arXiv research

Normal forms and moduli stacks for flat connections on complex manifolds.

We generalize Cartan's logarithmic derivative of a smooth map from a manifold into a Lie group $G$ to smooth maps into a homogeneous space $M=G/H$, and determine the global monodromy obstruction to reconstructing such maps from infinitesimal data. The logarithmic derivative of the embedding of a submanifold $Σ\subset M…

Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.

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

The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the function. In particular, the nonzero level sets of a homogeneous polynomial are proper…

Study flat connections with logarithmic singularities on complex plane curves.

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

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves…

We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed -- the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a sequence of subsets $A_n$ of a homogeneous space $G/Γ$ ($G$ a semisimple Lie group…

We study the problem of the existence and the holomorphicity of the Monge-Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a …

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

New algorithm tackles heterogeneous curvature in online convex optimization.

Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.

Study BF invariants using simple type concepts.

In this paper we extend our previous work on singularities of Monge-Ampère foliations to the case of pseudoconvex finite type domains. We are able to answer the questin of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge-Ampère equation completely in dimension 2 . We are also able to general…

The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow define…

In an effort to better understand the different ways in which the discount factor affects the optimization process in reinforcement learning, we designed a set of experiments to study each effect in isolation. Our analysis reveals that the common perception that poor performance of low discount factors is caused by (to…

An extension of the ambient metric construction of Fefferman-Graham to infinite order in even dimensions is described. The main ingredients are the introduction of "inhomogeneous ambient metrics" with asymptotic expansions involving the logarithm of a defining function homogeneous of degree 2, and an invariant procedur…

Recently, prediction markets have shown considerable promise for developing flexible mechanisms for machine learning. In this paper, agents with isoelastic utilities are considered. It is shown that the costs associated with homogeneous markets of agents with isoelastic utilities produce equilibrium prices correspondin…

Study on geodesic distances on SE(3)/SO(2) in machine learning.

We investigate the infinitesimal invariants of an immersed submanifold $Σ$ of a Klein geometry $M\cong G/H$, and in particular an invariant filtration of Lie algebroids over $Σ$. The invariants are derived from the logarithmic derivative of the immersion of $Σ$ into $M$, a complete invariant introduced in the companion…

Let $U \subset \mathbb A^n$ be an open subset of real affine space. We consider functions $F: U \to \mathbb R$ with non-degenerate Hessian such that the first or the third derivative of $F$ is parallel with respect to the Levi-Civita connection defined by the Hessian metric $F"$. In the former case the solutions are gi…

New metrics defined for full-rank correlation matrices, ensuring unique operations.

New method achieves nearly horizon-free offline reinforcement learning for tabular and linear MDPs.

Study optimal portfolio strategy with sporadic bankruptcy for isoelastic utility.

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…

New entropy flow method extends generalization bounds for all Markov algorithms.

We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an inf…

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.

Logarithmic connections on principal bundles over normal varieties are studied.

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.

Geometrically convex return risk measures on AM-algebras

New framework for logarithmically divergent integrals on manifolds with corners.

In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation function…

Directly proves logarithmic systolic growth for all hyperbolic surfaces.

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.

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…

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.

Nonparametric tests via kernel embedding of distributions have witnessed a great deal of practical successes in recent years. However, statistical properties of these tests are largely unknown beyond consistency against a fixed alternative. To fill in this void, we study here the asymptotic properties of goodness-of-fi…

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