arXiv research

The paper introduces various canonical parameterizations for 2D-curved shapes.

A new method scales CCA parameters by input to learn more correlated representations.

Solves parameter non-identifiability in Bayesian LTI system identification.

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic $k$-vector field, analogous to the ordinary geodesic field and which …

Method learns symmetries in curves without augmentation.

A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…

We show that there is a family of pseudo-Anosov braids independently parameterized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length and conclude that the conjugacy problem remains exponential in the braid index under t…

HOPE improves SSMs for long-memory tasks with robust initialization and training.

Geometrically classifies maps from R^0|2 to any manifold, unifying theories.

Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type locally admits geometrically determined parameters - canonical parameters. In such pa…

We propose a new technique, Singular Vector Canonical Correlation Analysis (SVCCA), a tool for quickly comparing two representations in a way that is both invariant to affine transform (allowing comparison between different layers and networks) and fast to compute (allowing more comparisons to be calculated than with p…

LNNs learn Lagrangians without canonical coordinates, conserving energy and relativity.

FedAvg converges linearly to global minimum in federated learning with partial participation.

In this paper, we present some theoretical work to explain why simple gradient descent methods are so successful in solving non-convex optimization problems in learning large-scale neural networks (NN). After introducing a mathematical tool called canonical space, we have proved that the objective functions in learning…

We present a particle flow realization of Bayes' rule, where an ODE-based neural operator is used to transport particles from a prior to its posterior after a new observation. We prove that such an ODE operator exists. Its neural parameterization can be trained in a meta-learning framework, allowing this operator to re…

Strata of $k$-differentials on smooth curves parameterize sections of the $k$-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the $κ$ and $ψ$ classes of moduli spaces of pointed smooth curves along with the tautological class $η$…

New method uses quotient predictor space for better PAC-Bayes bounds, reducing KL divergence and improving model performance.

The spaces of harmonic maps of the projective plane to the four-dimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree less than 6 it was shown that such spaces are path-connected and an explicit par…

New parameterization of Teichmüller spaces for complex Lie groups.

Given an effectively parameterized family $f:X\to S$ of canonically polarized manifolds, the Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $K_{X/S}$. We use a global elliptic equation to show that this metric is strictly positive everywhere and give estimates. The dire…

The aim of this survey is to give an overview on the geometry of Einstein maximal globally hyperbolic 2+1 spacetimes of arbitrary curvature, conatining a complete Cauchy surface of finite type. In particular a specialization to the finite type case of the canonicla Wick rotation-rescaling theory, previously developed b…

The paper proposes a new method for creating interpretable models using convex optimization.

Neural networks parameterize time-varying Markov dynamics in financial time series.

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of $k$-differentials on smooth curves which parameterize sections of the $k$-th power of the canonical line bund…

There are currently two parameterizations used to derive fixed kernels corresponding to infinite width neural networks, the NTK (Neural Tangent Kernel) parameterization and the naive standard parameterization. However, the extrapolation of both of these parameterizations to infinite width is problematic. The standard p…

We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let $(X,B)$ be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…

D3-brane solutions derived from Ricci-flat metrics on Kähler-Einstein surfaces.

A new GAN model $α$-GAN with tunable loss function addresses gradient vanishing and mode collapse issues.

New parameterization for $(1,1)$-knots simplifies their study.

Develops a new framework to analyze gradient flow regimes and derive explicit solutions.

Stochastic parameterizations account for uncertainty in the representation of unresolved sub-grid processes by sampling from the distribution of possible sub-grid forcings. Some existing stochastic parameterizations utilize data-driven approaches to characterize uncertainty, but these approaches require significant str…

The current paper discusses some new results about conformal polynomic surface parameterizations. A new theorem is proved: Given a conformal polynomic surface parameterization of any degree it must be harmonic on each component. As a first geometrical application, every surface that admits a conformal polynomic paramet…

Polynomially parameterizes knots and spheres, proving analogous results.

Conformal surface parameterization is useful in graphics, imaging and visualization, with applications to texture mapping, atlas construction, registration, remeshing and so on. With the increasing capability in scanning and storing data, dense 3D surface meshes are common nowadays. While meshes with higher resolution …

Two novel algorithms for conformal parameterization of multiply-connected surfaces.

The paper analyzes how over-parameterization affects GD convergence in matrix sensing problems.

Novel framework for policy optimization with general parameterization and linear convergence.

We introduce a new parameterization method for deep learning layers using spectral tensor train decomposition.

Machine learning models emulate and approximate complex mappings in model physics.

Develops a method for conformal parameterization of point clouds without fixed boundaries.

New method learns soliton dynamics from scattering data without assuming known equations.

Point cloud is the most fundamental representation of 3D geometric objects. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh st…

Least squares regression shows unexpected double descent in under-parameterized models.

The paper proposes methods for volumetric parameterization of 3D solid manifolds.

This is a survey of the theory of complex projective (CP^1) structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: (1) Using the Schwarzian derivative to parameterize the moduli space (2) Thurston's parameterization of the moduli space using grafting (3) …

Study shows how over-parameterized classifiers can still perform well on noisy data.

Improves adaptivity in sequence models by over-parameterizing.

Improves understanding of stochastic NGVI convergence rates.