arXiv research

Study of symplectic trivialization and reduction of bundles with symmetry and connection.

Taking configuration space as a Lie group, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew's symplecti…

A geometrization of Schmidt-Legendre transformation of the second order Lagrangians is proposed by building a proper Tulczyjew's triplet. The symplectic relation between Ostrogradsky-Legendre and Schmidt-Legendre transformations is obtained. Several examples are presented.

Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.

In the scientific literature there are basically two schools of formulating Lagrangian (or Hamiltonian) mechanics in the (Lie) algebroid setting: in terms of prolongations and in terms of Tulczyjew triples. Despite the fact that in both approaches we describe the same phenomena, so far no comparison between prolongatio…

Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.

We propose a geometric approach to dynamical equations of physics, based on the idea of the Tulczyjew triple. We show the evolution of these concepts, starting with the roots lying in the variational calculus for statics, through Lagrangian and Hamiltonian mechanics, and concluding with Tulczyjew triples for classical …

We show how to extend the construction of Tulczyjew triples to Lie algebroids via graded manifolds. We also provide a generalisation of triangular Lie bialgebroids as higher Poisson and Schouten structures on Lie algebroids.

Constructs a triple on an Atiyah algebroid with connection.

In this paper the notion of Tulczyjew's triples in classical mechanics is extended to classical field theories, using the so-called multisymplectic formalism, and a convenient notion of lagrangian submanifold in multisymplectic geometry. Accordingly, the dynamical equations are interpreted as the local equations defini…

The geometrical structure known as the Tulczyjew triple has proved to be very useful in describing mechanical systems, even those with singular Lagrangians or subject to constraints. Starting from basic concepts of variational calculus, we construct the Tulczyjew triple for first-order Field Theory. The important featu…

Proposes a contact dynamics framework using generalized geometries.

The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of th…

We show that there exists a natural Tulczyjew triple in the dynamics of objects for which the standard kinematic configuration space $TM$, i.e. the tangent bundle, is replaced with its $n$-th exterior power, i.e. the bundle of tangent $n$-vectors. In this framework, which is fully covariant, we geometrically derive pha…

New method samples triplets from data distributions for training Triplet networks.

We study safe screening for metric learning. Distance metric learning can optimize a metric over a set of triplets, each one of which is defined by a pair of same class instances and an instance in a different class. However, the number of possible triplets is quite huge even for a small dataset. Our safe triplet scree…

Efficiently augments triplet data for better data analytics.

Around mid-1970s W. M. Tulczyjew discovered an approach which brings the two formalisms under a common geometric roof: the dynamics of a particle with configuration space $X$ is determined by a Lagrangian submanifold $D$ of $TT^*X$ (the total tangent space of $T^*X$), and the description of $D$ by its Hamiltonian $H$: …

New method accelerates large margin metric learning for nearest neighbor classification.

This paper analyzes the stability and generalization of triplet learning algorithms.

As an effective way of metric learning, triplet loss has been widely used in many deep learning tasks, including face recognition and person-ReID, leading to many states of the arts. The main innovation of triplet loss is using feature map to replace softmax in the classification task. Inspired by this concept, we prop…

Novel method decorrelates batches of triplets for active metric learning.

Personalized activity recognition improves performance for diverse users.

In an independence model, the triplets that represent conditional independences between singletons are called elementary. It is known that the elementary triplets represent the independence model unambiguously under some conditions. In this paper, we show how this representation helps performing some operations with in…

Paper proposes semi-supervised learning with triplet Markov chains.

We consider the problem of classification in a comparison-based setting: given a set of objects, we only have access to triplet comparisons of the form "object $x_i$ is closer to object $x_j$ than to object $x_k$." In this paper we introduce TripletBoost, a new method that can learn a classifier just from such triplet …

Recent studies have highlighted that deep neural networks (DNNs) are vulnerable to adversarial examples. In this paper, we improve the robustness of DNNs by utilizing techniques of Distance Metric Learning. Specifically, we incorporate Triplet Loss, one of the most popular Distance Metric Learning methods, into the fra…

This paper shows how optimizing with hard negative examples improves image retrieval.

The work investigates deep generative models, which allow us to use training data from one domain to build a model for another domain. We propose the Variational Bi-domain Triplet Autoencoder (VBTA) that learns a joint distribution of objects from different domains. We extend the VBTAs objective function by the relativ…

Abstracts a construction of boundary triplets for self-adjoint elliptic problems.

Alternative approach to regularize time-dependent singular Lagrangian systems.

We analyze the semi-hard triplet loss using Edgeworth expansion for better understanding of its behavior.

Paper introduces new loss functions for Siamese networks using FDA.

A description of time-dependent Mechanics in terms of Lagrangian submanifolds of Dirac manifolds (in particular, presymplectic and Poisson manifolds) is presented. Two new Tulczyjew triples are discussed. The first one is adapted to the restricted Hamiltonian formalism and the second one is adapted to the extended Hami…

Develops optimal trading strategy for illiquid currency pairs.

Triplet networks are widely used models that are characterized by good performance in classification and retrieval tasks. In this work we propose to train a triplet network by putting it as the discriminator in Generative Adversarial Nets (GANs). We make use of the good capability of representation learning of the disc…

Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similar…

Deep metric learning has been demonstrated to be highly effective in learning semantic representation and encoding information that can be used to measure data similarity, by relying on the embedding learned from metric learning. At the same time, variational autoencoder (VAE) has widely been used to approximate infere…

Deep metric learning is often used to learn an embedding function that captures the semantic differences within a dataset. A key factor in many problem domains is how this embedding generalizes to new classes of data. In observing many triplet selection strategies for Metric Learning, we find that the best performance …

Learning from triplet comparison data has been extensively studied in the context of metric learning, where we want to learn a distance metric between two instances, and ordinal embedding, where we want to learn an embedding in an Euclidean space of the given instances that preserves the comparison order as well as pos…

The static of smooth maps from the two-dimensional disc to a smooth manifold can be regarded as a simplified version of the Classical Field Theory. In this paper we construct the Tulczyjew triple for the problem and describe the Lagrangian and Hamiltonian formalism. We outline also natural generalizations of this appro…

Many machine learning tasks such as clustering, classification, and dataset search benefit from embedding data points in a space where distances reflect notions of relative similarity as perceived by humans. A common way to construct such an embedding is to request triplet similarity queries to an oracle, comparing two…

We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, w…

In this work, we use the Sternberg phase space (which may be considered as the classical phase space of particles in gauge fields) in order to explore the dynamics of such particles in the context of Hamilton-Dirac systems and their associated Hamilton-Pontryagin variational principles. For this, we develop an analogue…

To investigate objects without a describable notion of distance, one can gather ordinal information by asking triplet comparisons of the form "Is object $x$ closer to $y$ or is $x$ closer to $z$?" In order to learn from such data, the objects are typically embedded in a Euclidean space while satisfying as many triplet …

Involutory Hopf group-coalgebras provide new invariants for 4-manifold bundles.

The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew's triples. The results are also extended to the case of discrete dynamic…

In order to design haptic icons or build a haptic vocabulary, we require a set of easily distinguishable haptic signals to avoid perceptual ambiguity, which in turn requires a way to accurately estimate the perceptual (dis)similarity of such signals. In this work, we present a novel method to learn such a perceptual me…