New algebraic-geometry method for Ribaucour transformations.
problem Classical differential geometry problems.
method Algebraic-geometry approach to constructing orthogonal nets.
result Obtains smooth orthogonal nets as Ribaucour transformations.
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48 results for “orthogonal nets”
A Clifford algebra model for M"obius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames are derived. These equations are discretized and the geometry of the occuring discrete nets and sphere congruences is disc…
Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth -surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit de…
We define systems of pre-extremals for the energy functional of regular rheonomic Lagrange manifolds and show how they induce well-defined Hamilton orthogonal nets. Such nets have applications in the modelling of e.g. wildfire spread under time- and space-dependent conditions. The time function inherited from such a Ha…
Principal binetsmath-ph
New discretizations of principal curvature lines discovered.
problem Discretizing principal curvature line parametrizations.
method Generalization of polar pairs of line congruences in the Lie quadric.
result New discretizations of orthogonal and Gauss-orthogonal parametrizations.
Discrete conjugate systems are quadrilateral nets with all planar faces. Discrete orthogonal systems are defined by the additional property of all faces being concircular. Their geometric properties allow one to consider them as proper discretization of conjugate, resp. orthogonal coordinate systems of classical differ…
Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a con…
Confocal conics form an orthogonal net. Supplementing this net with one of the following: 1) the net of Cartesian coordinate lines aligned along the principal axes of conics, 2) the net of Apollonian pencils of circles whose foci coincide with the foci of conics, 3) the net of tangents to a conic of the confocal family…
Convolutional nets require fewer samples than fully-connected nets for image classification.
problem Understanding why convolutional nets are more sample-efficient than fully-connected nets.
method Construction of a natural distribution and target function to demonstrate a sample complexity gap.
result Convolutional nets require samples for a single target function, while fully-connected nets require samples.
Supercyclidic netsmath.DG
Supercyclides are surfaces with a characteristic conjugate parametrization consisting of two families of conics. Patches of supercyclides can be adapted to a Q-net (a discrete quadrilateral net with planar faces) such that neighboring surface patches share tangent planes along common boundary curves. We call the result…
The paper explores discrete isothermic nets using checkerboard patterns in quadrilateral nets.
problem Defining and understanding discrete isothermic nets in quadrilateral nets.
method Using checkerboard patterns and discrete differential geometry to define and analyze isothermic nets.
result The class of isothermic nets is invariant under dualization and Moebius transformations.
New discrete cmc surfaces defined from sphere packings and combinatorics.
problem Creating constant mean curvature surfaces from discrete data.
method Discrete cmc surfaces defined via sphere packings and combinatorial patterns.
result Construction of discrete cmc surfaces from orthogonal ring patterns.
On a discretization of confocal quadrics. II. A geometric approach to general parametrizationsmath.DG
We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system…
Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.
problem Creating integrable discrete analogs of surface nets and conjugate nets.
method Interpreting classical differential geometry results through Bäcklund transformations and applying permutability properties.
result Integrable discrete analogs of asymptotic and conjugate nets are constructed.
RieszNet and ForestRiesz: Automatic Debiased Machine Learning with Neural Nets and Random Forestscs.LG
Automatic debiasing for causal and policy effects using Neural Nets and Random Forests.
problem Estimating causal and policy effects from high-dimensional or non-parametric regression functions.
method Automatic learning of Riesz representation using Neural Nets and Random Forests.
result Automatic debiasing method performs well compared to state-of-the-art algorithms.
Principal binets generalize curvature line surfaces to square lattices and are a discrete integrable system.
problem Discretizing curvature line surfaces on square lattices.
method Showed principal binets as a multi-dimensional consistent system.
result Principal binets generalize to higher-dimensional square lattices and are integrable.
A new pruning method reduces neural network computation without retraining.
problem Efficiently reduce neural network computation while maintaining accuracy.
method Structured directional pruning via perturbation orthogonal projection.
result Achieves state-of-the-art pruned accuracy without retraining.
Despite the phenomenal success of deep learning in recent years, there remains a gap in understanding the fundamental mechanics of neural nets. More research is focussed on handcrafting complex and larger networks, and the design decisions are often ad-hoc and based on intuition. Some recent research has aimed to demys…
The problem of determining the {\it Bonnet hypersurfaces in} , for , is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/o…
Deterministic neural nets have been shown to learn effective predictors on a wide range of machine learning problems. However, as the standard approach is to train the network to minimize a prediction loss, the resultant model remains ignorant to its prediction confidence. Orthogonally to Bayesian neural nets that indi…
The study examines how shallow neural nets converge to training samples or manifold points during diffusion.
problem Understanding when and how shallow neural nets converge to training samples or manifold points during diffusion.
method Analysis of shallow ReLU neural network denoisers trained with minimal norm, comparing score flow and diffusion flow.
result Probability flow converges to training points, sums of training points, or manifold points, depending on the diffusion time scheduler.
Batch normalization has become ubiquitous in many state-of-the-art nets. It accelerates training and yields good performance results. However, there are various other alternatives to normalization, e.g. orthonormalization. The objective of this paper is to explore the possible alternatives to channel normalization with…
GAMI-Tree uses model-based trees to fit low-order fANOVA models.
problem Fitting interpretable fANOVA models with low-order interactions.
method GAMI-Tree uses model-based trees and a new interaction filtering method.
result GAMI-Tree outperforms EBM and GAMI-Net in predictive performance and interpretability.
