy and efficiency for large-scale data.
First we provide a simple set of sufficient conditions for the weak convergence of scaled affine processes with state space R + × R d R_+ \times R^d R + × R d . We specialize our result to one-dimensional continuous state branching processes with immigration. As an application, we study the asymptotic behavior of least squares estimators…
Estimates box dimension of fractal interpolation surfaces using oscillation vectors.
problem Estimating the complexity of fractal interpolation surfaces.
method Defined vertical scaling matrices and used them to relate oscillation vectors of different levels.
result Obtained the box dimension of generalized affine fractal interpolation surfaces.
Properties of low-variability periods in the time series are analysed. The theoretical approach is used to show the relationship between the multi-scaling of low-variability periods and multi-affinity of the time series. It is shown that this technically simple method is capable of reveling more details about time-seri…
The paper improves IPS for modern optimization, scaling and regularizing it.
problem Improving iterative proportional scaling for modern optimization.
method Coordinate descent, majorization-minimization, optimization techniques, regularized variants.
result IPS can deliver coefficient estimates and handle log-affine models.
The paper constructs new structures for manifolds using connections and combinations.
problem Understanding smooth manifolds with precise infinitesimal affine structures.
method Constructing new infinitesimal structures for higher-order neighbourhoods of the diagonal.
result Any symmetric affine connection on a manifold extends to a second-order infinitesimally affine structure.
Algorithm tackles large-scale portfolio optimization with higher moments, improving computational efficiency.
problem Optimizing portfolios with higher moments (variance, skewness, kurtosis) for large asset universes is computationally infeasible.
method Developed a structure-exploiting algorithm based on Yau's affine-normal descent, working directly with return matrix.
result Algorithm avoids explicit higher-order tensors and exploits quartic structure for efficient computation.
We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play …
Researchers found invariant metric connections on Berger spheres that are Einstein with skew torsion.
problem Determining invariant metric affine connections on Berger spheres that are Einstein with skew torsion.
method Explicitly determined and expressed connections in both Riemannian and Lorentzian signatures.
result Every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstein with skew-torsion up to S 3 \mathbb{S}^3 S 3 . Study compares geometric approaches for shape and deformation statistics.
problem Characterizing statistical models of shapes and deformations.
method Information geometry and Wasserstein geometry.
result Wasserstein estimator is robust against waveform perturbation.
The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.
problem Calculating area Siegel--Veech constants for affine invariant submanifolds of REL zero.
method Using volumes of the principal boundary strata and intersection theory.
result Proves a conjectural formula for the area Siegel--Veech constant in the case of REL zero.
Study heat kernel coefficients in Bianchi IX gravity models.
problem Analyzing gravitational wave properties in Bianchi IX models.
method Algebraic geometry and motives applied to heat kernel coefficients.
result Coefficients are algebro-geometric periods of motives.
Method identifies latent variables from high-dimensional data with piecewise affine mixing.
problem Identifying latent variables from high-dimensional observations with dependencies and piecewise affine transformations.
method Proposes a two-stage method with sparsity and Gaussianity regularization.
result Effectively recovers ground-truth latent variables from synthetic and image data.
A new ranking algorithm learns data affinity and ranking scores simultaneously.
problem Retrieving similar objects in large databases is challenging.
method Proposes a ranking algorithm that learns data affinity and ranking scores simultaneously, using adaptive neighbors and smoothness constraints.
result The proposed algorithm outperforms existing methods in synthetic and real datasets.
New method constructs potential functions for Kähler-Einstein metrics.
problem Constructing potential functions for Kähler-Einstein metrics on pseudoconvex domains.
method Method of potential scaling.
result Existence of 1-parameter family of automorphisms for certain pseudoconvex domains.
Exact LAD line fitting via PALB with linear scaling and speed.
problem Robust line fitting for data with outliers.
method Piecewise Affine Lower-Bounding (PALB) method using supporting lines and subdivision scheme.
result Empirical log-linear scaling and significantly faster than LP and IRLS methods.
New shape representation for airfoils improves design and manufacturing.
problem Designing and manufacturing airfoils efficiently and accurately.
method Combining physics-based and data-driven techniques on a Grassmannian manifold.
result Rich set of novel 2D airfoil deformations not previously captured.
Efficient superpixel method for real-time segmentation.
problem Real-time superpixel generation for computer vision tasks.
method Two-stage graph-based framework with Deep Affinity Learning and Hierarchical Entropy Rate Segmentation.
result HERS produces superpixels in near real-time.
We consider the Nordic electricity spot market from mid 1992 to the end of year 2000. This market is found to be well approximated by an anti-persistent self-affine (mean-reverting) walk. It is characterized by a Hurst exponent of H ≃ 0.41 H\simeq 0.41 H ≃ 0.41 over three orders of magnitude in time ranging from days to years. We argu…
This paper studies gradient flows for sampling using various metrics and their affine invariance.
problem Sampling from probability distributions with unknown normalizations.
method Gradient flows in the space of probability measures, focusing on Kullback-Leibler divergence and affine invariance of metrics.
result Gradient flows of Kullback-Leibler divergence do not depend on the normalization constant, and affine invariance is achieved for certain metrics.
