Unified determinants via a single equation.
problem Defining determinants with all known properties.
method Proposing a single equation implying all known properties of determinants.
result Unified definition of determinants with all properties.
Study one-dimensional topological theories with linear generating functions.
problem Understanding one-dimensional topological theories with defects.
method Construct bases of hom spaces for decorated unoriented one-dimensional cobordisms.
result Gram determinant and linear generating functions constructed.
StarNet trains deep models without gradients using linear equations.
problem Training deep generative models with gradients.
method Solving determined systems of linear equations.
result Least-square bounds for latent codes and model parameters.
New method for linearly determining Lie groups from data.
problem Linear determination of Lie groups from data.
method Polar representation and isotropy subgroup data.
result Linear determination possible in general reducible case.
We determine the number of connected components of the moduli space for representations of a surface group in the general linear group.
Develops goodness-of-fit tests for noisy submanifold samples.
problem Testing non-linear models on noisy submanifold samples.
method Non-linear least-square problem solution and χ2 distribution application. result Residual follows χ2 distribution with parameters related to model order and dimension. The paper classifies symplectic forms on R^4 and determines invariants under symplectomorphisms.
problem Classifying symplectic forms on R^4 under symplectomorphisms.
method Using pfaffian and sum function invariants, the paper provides a complete description of orbit spaces and determines global invariants.
result The paper provides a complete classification of symplectic forms on R^4 under symplectomorphisms, providing necessary conditions for intertwining.
New rings reveal surprising prime colorings.
problem Determining which primes can color rainbow rings.
method Linear algebra eigenvalues and knot theory colorability.
result Almost all primes admit 0, 1, or infinite colorings.
An approach is proposed to determine structural shift in time-series assuming non-linear dependence of lagged values of dependent variable. Copulas are used to model non-linear dependence of time series components.
Study on determining metrics via Dirichlet-to-Neumann map for harmonic maps.
problem Determining Riemannian metrics from boundary measurements.
method Higher linearization method, integral identities, energy rigidity.
result Metrics on the target manifold are equal if the target is analytic.
Study shows boundary measurements can determine transversal singularities in anisotropic geometries.
problem Determining transversal singularities in anisotropic geometries from boundary measurements.
method Geometric condition on transversal manifold and FBI type transform.
result Recovery of transversal singularities in the linearized problem.
Covariance in physics and CNNs share similarities, with simple assumptions uniquely determining convolution forms.
problem Understanding the similarities between physics and CNNs using covariance.
method Examined similarities and differences, and demonstrated that simple assumptions lead to unique convolution forms.
result Simple assumptions of covariance, locality, linearity, and weight sharing uniquely determine convolution forms.
We present an alternative to the pseudo-inverse method for determining the hidden to output weight values for Extreme Learning Machines performing classification tasks. The method is based on linear discriminant analysis and provides Bayes optimal single point estimates for the weight values.
We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of linear combinations of Chern numbers with that property and prove its optimality in…
Determines conditions for ribbon cobordisms between lens spaces.
problem Conditions for ribbon rational homology cobordisms between lens spaces.
method Analyzes ribbon cobordisms and uses properties of lens spaces and linear lattices.
result If a lens space admits a ribbon rational homology cobordism to a different lens space, it must be homeomorphic to L(n,1), up to orientation-reversal. Study Gram determinants in knot theory, focusing on a Möbius band determinant.
problem Closed formula for the Gram determinant of type (Mb)1. method Survey of Gram determinants, focusing on a Möbius band determinant.
result Speculation on closed formula for (Mb)1 Gram determinant. We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduc…
Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.
problem Linear perturbations of Spin(7) metrics.
method Applying the method of linear perturbations to Spin(7)-structures.
result Only rank one nilpotent matrices determine nontrivial perturbations.
The paper calculates bounds on the local Lipschitz constants of neural network layers.
problem Understanding the Lipschitz constants of neural network layers for robustness analysis.
method Analytical approach to determine upper bounds on local Lipschitz constants of affine-ReLU functions.
result The method produces tighter bounds than the standard conservative bound, especially for small perturbations.
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.
In the two parts of this paper we solve a problem of De Rham, proving that Reidemeister torsion invariants determine topological equivalence of linear G-representations, for G a finite cyclic group. Methods in controlled K-theory and surgery theory are developed to establish, and effectively calculate, a necessary and …
Defines linear weightings for vector bundles and explores their applications.
problem Understanding and extending the concept of weightings in vector bundles.
method Constructs weighted normal bundles and deformation spaces; explains the relationship between weightings and differential operators.
result Captures the rescaled spinor bundle and related constructions.
We analyze concave idleness functions of graph curvature.
problem Understanding curvature functions of graphs.
method We study the concavity and piecewise linearity of Ollivier-Ricci idleness functions of graphs.
result Idleness functions are concave and piecewise linear with at most 3 linear parts.
We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and tho…
Estimates for polynomial operators using determinant majorization and subharmonics.
problem Bounding solutions of polynomial operators on Euclidean domains.
method Combines Alexandrov estimate and determinant majorization, using subharmonics and semiconvex approximation.
result Includes classical Alexandrov-Bakelman-Pucci estimate for linear operators.