Study curvature loci of 3-manifolds in R^6 and R^5.
problem Characterize curvature loci of 3-manifolds in different dimensions.
method Refine affine classification of real nets of quadrics, study singularities, and analyze systems of ternary cubics.
result Obtain generic curvature loci and singularities of 3-manifolds.
DFSOS improves sparse discriminant analysis for high-dimensional data.
problem Sparse discriminant analysis in high-dimensional settings with feature selection.
method Deflation-Free Sparse Optimal Scoring (DFSOS) using Bregman iteration and orthogonality-constrained optimization.
result DFSOS achieves comparable or better classification accuracy than deflation-based methods.
By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Li…
ICCNLS models complex relationships as convex and concave components.
problem Complex input-output relationships with affine ambiguity.
method Sub-gradient constrained affine functions, global orthogonality constraints, L1, L2, and elastic net regularisation.
result Improved predictive accuracy and model simplicity compared to conventional methods.
Readouts for Echo-state Networks Built using Locally Regularized Orthogonal Forward Regressionstat.ML
Echo state network (ESN) is viewed as a temporal non-orthogonal expansion with pseudo-random parameters. Such expansions naturally give rise to regressors of various relevance to a teacher output. We illustrate that often only a certain amount of the generated echo-regressors effectively explain the variance of the tea…
We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, -nets and conical…
The study examines how gamma positivity and PL homeomorphism types affect simplicial spheres.
problem Understanding gamma positivity and its relation to PL homeomorphism types in simplicial spheres.
method Using edge contractions and the link condition as proxies for flagness, the study analyzes the effect of gamma positivity on simplicial spheres.
result The link condition has a trivial effect on gamma vectors of high-dimensional simplicial spheres with nonnegative gamma vectors.
Paper proposes a new deflation varimax method for vintage factor analysis.
problem Finding a scientifically meaningful low-dimensional representation of data.
method Deflation varimax procedure for orthogonal matrix rotation.
result The proposed method achieves minimax optimal factor loading estimation.
In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a …
Discretizes special surfaces using Koenigs nets.
problem Integrable structure of special surfaces.
method Discretisation via Koenigs nets.
result Preserves integrable structure in discretization.
We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel.…
Classifies nets with area-preserving transformations into two types.
problem Classifying nets with area-preserving transformations.
method Classification using Combescure transformations and isotropic metric duality.
result Found two classes of nets: cone nets and Koenigs nets.
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.
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation g…
Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.
problem Discretizing surfaces with spherical curvature lines.
method Lie-geometric discretisation in terms of principal contact element nets.
result Circular nets with two families of spherical parameter lines are related to Darboux cyclides.
Smart Close-out Nettingq-fin.RM
Smart Close-out Netting aims to automate close-out netting processes.
problem Inefficiencies in close-out netting processes for financial institutions.
method Standardisation and automation of legal and regulatory processes using a data-driven framework and controlled natural language.
result Standardisation and automation can improve close-out netting processes for prudentially regulated financial institutions.
This paper considers the power of deep neural networks (deep nets for short) in realizing data features. Based on refined covering number estimates, we find that, to realize some complex data features, deep nets can improve the performances of shallow neural networks (shallow nets for short) without requiring additiona…
Integrable nets described with curvature relations to pseudospherical surfaces.
problem Describing integrable curve nets and their geometric properties.
method Overview of second-order invariants, specific example of concordant nets, and construction of pseudospherical surfaces.
result Concordant Chebyshev nets correspond to pairs of pseudospherical surfaces.
Local Lipschitz property of the map which puts in correspondence to N--net its Chebyshev centermath.MG
We prove local Lipschitz property of the map which puts in correspondence to each --net different from --net its Chebyshev center. If dimension of Eucledean or Lobachevskii space is greater than 1 and net consists of more than 2 points we show that this map is not Lipschits in a neighbourhood of the space of …
Sign equivariant networks improve model expressiveness for spectral geometric learning.
problem Limited expressiveness of sign invariant models for tasks like graph link prediction.
method Developed sign equivariant neural network architectures based on new analytic sign equivariant polynomials.
result Sign equivariant models achieve theoretical benefits in spectral geometric learning tasks.
Unified framework for U-Net design and analysis.
problem Understudied design and architecture of U-Nets.
method Theoretical results, Multi-ResNets, function constraints encoding.
result Competitive and superior performance in various tasks.
Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary quadrilaterals are skew. We classify the simply connected affine A-nets that can be ext…
The paper examines special Q-nets that terminate after a finite number of Laplace steps.
problem Understanding the termination of Laplace sequences in Q-nets.
method Analyzing discrete Koenigs nets and their Laplace sequences.
result For certain Koenigs nets, Laplace sequences terminate after a finite number of steps.
Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
Generic metrics make geodesic nets dense.
problem Density of geodesic nets under generic metrics.
method Proving density for Baire-generic metrics.
result Union of geodesic nets images is dense.