arXiv research

Logarithmic connections on complex manifolds with trivial tangent bundle.

We pursue the study of holomorphic Cartan geometry with singularities. We introduce the notion of logarithmic Cartan geometry on a complex manifold, with polar part supported on a normal crossing divisor. In particular, we show that the push-forward of a Cartan geometry constructed using a finite Galois ramified coveri…

We shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a $C^\infty$ logarithmic symplectic structure has unobstruc…

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…

Research examines correlations of complex logarithms of lattice points, showing level repulsion and Poissonian behavior.

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…

Paper shows FB and FC are equally hard up to logarithmic factors.

Study logarithmic flat connections on principal bundles using Lie groupoids.

Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.

We prove a global residual formula in terms of logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normal crossing divisors. We also give a formula for the total sum of the logarithmic indices if the singular set of the foliation is contained in the invaria…

Applying logarithmic transformations along 2-tori, we construct a generalized complex structure J_n with n type changing luci for every $n\geq 0$ on genus 1-Lefschetz fibrations with a cusp neighborhood, which include elliptic surfaces with non-zero euler characteristic. Applying a technique of broken Lefschetz fibrati…

Near-logarithmic regret per switch achieved for mixable/exp-concave losses.

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connectio…

Study flat connections with logarithmic singularities on complex plane curves.

Localizes Wodzicki residue for logarithm of differential operators.

Study Higgs bundles on curves with punctures, extending spectral correspondence.

Study quadratic one-forms on logarithmic Higgs bundles on pointed curves.

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

We develop a new theoretical framework, the \emph{envelope complexity}, to analyze the minimax regret with logarithmic loss functions and derive a Bayesian predictor that adaptively achieves the minimax regret over high-dimensional $\ell_1$-balls within a factor of two. The prior is newly derived for achieving the mini…

Formula for sections on complex manifolds with non-isolated components.

The logarithmic Riemann surface Sigma_{log} is a classical holomorphic 1-manifold. It lives into R^4 and induces a covering space of C - 0 defined by exp. This paper suggests a geometric construction of it, derived as the limit of a sequence of vector fields extending exp suitably to embeddings of C into R^3, which tur…

We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.

We analyze the problem of sequential probability assignment for binary outcomes with side information and logarithmic loss, where regret---or, redundancy---is measured with respect to a (possibly infinite) class of experts. We provide upper and lower bounds for minimax regret in terms of sequential complexities of the …

The paper proves a logarithmic partial derivative lemma and applies it to several geometric problems.

Proves a Baum--Bott formula for foliations by curves with logarithmic terms.

Transformers capture combinatorial tasks with bounded error and logarithmic sample dependence.

The paper analyzes the statistical cost of tuning kernel hyperparameters in robust regression.

GOCPD detects change points by maximizing the probability of two independent models.

In this paper, we investigate representations of $\operatorname{At}(N)$, the Atiyah algebroids of a holomorphic line bundles $N$ over a complex manifold $Y$. In particular, we relate $\operatorname{At}(N)$-modules with logarithmic connections through two functors. On the one hand, we use these functors to the define in…

Improved model-free RL algorithm with reduced sample complexity.

Optimal ReLU networks can memorize any separable set of points with a small number of parameters.

Study hypothesis testing under quantized samples with communication constraints, achieving near-optimal sample complexity.

Paper analyzes and improves KL-regularized RL for LLMs with logarithmic regret.

We introduce blow-up and blow-down operations for generalized complex 4-manifolds. Combining these with a surgery analogous to the logarithmic transform, we then construct generalized complex structures on nCP2 # m \bar{CP2} for n odd, a family of 4-manifolds which admit neither complex nor symplectic structures unless…

Optimal unimodal fitting for linear loss functions in a sequential, efficient manner.

We consider the flows generated by generic gradients of Morse maps of a closed connected manifold $M$ to a circle. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called…

We present a logarithmic-scale efficient convolutional neural network architecture for edge devices, named WaveletNet. Our model is based on the well-known depthwise convolution, and on two new layers, which we introduce in this work: a wavelet convolution and a depthwise fast wavelet transform. By breaking the symmetr…

Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.

Improved GNN simulation of WL test with exponentially lower complexity.

Study 2-complexes' homology properties and torsion growth.

Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cota…

This work improves the convergence theory of diffusion models for generating samples from complex distributions.

Study shows sample complexity for learning optimal policies in SSP with generative model.

Study Kähler-Einstein potentials on stable varieties near singularities

Paper constructs an invariant for a specific type of complex manifolds.

Study rigidity by logarithmic capacity and related functions.

New algorithm reduces online logistic regression regret without exponential constant.

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