arXiv research

Strictly enforcing orthonormality constraints on parameter matrices has been shown advantageous in deep learning. This amounts to Riemannian optimization on the Stiefel manifold, which, however, is computationally expensive. To address this challenge, we present two main contributions: (1) A new efficient retraction ma…

This paper establishes inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.

This work improves disentanglement in latent space models without sacrificing generation quality.

Deformations of singular Cayley submanifolds studied.

In this article we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a C…

Desingularizes conically singular Cayley submanifolds.

Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.

The study examines discrete curvature notions on Cayley graphs of certain groups.

Study Cayley fibrations on Bryant-Salamon $Spin(7)$ manifolds.

Given a Kaehler manifold of complex dimension 4, we consider submanifolds of (real) dimension 4, whose Kaehler angles coincide. We call these submanifolds Cayley. We investigate some of their basic properties, and prove that (a) if the ambient manifold is a Calabi-Yau, the minimal Cayley submanifolds are just the Cayle…

Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when…

The study introduces Cayley--Abels--Rosendal graphs for Polish groups.

Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.

We define Cayley structures as a field of Cayley's ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations. In particular, for Cayley structures an extension of certain notions defined for inde…

Cayley cones in the octonions $\mathbb{O}$ that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well known twistor fibration $G(2,8) -> S^6$ is used to prove the existence of immersed higher-genus pseudoholomorphic curves in $\gro$. Equivale…

The Cayley graph of quandles reveals structural properties and is studied for various classes.

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras $\mathfrak{f}_4$ and $\mathfrak{f}^*_4$, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we cla…

In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculate…

Introduces Lorentzian Cayley form solving geometric puzzle.

Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.

Functions of several octonion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the ${\tilde {\partial}}$-equations are studied. More generally functions of several Cayley-Dickson variables are considered. Integral formulas of the Martinelli-Bochner,…

We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.

The paper studies properties of Artin monoid Cayley graphs and their quasi-isometry to Deligne complexes.

Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R^8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley 4-form. We also…

Hill-ADAM optimizes loss landscapes by exploring state space deterministically.

We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This rec…

EAdam optimizer improves performance by adjusting ε in Adam.

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

This paper develops graph theory for racks and quasigroups.

Groups can act on 3-manifolds if their Cayley complex can embed in specific types of 3-manifolds.

Adam's generalization performance is improved by batch size and weight decay in neural networks.

AdamS uses momentum as a denominator to optimize LLMs efficiently.

Adam optimizes linear classifiers with separable data.

Simpler, parameter-free AdaGrad and Adam variants with convergence guarantees.

A new memory-efficient Adam variant reduces second moments when feasible.

Adam optimizer converges to zeros of a new vector field, not just gradient zeros.

AdaX improves Adam by exponentially accumulating past gradients, leading to better performance in machine learning tasks.

AdaSGD combines SGD and Adam benefits, eliminating the need for transition.

Adaptive optimization algorithms, such as Adam and RMSprop, have shown better optimization performance than stochastic gradient descent (SGD) in some scenarios. However, recent studies show that they often lead to worse generalization performance than SGD, especially for training deep neural networks (DNNs). In this wo…

Improved Adam for time series forecasting with distributional drift.

Extends Adams' theorem to periodic cohomology.

In this article, we consider Cayley deformations of a compact complex surface in a Calabi--Yau four-fold. We will study complex deformations of compact complex submanifolds of Calabi--Yau manifolds with a view to explaining why complex and Cayley deformations of a compact complex surface are the same. We in fact prove …

DualAdam improves generalization of Adam by integrating its update mechanisms.

Adam converges to stationary points under relaxed conditions.

In this paper we introduce the area of 2-ruled 4-folds in R^n (n=7 or 8), that is, submanifolds M of R^n that admit a fibration over some 2-fold Sigma such that each fibre is an affine 2-plane in R^n. This is motivated by the paper math.DG/0012060 by Joyce on ruled special Lagrangian 3-folds in C^3 and the work of the …

We construct a Kaehler structure on the punctured cotangent bundle of the Cayley projective plane whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and we show that the geodesic flow action is holomorphic and is expressed in a quite explicit form. We also give an embedding of the pun…

Laprop separates Adam's momentum and adaptivity to improve stability and speed.

The ADAM optimizer is exceedingly popular in the deep learning community. Often it works very well, sometimes it doesn't. Why? We interpret ADAM as a combination of two aspects: for each weight, the update direction is determined by the sign of stochastic gradients, whereas the update magnitude is determined by an esti…