Affine actions fail for Hitchin linear parts, except flat pseudo-Riemannian cases.
problem Properly discontinuous affine actions of surface groups with Hitchin linear part.
method Analysis of representations and pseudo-Riemannian metrics.
result Hitchin linear parts in SO(n,n−1) lead to non-properly discontinuous affine actions. The paper analyzes deep neural networks using rectified linear units.
problem Understanding the individual affine linear representations of deep neural networks.
method Signal processing perspective, atomic decompositions, Lipschitz regularity estimation.
result Conditions for stabilizing learning in deep neural networks without network depth constraints.
In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations and 1-cohomology classes of the algebra. We classify the affine representations of…
New algorithm reveals piecewise affine structure of neural networks.
problem Lack of strong guarantees on deep neural networks' behavior in safety-critical applications.
method Developed a novel algorithm to compute the piecewise affine form of neural networks.
result Computed piecewise affine representations of neural networks with rectified linear unit activations.
We define the notion of affine Anosov representations of word hyperbolic groups into the affine group SO0(n+1,n)⋉R2n+1. We then show that a representation ρ of a word hyperbolic group is affine Anosov if and only if its linear part Lρ is Anosov in SO0(n+1,n) with …
The paper generalizes deformation results for Fuchsian representations and shows proper affine actions.
problem Infinitesimal deformations of Fuchsian representations do not act properly in certain directions.
method Using results from Labourie--Wentworth, Potrie--Sambarino, and Smilga, the authors introduce affine versions of cross ratios and triple ratios, Margulis invariants, and relate them to infinitesimal Jordan projections.
result A general criterion for existence of proper affine actions in terms of Margulis invariant spectra.
We define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of SL(2,R). We prove a fuchsian affine action of a surface group is never proper.
We study affine Jacobi structures on an affine bundle π:A→M, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A+=⋃p∈MAff(Ap,R) of affine functionals. Som…
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.
Groups with hyperbolic properties don't have strong Property (T).
problem Proving groups with hyperbolic properties don't have strong Property (T).
method Constructing an unbounded affine representation with polynomial growth.
result Groups with hyperbolic properties do not have strong Property (T).
Study partially hyperbolic flows on flat bundles, proving equivalence for complete affine manifolds.
problem Characterize partially hyperbolic representations of fundamental groups of manifolds.
method Representation theory techniques, focusing on holonomy representations and their properties.
result Show equivalence between partially hyperbolic representations and P-Anosov representations for complete affine manifolds. The paper studies Anosov holonomy groups in complete affine manifolds.
problem Characterizing Anosov holonomy groups in complete affine manifolds.
method Representation theory and coarse geometry techniques.
result Complete affine manifolds with Anosov holonomy groups have specific geometric properties.
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.
Characterizes flat affine symplectic Lie groups and their properties.
problem Characterizing flat affine symplectic Lie groups.
method Using symplectic étale affine representations and central translations.
result Obtains nontrivial examples of flat affine symplectic Lie groups in every even dimension.
Survey on affine Anosov representations and their implications.
problem Generalizing Anosov representations to affine settings.
method Discussion and survey of existing work.
result Implications of affine Anosov representations.
Constructs special Kähler structures on Lie groups.
problem Creating special Kähler structures on Lie groups.
method Introducing twisted cartesian product and double extension process.
result Characterizes left invariant flat special Kähler structures.
The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes…
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.
Fractional processes have gained popularity in financial modeling due to the dependence structure of their increments and the roughness of their sample paths. The non-Markovianity of these processes gives, however, rise to conceptual and practical difficulties in computation and calibration. To address these issues, we…
For spherical Tits buildings of the classical types there are well-known explicit descriptions as flag complexes. Similarly for affine buildings of the classical types there are explicit constructions in terms of lattices. In this article we generalize the flag complex description to twin cities, a generalization of tw…
New insights into surface group actions and entropy.
problem Understanding proper affine actions of surface groups.
method Explicit neighborhoods in quasifuchsian space and critical points of entropy.
result Critical points of entropy lie on the Fuchsian locus.
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 defines cocycles for positive Anosov representations and constructs affine actions with bounded fundamental domains.
problem Positive Anosov representations into SO(2n,2n−1). method Definition of cocycles and construction of affine actions with fundamental domains.
result Quotient manifolds are homeomorphic to handlebodies.