Let σ t ( x ) σ_t(x) σ t ( x ) denote the implied volatility at maturity t t t for a strike K = S 0 e x t K=S_0 e^{xt} K = S 0 e x t , where $x\in\bbR$ and S 0 S_0 S 0 is the current value of the underlying. We show that σ t ( x ) σ_t(x) σ t ( x ) has a uniform (in x x x ) limit as maturity t t t tends to infinity, given by the formula σ ∞ ( x ) = 2 ( h ∗ ( x ) 1 / 2 + ( h ∗ ( x ) − x ) 1 / 2 ) σ_\infty(x)=\sqrt{2}(h^*(x)^{1/2}+(h^*(x)-x)^{1/2}) σ ∞ ( x ) = 2 ( h ∗ ( x ) 1/2 + ( h ∗ ( x ) − x ) 1/2 ) , for…
In 1960 Reifenberg proved the topological disc property. He showed that a subset of R n R^n R n which is well approximated by m m m -dimensional affine spaces at each point and at each (small) scale is locally a bi-Hölder image of the unit ball in R m R^m R m . In this paper we prove that a subset of R 3 R^3 R 3 which is well approximated b…
Subspace clustering methods based on ℓ 1 \ell_1 ℓ 1 , ℓ 2 \ell_2 ℓ 2 or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, ℓ 1 \ell_1 ℓ 1 regularization is guaranteed t…
Efficiently clusters large datasets with a subset of landmarks.
problem High computational complexity in subspace clustering for large-scale datasets.
method Selects a subset of landmarks to reduce the clustering problem to linear time.
result Subspace clustering method runs in linear time with respect to the size of the original data.
Automated spectral clustering algorithm discovers clusters without parameter tuning.
problem Automatic spectral clustering for multi-scale data.
method Heuristic iterative eigengap search with global and local scaling.
result Discover different patterns with accuracy >90% in most cases.
SwISS improves scalability of Bayesian inference for large datasets.
problem Scalability issues in Bayesian inference for large datasets.
method Divide-and-conquer approach with SwISS for recombining sub-posterior samples.
result SwISS accurately approximates the original posterior distribution.
Study proves existence, uniqueness, and stability for specific stochastic Volterra equations.
problem Analyzing existence, uniqueness, and stability of affine stochastic Volterra equations with L 1 L^1 L 1 -kernels. method Approximations with L 2 L^2 L 2 -kernels, stability result, duality argument, deterministic Riccati--Volterra integral equation. result Established weak uniqueness for the equations using Fourier--Laplace transform and a deterministic Riccati--Volterra integral equation.
UCoS avoids forward model evaluations in sampling for large-scale linear inverse problems.
problem Efficient sampling from posterior distributions in large-scale linear inverse problems.
method UCoS approach that learns a task-dependent score function offline and uses affine transformations to derive the conditional score.
result UCoS eliminates the need for forward model evaluations during sampling, making it more efficient.
Study geodesic and affine Killing completeness in homogeneous affine surfaces.
problem Geodesic and affine Killing completeness in homogeneous affine surfaces.
method Examined using the solution space of the quasi-Einstein equation.
result Characterized geodesic and affine Killing completeness in homogeneous affine surfaces.
Almost Zoll affine surface found on cylinder.
problem Finding surfaces with special geodesic properties.
method Exhibited an affine structure on a cylinder.
result Affine structure on cylinder is almost Zoll.
We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the affine structure of an affine manifold. We found various conditions which make the…
This paper explores gradient flows for sampling distributions without normalization constants.
problem Sampling from distributions with unknown normalization constants.
method Gradient flows in the space of probability measures, focusing on Kullback-Leibler divergence, Fisher-Rao metric, and affine invariance.
result Gradient flows derived from Kullback-Leibler divergence do not depend on the normalization constant.
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.
Study shows spectral action coefficients are periods in specific spacetimes.
problem Understanding spectral action coefficients in Robertson-Walker spacetimes.
method Analyzes asymptotic expansion coefficients as periods of mixed Tate motives.
result Coefficients are periods involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces.
The paper introduces a new geometric capacity and proves inequalities related to it.
problem Developing a new geometric capacity and comparing it to classical quantities.
method Introducing the general p p p -affine capacity and proving its properties and inequalities. result Sharp geometric inequalities for the general p p p -affine capacity are derived. Study affine curvature lines on surfaces in 3D space.
problem Understanding the behavior of affine curvature lines on surfaces.
method Analyzing binary differential equations and topological models.
result Obtained topological models and generic behavior of affine curvature lines.
LCRSR recovers latent row space for multi-view clustering.
problem Efficiently recover latent representation from multiple views.
method LCRSR assumes latent representation from multiple views, recovers row space, and determines subspace membership.
result LCRSR recovers complete subspace structure efficiently.
Study of time-inhomogeneous affine processes in finance.
problem Understanding and modeling financial processes with time-varying parameters.
method Developed a theory for time-inhomogeneous affine processes and applied it to financial market models.
result Affine processes can be modified to include real-valued processes, improving model flexibility.
An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformati…
In this paper, we show that a compact affine manifold endowed with an Affine Anosov transformation is finitely covered by a complete affine nilmanifold.
New affine Szabó connections defined on smooth manifolds.
problem Defining and characterizing affine Szabó connections on manifolds.
method Introduced affine Szabó connections and proved their equivalence to cyclic parallelism on 2D affine manifolds.
result Characterization of locally homogeneous affine Szabó surfaces.
Proofs for decomposing branched affine surfaces into triangles and cylinders.
problem Decomposing branched affine surfaces into simpler geometric shapes.
method Proof of Veech's theorem and introduction of invariant α.
result Any pair of decompositions can be connected by flips.
Affine maps on tori preserve lines.
problem Characterizing affine automorphisms on tori.
method Analogous to Euclidean space, established for tori.
result Affine maps on tori preserve lines.