arXiv research

Logarithmic connections on principal bundles over normal varieties are studied.

Study of logarithms in SVD-closed subgroups of unitary group.

Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.

Study logarithmic flat connections on principal bundles using Lie groupoids.

In this paper, we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure; we prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kahler differentials; therefore, it induces a differential complex from which we derive the notio…

The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). Th…

Logarithmic connections on complex manifolds with trivial tangent bundle.

In the framework of Abstract Differential Geometry, we show that to a given principal sheaf and a representation of its stuctural sheaf in $A^n$, where A is a sheaf of associative, commutative, unital algebras (over R or C), we associate a vector sheaf. Moreover, under some natural assumptions on the compatibility of t…

Local connection forms provide a very useful tool for handling connections on principal bundles, because they ignore any complexities of the total space and, essentially, involve only two fundamental features of the structure group, namely the adjoint representation and the left (logarithmic) differential. The main res…

Paper optimizes private PCA for covariance estimation in statistics.

Algorithm learns optimal coordination for strategic agents in uncertain settings.

This paper provides theoretical guarantees for SPCA using the Elastic Net.

Gradient descent recovers principal components of overparametrized asymmetric matrices without explicit regularization.

We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and Sarnak by Katz, Schaps, and Vishne, which uses principal "congruence" subgroups o…

A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.

We consider the two logarithmic strain measures\[ω_{\rm iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and }\quad ω_{\rm vol}=|\mathrm{tr}(\log U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,\]which are isotropic invariants of the Hencky strain tensor $\log U$, and show that they can be uniquely char…

Kernel principal component analysis (KPCA) provides a concise set of basis vectors which capture non-linear structures within large data sets, and is a central tool in data analysis and learning. To allow for non-linear relations, typically a full $n \times n$ kernel matrix is constructed over $n$ data points, but this…

New geometric framework for positive semidefinite matrices of fixed rank.

Paper addresses eigenvector perturbation in small eigen-gap scenarios.

As described in this paper, we study market-wide price co-movements around crashes by analyzing a dataset of high-frequency stock returns of the constituent issues of Nikkei 225 Index listed on the Tokyo Stock Exchange for the three years during 2007--2009. Results of day-to-day principal component analysis of the time…

Using adiabatic limits of Eta invariants, Rho invariants of the total space of a fiber bundle are investigated. One concern is to formulate the aspects of local index theory for families of Dirac operator in terms of the odd signature operator, and place known results in a context which permits the treatment of Rho inv…

The paper develops a method for optimal projection selection in high-dimensional classification.

A new method calculates fractional moments using the moment-generating function.

Fourier PCA is Principal Component Analysis of a matrix obtained from higher order derivatives of the logarithm of the Fourier transform of a distribution.We make this method algorithmic by developing a tensor decomposition method for a pair of tensors sharing the same vectors in rank-$1$ decompositions. Our main appli…

Low-precision streaming PCA estimates the leading eigenvector with limited precision.

Study rigidity by logarithmic capacity and related functions.

Study real logarithms of semi-simple matrices, focusing on differential structure.

Study excess logarithmic residues for foliations to bound invariant hypersurfaces and test log canonicity.

Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite, number of loss functions. In this paper, we propose a novel Riemannian extension of the Euclidean stochastic variance reduced gradient algorithm (R-SVRG) to a compact manifold search space. To this e…

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness…

Local logarithmic Brunn-Minkowski holds for zonoids.

New framework for logarithmically divergent integrals on manifolds with corners.

Directly proves logarithmic systolic growth for all hyperbolic surfaces.

The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied by Slomczynski and Zastawniak (Ann. Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability space…

Logarithmic separation profile in hyperbolic groups shows hierarchical structure.

Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.

Investment and consumption strategy optimized under uncertain conditions.

The paper constructs a Saito basis for a specific class of divisors and applies it to logarithmic Poisson geometry.

Extended logarithm for solvable elements in mapping class groups.

In this work, we give a formula for the logarithmic invariant of knots in terms of certain derivatives of the colored Jones invariant. This invariant is related to the logarithmic conformal field theory, and was defined by using the centers in the radical of the restricted quantum group at root of unity. A relation bet…

Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.

Paper finds necessary condition for logarithmic Minkowski problem in higher dimensions.

We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a "logarithmic cone", the surfaces are embedded.

Bandit algorithms struggle with consistent performance and robustness.

Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.

We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a m…

Logarithmic Sobolev inequality proven for non-compact self-shrinkers.