Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.
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35 results for “semi-norms”
Volume entropy semi-normmath.GT
Innovates volume entropy semi-norm, proving equivalence to simplicial volume.
problem Equivalence of volume entropy and simplicial volume in real homology.
method Introduces volume entropy semi-norm and proves its equivalence to simplicial volume.
result Equivalence of volume entropy semi-norm and simplicial volume in every dimension.
The study determines dominations between manifold products and semi-norm finiteness.
problem Understanding dominations between different products of manifolds.
method Analyzing the finiteness of product-associated semi-norms on fundamental classes.
result Partial answers to M. Gromov's questions on manifold product dominations and semi-norms.
A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. …
Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical -semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in …
We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a finite-dimensional simplex. This function is a generalisation, in the framework of branched surfaces, of the Thurston semi-no…
Alexander and Thurston norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groupsmath.GR
We investigate Friedl-Lück's universal -torsion for descending HNN extensions of finitely generated free groups, and so in particular for -by- groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for -manifold groups. We prove…
Sharp uniqueness result for Q-curvature type equation on S^6.
problem Proving uniqueness of axially symmetric solutions to a Q-curvature type equation on S^6.
method New estimates on semi-norm and Gegenbauer coefficients.
result Axially symmetric solutions are constants for a specific range of α.
The paper surveys pressure metrics in geometry and dynamics.
problem Understanding pressure metrics in various deformation spaces.
method Survey and discussion of pressure semi-norms and their degeneracy loci.
result Discussion of pressure semi-norms and their degeneracy loci in quasi-Blaschke products.
In a previous paper, the second author defined integer-valued functions delta_n on the first cohomology of a 3-manifold, generalizing McMullen's Alexander norm. It was shown that these functions give lower bounds on the Thurston norm. In this paper, we reformulate these invariants in terms of Reidemeister torsion over …
Approximation Rates in Fréchet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliersmath.NA
Study approximates operator learning for PDEs using Fourier multipliers.
problem Approximating operator behavior for PDE simulations.
method Approximation of operator symbols in Fourier domain using semi-norms.
result Identifies conditions for achieving predefined approximation error.
Explicitly constructs efficient cycles in quotient groups.
problem Efficient cycles in quotient groups.
method Elementary construction of Gromov's description of diffusion.
result Explicit -efficient cycles in quotient groups.
The paper studies volume classes and Borel classes for dense group representations.
problem Understanding volume and Borel classes for dense representations of discrete groups.
method Utilizes tools from Kleinian groups and properties of hyperbolic manifolds.
result Volume classes are linearly independent and have additional properties.
Study simplicial volume for fixed fundamental groups, finding gaps.
problem Understanding simplicial volume for manifolds with fixed fundamental group.
method Relate gap problem to rationality questions in bounded (co)homology.
result Show existence of gaps in simplicial volume spectrum at zero.
Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Pressure Metricmath.DS
The paper studies Blaschke products, proving uniformization and non-degeneracy of pressure metrics.
problem Analytic aspects of Blaschke products and their moduli space.
method Definition of complex structure and proof of uniformization theorem.
result Pressure semi-norms are non-degenerate outside the super-attracting locus.
We consider the minimum error entropy (MEE) criterion and an empirical risk minimization learning algorithm in a regression setting. A learning theory approach is presented for this MEE algorithm and explicit error bounds are provided in terms of the approximation ability and capacity of the involved hypothesis space w…
New constructions show manifold volumes are dense in non-negative reals.
problem Understanding the spectrum of simplicial volumes in manifolds.
method Group homology constructions and manifold constructions using cross-products and Thom realisation.
result The set of simplicial volumes of orientable closed connected manifolds is dense in for dimensions > 3, and every non-negative rational number is a simplicial volume for dimension 4.
Convex learning for diverse invariances in semi-inner-product space.
problem Efficiently learning invariant representations for a wide range of invariances.
method Developed a convex representation learning algorithm for generalized invariances modeled as semi-norms, introducing Euclidean embeddings for kernel representers in a semi-inner-product space.
result Accurate invariant representations learned efficiently and effectively, validated by experiments.
