arXiv research

Study proposes a functional for LCK metrics on complex manifolds.

Study continuity of complex Sobolev functions, with applications to Kaehler metrics.

Characterizes complex Hessian equations for bounded energy functions.

Complex functional maps link tangent bundles, preserving orientation and angles.

Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.

We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistors spaces of comple…

We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We also provide some applications to the regularization of J-plurisubharmonic func…

In the previous papers \cite{L1, L2} the author constructed Mabuchi and Aubin-Yau functionals over any complex surfaces and three-folds, respectively. Using the method in \cite{L2}, we construct those functionals over any complex manifolds of the complex dimension bigger than or equal to 2.

Complexity measures for neural nets with general activations using path-based norms.

The paper explores complex Poisson structures on smooth functions in complex manifolds.

New neural network models for complex functional data analysis.

Paper proposes LANN to measure model complexity of neural networks with curve activation functions.

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold $(M,ω)$ with $c_{1}|_{π_{2}(M)}=[ω]|_{π_{2}(M)…

The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction to a fundamental domain of the real-valued Morse function on the pullback infinite cyclic cover.

Paper constructs Thom-Smale complex using instantons from Morse functions.

Improves sampling, rounding, and integration of logconcave functions.

The paper calculates the number of closed cycles in a specific complex group.

Maxout networks show similar complexity issues as ReLU networks.

CVNNs improve performance in tasks with complex-valued inputs.

We show that the Rademacher complexity of any $\mathbb{R}^{K}$-valued function class composed with an $\ell_{\infty}$-Lipschitz function is bounded by the maximum Rademacher complexity of the restriction of the function class along each coordinate, times a factor of $\tilde{O}(\sqrt{K})$.

The study links Ricci curvature and convexity in complex tori.

The paper uses complex-valued functions to simplify plane differential geometry and kinematics.

Unified theory of deep neural networks with diverse activations.

Extends holomorphic functions on complex manifolds to larger spaces.

We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the ho…

It is well-known that non-constant holomorphic functions do not exist on a compact complex manifold. This statement is false for a supermanifold with a compact reduction. In this paper we study the question under what conditions non-constant holomorphic functions do not exist on a compact homogeneous complex supermanif…

In this paper we construct Mabuchi $\mathcal{L}^{\rm M}_ω$ functional and Aubin-Yau functionals $\mathcal{I}^{\rm AY}_ω, \mathcal{J}^{\rm AY}_ω$ on any compact complex three-folds. The method presented here will be used in the forthcoming paper \cite{L1} on the construction of those functionals on any compact complex m…

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical point…

The paper introduces new functionals and equations for complex vector bundles.

We show how the space of complex spin structures of a closed oriented three-manifold embeds naturally into a space of quadratic functions associated to its linking pairing. Besides, we extend the Goussarov-Habiro theory of finite type invariants to the realm of compact oriented three-manifolds equipped with a complex s…

The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M de…

Complex-valued neural networks can approximate any continuous function.

Let f be a Morse map from a closed manifold to a circle. S.P.Novikov constructed an analog of the Morse complex for f. The Novikov complex is a chain complex defined over the ring of Laurent power series with integral coefficients and finite negative part. This complex depends on the choice of a gradient-like vector fi…

Improved generalization bounds for CNNs using Rademacher complexity.

Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.

Study approximates unknown function levels with queries.

SGD learns neural networks with a complexity measure called leap.

New bounds on ReLU networks for low-regular functions.

Novel framework for teaching complexity in machine teaching models.

We provide tight upper and lower bounds on the complexity of minimizing the average of $m$ convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient de…

Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.

New complexity measure explains neural network generalization gap.

New complexity measure ADL connects to classical complexity measures.

Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids o…

The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.

The paper analyzes risk bounds and Rademacher complexity in batch RL.

Introduces HTV to measure function complexity in learning schemes.

Transformer architecture struggles with complex tasks due to limitations in function composition.