arXiv research

When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equat…

Generates natural product-like compounds using GPT models.

We derive both {\em local} and {\em global} generalized {\em Bianchi identities} for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the {\em a priori} introduction of a connection. The proof is based on a {\em global} decom…

By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in …

Machine learning finds new natural laws from noisy data.

The paper proposes a method to train time-varying generative models using natural gradients.

We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold $(M,g)$ and we find the conditions under which the Riemannian manifold $(TM,G)$, where $TM$ is the tangent bundle of $M$ and $G$ is the general natural lifted metric of $g$, has constant sectio…

Paper proves naturally reductive property is inaudible for certain manifolds.

In optimization, the natural gradient method is well-known for likelihood maximization. The method uses the Kullback-Leibler divergence, corresponding infinitesimally to the Fisher-Rao metric, which is pulled back to the parameter space of a family of probability distributions. This way, gradients with respect to the p…

We continue the study of the anti-Hermitian structures of general natural lift type on the tangent bundles. We get the conditions under which these structures are in the eight classes obtained by Ganchev and Borisov. We complete the characterization of the general natural anti-Kahlerian structures on the tangent bundle…

Natural gradient descent avoids the magic of model parametrization, leading to different optimization outcomes.

Interactive Fiction games are text-based simulations in which an agent interacts with the world purely through natural language. They are ideal environments for studying how to extend reinforcement learning agents to meet the challenges of natural language understanding, partial observability, and action generation in …

A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-ad…

The main result of this paper is that every naturally reductive space can be explicitly constructed from the construction in \cite{Storm2018}. This gives us a general formula for any naturally reductive space and from this we prove reducibility and isomorphism criteria.

Introduces a natural parallel translation for navigation data.

We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant sectional curvature and we find some expressions for the natural lifted metric $G…

Proposes a new framework for environmental CVA with robust wrong-way risk.

New Einstein metrics found on SU(N) without being naturally reductive.

Study natural and conjugate mates of Frenet curves in Lie groups.

New definition of naturally reductive Finsler manifolds using geodesic graphs.

Proposes model-based robust deep learning to handle natural variation in data.

Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their met…

We construct a natural generalized complex structure on the total space of any bundle endowed with a Chern connection and whose typical fibre is a homogeneous symplectic manifold. This extends known constructions of generalized complex structures on Lie groups and leads to natural examples of holomorphic maps between g…

Generative Adversarial Networks (GAN) have demonstrated impressive results in modeling the distribution of natural images, learning latent representations that capture semantic variations in an unsupervised basis. Beyond the generation of novel samples, it is of special interest to exploit the ability of the GAN genera…

New method for natural policy gradients converges linearly.

We assume a vector bundle $p: E\to M$ with a general linear connection $K$ and a classical linear connection $\Lam$ on $M$. We prove that all classical linear connections on the total space $E$ naturally given by $(\Lam, K)$ form a 15-parameter family. Further we prove that all connections on $J^1 E$ naturally given by…

Improved robot navigation using multi-head attention for natural language instructions.

We propose some natural generalizations of Reidemeister moves that do not increase the number of crossings in the generated diagrams. Experimentations make us conjecture that this class of monotonic moves is complete for computing canonical forms and then deciding isotopy.

A new method generates natural-looking adversarial examples by bounding internal activation values.

New approach improves adversarial robustness without sacrificing natural generalization.

Is cognition a collection of loosely connected functions tuned to different tasks, or can there be a general learning algorithm? If such an hypothetical general algorithm did exist, tuned to our world, could it adapt seamlessly to a world with different laws of nature? We consider the theory that predictive coding is s…

Study flag curvature in homogeneous Finsler spaces with a specific metric.

We consider the second variational derivative of a given gauge-natural invariant Lagrangian taken with respect to (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. By requiring such a second variational derivative to vanish, {\em via} the Second Noether Theorem we find t…

Natural gradient descent is a robust optimization method for machine learning.

We study the conditions under which a Kählerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for $(T^*M,G,J)$ to be a Kählerian manifold, is expres…

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left-invariant metric is generated by a certain naturally defined inner product.

In order to study tensor fields of type (0,2) on manifolds and fibrations we introduce the notion of s-spaces. With the help of these objects we generalized the concept of natural tensor without making use of the theory of natural operators and differential invariants.

Variational inference transforms posterior inference into parametric optimization thereby enabling the use of latent variable models where otherwise impractical. However, variational inference can be finicky when different variational parameters control variables that are strongly correlated under the model. Traditiona…

Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.

Generative Adversarial Networks (GANs) have gathered a lot of attention from the computer vision community, yielding impressive results for image generation. Advances in the adversarial generation of natural language from noise however are not commensurate with the progress made in generating images, and still lag far …

Hardness proven for neural networks with natural weights.

The reduction theorems for general linear and classical connections are generalized for operators with values in higher order gauge-natural bundles. We prove that natural operators depending on the $s_1$-jets of classical connections, on the $s_2$-jets of general linear connections and on the $r$-jets of tensor fields …

New insights into natural exponential families improve regret bounds for bandit problems.

Introduces new deformation classes in generalized Kähler geometry.

A central challenge in neuroscience is to understand neural computations and circuit mechanisms that underlie the encoding of ethologically relevant, natural stimuli. In multilayered neural circuits, nonlinear processes such as synaptic transmission and spiking dynamics present a significant obstacle to the creation of…

A characterization of the foliation by spacelike slices of an $(n+1)$-dimensional spatially closed Generalized Robertson-Walker spacetime is given by means of studying a natural mean curvature type equation on spacelike graphs. Under some natural assumptions, of physical or geometric nature, all the entire solutions of…

iPrompt uses LLMs to generate natural-language explanations of data patterns.

We give an alternative proof for the Mathai-Quillen formula for a Thom form using its natural behaviour with respect to fiberwise integration. We also study this phenomenon in general context.