For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of its integer points. We give an interpretation of these points in terms of certain coorientations of the original collection of curves. …
New Thurston norm defined for a specific type of groups using -invariants.
problem Measuring splitting complexity of integral characters in coherent right-angled Artin groups.
method Defining splitting complexity via -Euler characteristic and using Friedl--Lück's -polytope.
result A Thurston-type semi-norm defined for measuring splitting complexity of integral characters.
Estimates inner products between nonparametric distributions using Fourier basis.
problem Estimating inner products between two nonparametric distributions.
method Proposes estimators for inner products and induced norms, proves mean squared error bounds and minimax lower bounds.
result Proposed estimators are rate-optimal over Fourier ellipsoids.
We combine Gromov's amenable localization technique with the Poincaré duality to study the traversally generic vector flows on smooth compact manifolds with boundary. Such flows generate well-understood stratifications of by the trajectories that are tangent to the boundary in a particular canonical fashion. Sp…
Harmonic functions on one side of a quasicircle on a compact Riemann surface can be uniquely extended to the other side.
problem Transmission of harmonic functions across a quasicircle boundary on compact Riemann surfaces.
method Analyzing the properties of quasicircles and applying them to harmonic function boundary values.
result A unique harmonic function can be defined on the other side of a quasicircle boundary, preserving boundary values.
The paper generalizes product inequalities for random vectors and their applications.
problem Understanding concentration of measure for products of random vectors.
method Develops expressions for the concentration of functionals of random vectors based on product norms.
result Provides generalized Hanson-Wright inequalities and applications to random matrices.
New algorithms improve robust reinforcement learning under uncertainty.
problem Robust reinforcement learning in MDPs with contamination.
method Non-asymptotic convergence analysis of -learning and actor-critic methods.
result Efficient algorithms learn robust policies with minimal samples.
Study variance-reduced method for estimating fixed points in Banach spaces.
problem Estimating fixed points of contractive operators in Banach spaces with noisy evaluations.
method Variance-reduced stochastic approximation scheme in Banach spaces.
result Establish non-asymptotic bounds for operator defect and estimation error.
The paper analyzes Laplace learning for Gaussian measure data in infinite dimensions, proving convergence.
problem Analyzing Laplace learning for infinite-dimensional Gaussian measure data.
method Minimizes Dirichlet energy on a graph constructed from the full dataset.
result Proves pointwise convergence of the graph Dirichlet energy for Gaussian measure data.
New method reduces sample complexity for robust reinforcement learning.
problem Finite sample analysis in robust reinforcement learning.
method Stochastic approximation framework with controlled bias, using MLMC techniques and geometric truncation.
result Order-optimal sample complexity of for robust policy evaluation.
Given a free group of rank with a fixed set of free generators we associate to any homomorphism from to a group with a left-invariant semi-norm a generic stretching factor, , which is a non-commutative generalization of the translation number. We concentrate on the situation when $φ:F…
For the problems of low-rank matrix completion, the efficiency of the widely-used nuclear norm technique may be challenged under many circumstances, especially when certain basis coefficients are fixed, for example, the low-rank correlation matrix completion in various fields such as the financial market and the low-ra…
Study geodesic flows on manifolds with boundary using Gromov's amenable localization.
problem Estimate the numbers of connected components of geodesic flow-generated strata.
method Combining Gromov's amenable localization with Poincaré duality.
result Lower estimates of connected components in terms of homology.
Causal holography reconstructs manifold properties from scattering data.
problem Reconstructing manifold properties from scattering data.
method Introducing metrics of gradient type and using scattering maps.
result Reconstruction of manifold's homology, Gromov simplicial semi-norm, and fundamental group.
Improved bounds for neural network approximations of functions.
problem Bounding the width of neural networks for function approximation.
method Extending Radon-based norms to bounded open sets and deriving new approximation bounds.
result Improved sparse approximation bounds for neural networks.
New geometric inequalities for convex bodies derived from Log-Brunn-Minkowski conjecture.
problem Proving geometric inequalities for convex bodies.
method Analyzing semi-norms and symmetric convex bodies, using integral inequalities.
result Characterization and improvement of geometric inequalities involving convex bodies.
Learning Clique Forestsstat.ML
MFCF algorithm learns conditional dependency structure from sparse data.
problem Learning conditional dependency structure from sparse and noisy data.
method Repeated application of clique expansion to produce clique forest and MRF.
result MFCF outperforms Graphical Lasso for covariance selection models.