Given a 2-stranded tangle in a $\ZZ/2$ homology ball, T⊂Y, we investigate the character variety R(Y,T) of conjugacy classes of traceless SU(2) representations of π1(Y∖T). In particular we completely determine the subspace of binary dihedral representations, and identify all of R(Y,T) for many t…
This paper explores evolving spheres to Hopf spheres using integer coefficients.
problem Conditions for evolving rotationally symmetric spheres to Hopf spheres.
method Integer linear combination of radii of curvature.
result The fate of an initial sphere is determined by the local geometry of umbilic points.
The paper calculates automorphisms of Weil algebras, focusing on one-component groups.
problem Calculating automorphisms of Weil algebras.
method Direct calculation method detailed for R-algebra automorphisms. result Two cases of determinant values for linear part of one-component groups.
Paper converts deep networks to flat, equivalent kernel machines.
problem Capacity control and uniform convergence in deep learning.
method Push-forward transformation from deep networks to indefinite kernel machines.
result Flat network weights are Lp-norm regularized (0<p<1).
Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in R2. method Determined Morse indices and kernels of the linearized operator of shrinkers. Constructed flows using unstable eigenfunctions.
result Existence of ancient flows with finite entropy.
Deep learning approach to inverse optimization.
problem Determine parameters of an optimization process from observations.
method Unroll iterative optimization, use backpropagation to learn parameters.
result Learn coefficients for cost vector and constraints in linear programs.
Paper proves linearity of solutions to degenerate elliptic equations in 3D.
problem Determining linearity of degree-one homogeneous solutions to degenerate elliptic equations in 3D.
method Analyzes degenerate ellipticity condition and uses geometric properties of geodesic arcs.
result Proves linearity of solutions under specific degenerate ellipticity condition.
The paper investigates compatible linear connections on Randers spaces and finds a unique extremal connection.
problem Investigating compatible linear connections on Randers spaces.
method Transformed compatibility equations by taking torsion components as variables and determined when these equations have solutions.
result Characterized Randers spaces as non-Riemannian generalized Berwald spaces with a positive constant norm of perturbing term.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
DGPs learn from multiple tasks using shared and private latent processes.
problem Improving learning performance and information transfer between tasks.
method Non-linear mixtures of latent processes with shared and task-specific components, using hard or soft sharing.
result DGPs outperform other multi-task learning models across various settings.
We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally g…
As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian manifold determined by its Levi-Civita connection? It is shown that for a generic…
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.
problem Cohomology of the Regge complex in three dimensions.
method Constructing a discrete version of linearized Riemann-Cartan geometry on any triangulation.
result The cohomology of the Regge complex is isomorphic to the infinitesimal-rigid-body-motion-valued de~Rham cohomology.
The paper calculates the Saito determinant for Coxeter discriminant strata.
problem Calculating the Saito determinant for specific geometric strata.
method Using the Saito flat metric and Lie derivatives, the paper finds the determinant of the metric restricted to Coxeter discriminant strata.
result The determinant of the Saito metric on Coxeter discriminant strata is proportional to a product of linear factors in flat coordinates.
In this paper we prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 4. This is accomplished by introducing a smaller yet equivalent line…
Neural networks and linear systems linked, revealing training loss and kernel limitations.
problem Exploring the training loss and limitations of neural networks and their kernels.
method Drawing connections between neural networks and under-determined linear systems, providing lower bounds, and analyzing gradient descent.
result Zero training loss achievable for neural networks under certain conditions, but not for ReLU kernels.
We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperb…
The paper analyzes methods for sparse Bayesian regression in nonlinear system identification.
problem Learning sparse models in Bayesian regression with nonlinear applications.
method Two classes of methods: regularization and thresholding based, built on automatic relevance determination (ARD).
result Analytical demonstration of favorable performance with sparse solutions in linear problems.
Using the Selberg trace formula, we show that for a hyperbolic 2-orbifold, the spectrum of the Laplacian acting on functions determines, and is determined by, the following data: the volume; the total length of the mirror boundary; the number of conepoints of each order, counting a mirror corner as half a conepoint; an…
The article describe the model, derivation, and implementation of variational Bayesian inference for linear and logistic regression, both with and without automatic relevance determination. It has the dual function of acting as a tutorial for the derivation of variational Bayesian inference for simple models, as well a…
Invertible networks help explain decisions and identify important features.
problem Interpreting and explaining the decisions of black-box neural networks.
method Two-stage approach: invertible transformation to feature space and linear classifier. Determining decision boundaries and feature importance using local linear models.
result Ability to explain decisions and identify important features in neural networks.
Paper presents a novel orbit determination method for spacecraft clusters.
problem Orbit determination for spacecraft clusters with noisy and non-linear observations.
method Kernel embedding techniques for learning orbits from range-rate observations.
result The method can accurately estimate orbits and identify individual satellites.
Optimizes machine learning by approximating log determinants efficiently.
problem Computational expense in calculating log determinants for large data sets.
method Demonstrates the optimality of Maximum Entropy methods in approximating log determinants.
result Reduction of KL divergence between proposal and true eigenvalue distribution by adding more moments.