A scattering transform defines a signal representation which is invariant to translations and Lipschitz continuous relatively to deformations. It is implemented with a non-linear convolution network that iterates over wavelet and modulus operators. Lipschitz continuity locally linearizes deformations. Complex classes o…
Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so…
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.
Study of tangent cones at infinity for algebraic sets.
problem Characterizing algebraic sets based on their tangent cones at infinity.
method Definition and analysis of tangent cones C4,∞(X) and C5,∞(X), proving properties and relations. result Affine linear subspace characterization based on C5,∞(X)'s dimension. 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.
The paper studies affine manifolds with linear foliations and their topological properties.
problem Characterizing the topology of affine manifolds with linear foliations.
method Analyzing the structure of affine manifolds and their foliations, proving topological properties.
result Compact affine manifolds with linear foliations are homeomorphic to the torus under certain conditions.
Surface groups can't act properly on affine space if their linear part is Hitchin.
problem Proper affine actions of surface groups with specific linear parts.
method Independent proof of a theorem by Danciger and Zhang.
result Surface groups with Hitchin linear part cannot act properly on affine space.
The abstract investigates how volume ratios relate to curvature in geometric surfaces.
problem Understanding the relationship between curvature and volume in geometric surfaces.
method Investigates the geometric meaning of a quantity related to curvature and volume ratios.
result Shows how the ratio of Gaussian curvature to a volume function can be represented as a function of volumes.
Heegaard Floer homology connects to polynomial representations of Hecke algebras.
problem Understanding polynomial representations of double affine Hecke algebras.
method Using higher-dimensional Heegaard Floer homology and topological interpretations.
result Recovery of polynomial representations from Heegaard Floer homology.
Constructs positive energy representations from Toda equations Stokes data.
problem Creating positive energy representations of affine algebras.
method Using Stokes data of tt*-Toda equations to construct representations.
result Illustrates construction with examples in conformal field theory.
This is a review with examples concerning the concepts of affine (in particular, constant and linear) vector fields and fundamental vector fields on a manifold. The affine, linear and constant vector fields on a manifold are shown to be in a bijective correspondence with the fundamental vector fields on it of respectiv…
Researchers study rational and pretzel knots using affine group representations.
problem Understanding the structure and properties of rational and pretzel knots.
method Constructing representations of knot groups into the affine group AGL(1,ℂ) via a TQFT valued in spans of singular vector bundles.
result Closed-form expressions for Alexander polynomials and bounds on their zeros for rational and pretzel knots.
New model for Knightian uncertainty with jumps.
problem Knightian uncertainty and non-linear jumps.
method Probabilistic construction of non-linear affine processes with jumps.
result Tractable model for Knightian uncertainty with sublinear expectations.
Paper shows affine constraint is unnecessary for high-dimensional data.
problem The necessity of an affine constraint in affine subspace clustering.
method Theoretical and empirical analysis of conditions for correctness of affine subspace clustering methods.
result Affine constraint has negligible effect on clustering performance for high-dimensional data.
New framework for probabilistic linear solvers reduces manual effort.
problem Manual implementation of probabilistic iterative methods is laborious.
method Affine Tracing: Automatically constructs PIMs from standard implementations.
result Any realistic affine PIM is calibrated, motivating their adoption.
CF-INNs can approximate any invertible function, resolving a long-standing problem.
problem Whether CF-INNs can approximate any invertible function.
method Demonstrated CF-INNs are universal approximators for invertible functions by showing a convenient criterion.
result CF-INNs are universal approximators for invertible functions.
We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programm…
We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly projective invariant. Moreover the maximal dimension is shown to be achieved if and on…
New measures link neural representation geometry to decoding ability.
problem Understanding how neural representations relate to decoding ability.
method Showed that popular similarity measures can be interpreted from a decoding perspective.
result Proved that measures like CKA and CCA quantify alignment between optimal linear readouts.
Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…
The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topolo…
A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.
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…
The complete classification of representations of the Trefoil knot group G in S^{3} and SL(2,R), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the non cyclic groups of affine Euclidean isometries generat…