arXiv subjects

Alexander Fauck

Publications and source records attributed to Alexander Fauck.

4 recordsLinked to original sources

arXiv2023

ELRA: Exponential learning rate adaption gradient descent optimization method

We present a novel, fast (exponential rate adaption), ab initio (hyper-parameter-free) gradient based optimizer algorithm. The main idea of the method is to adapt the learning rate $α$ by situational awareness, mainly striving for orthogonal neighboring gradients. The method has a high success and fast convergence rate and does not rely on hand-tuned parameters giving it greater universality. It can be applied to problems of any dimensions n and scales only linearly (of order O(n)) with the dimension of the problem. It optimizes convex and non-convex continuous landscapes providing some kind of gradient. In contrast to the Ada-family (AdaGrad, AdaMax, AdaDelta, Adam, etc.) the method is rotation invariant: optimization path and performance are independent of coordinate choices. The impressive performance is demonstrated by extensive experiments on the MNIST benchmark data-set against state-of-the-art optimizers. We name this new class of optimizers after its core idea Exponential Learning Rate Adaption - ELRA. We present it in two variants c2min and p2min with slightly different control. The authors strongly believe that ELRA will open a completely new research direction for gradient descent optimize.

cs.LG↗

arXiv2020

Computing the Rabinowitz Floer homology of tentacular hyperboloids

We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids $Σ\simeq S^{n+k-1}\times\mathbb{R}^{n-k}$. Using an embedding of a compact sphere $Σ_0\simeq S^{2k-1}$ into the hypersurface $Σ$, we construct a chain map from the Floer complex of $Σ$ to the Floer complex of $Σ_0$. In contrast to the compact case, the Rabinowitz Floer homology groups of $Σ$ are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.

math.SG↗

arXiv2020

On manifolds with infinitely many fillable contact structures

We introduce the notion of asymptotically finitely generated contact structures, which states essentially that the Symplectic Homology in a certain degree of any filling of such contact manifolds is uniformly generated by only finitely many Reeb orbits. This property is used to generalize a famous result by Ustilovsky: We show that in a large class of manifolds (including all unit cotangent bundles and all Weinstein fillable contact manifolds with torsion first Chern class) each carries infinitely many exactly fillable contact structures. These are all different from the ones constructed recently by Lazarev. Along the way, the construction of Symplectic Homology is made more general. Moreover, we give a detailed exposition of Cieliebak's Invariance Theorem for subcritical handle attaching, where we provide explicit Hamiltonians for the squeezing on the handle.

math.SG↗

arXiv2016

Rabinowitz-Floer homology on Brieskorn manifolds

This thesis considers fillable contact structures on odd-dimensional manifolds. For that purpose, Rabinowitz-Floer homology (RFH) is used which was introduced by Cieliebak and Frauenfelder in 2009. A major part of the thesis is devoted to technical problems in the definition of RFH. In particular, it is shown that the moduli spaces involved are cut out transversally. Moreover, it is proved that RFH is essentially invariant under subcritical handle attachment. Finally, RFH is calculated for some Brieskorn manifolds. The obtained results are then used to show for every manifold, which supports fillable contact structures, that there exist either infinitely many different fillable contact structures, or one contact structure with infinitely many different fillings or for every fillable contact structure holds that RFH is infinite dimensional in every degree.

math.SG↗