arXiv research

Unified tensor model disentangles object appearance factors.

In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavour (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of disc…

A model separates visual style from digit type on MNIST and facial features from shape on CelebA.

We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is i…

Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.

In this contribution we present an intrinsic description of time-variant Port Hamiltonian systems as they appear in modeling and control theory. This formulation is based on the splitting of the state bundle and the use of appropriate covariant derivatives, which guarantees that the structure of the equations is invari…

Data living on manifolds commonly appear in many applications. Often this results from an inherently latent low-dimensional system being observed through higher dimensional measurements. We show that under certain conditions, it is possible to construct an intrinsic and isometric data representation, which respects an …

Classically time is kept fixed for infinitesimal variations in problems in mechanics. Apparently, there appears to be no mathematical justification in the literature for this standard procedure. This can be explained canonically by unveiling the intrinsic mathematical structure of time in Lagrangian mechanics. Moreover…

We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor field defining a parallelism structure on M. Also we give a novel and intrinsic pr…

Proofs Fisher-Rao distance on Gaussian covariance manifold.

We systematically analyse the necessary and sufficient conditions for the preservation of supersymmetry for bosonic geometries of the form R^{1,9-d} \times M_d, in the common NS-NS sector of type II string theory and also type I/heterotic string theory. The results are phrased in terms of the intrinsic torsion of G-str…

A new method classifies almost contact metric manifolds using intrinsic endomorphisms.

This paper explores intrinsic rewards to improve learning from multiple value functions.

We equip the whole tangent space $TM$ to a hyperbolic manifold $M$ (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of $M$ extend to isometries of $TM$ by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…

New tensors capture intrinsic embedding data of conformal hypersurfaces.

Zero-Shot Learning (ZSL) aims at classifying unlabeled objects by leveraging auxiliary knowledge, such as semantic representations. A limitation of previous approaches is that only intrinsic properties of objects, e.g. their visual appearance, are taken into account while their context, e.g. the surrounding objects in …

New method detects anomalies in systems influenced by their environment.

Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e., frame independent) of the first and second Cartan structure equations. Also, the…

We consider the "intrinsic" symmetry group of a two-component link $L$, defined to be the image $Σ(L)$ of the natural homomorphism from the standard symmetry group $\MCG(S^3,L)$ to the product $\MCG(S^3) \cross \MCG(L)$. This group, first defined by Whitten in 1969, records directly whether $L$ is isotopic to a link $L…

This work introduces a protocol to automatically select the correct range of scales for meaningful Intrinsic Dimension estimation.

Reinforcement learning algorithms rely on carefully engineering environment rewards that are extrinsic to the agent. However, annotating each environment with hand-designed, dense rewards is not scalable, motivating the need for developing reward functions that are intrinsic to the agent. Curiosity is a type of intrins…

Some intrinsic tools from the formal theory of variational equations are being demonstrated at work in application to one concrete example of the third-order evolution equation of free relativistic top in three-dimensional space-time. The main goal is to introduce a combined approach consisting in the simultaneous util…

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer th…

We give a relationship that yields an effective geometric way of evaluating mean curvature of surfaces. The approach is reminiscent of the Gauss's contour based evaluation of intrinsic curvature. The presented formula may have a number of potential applications including estimating the normal vector and mean curvature …

Paper extends Bayesian Cramér-Rao bound with geometric considerations.

Survey on geometric foundations of data reduction methods.

We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regard…

Deep neural networks (DNNs) are computationally/memory-intensive and vulnerable to adversarial attacks, making them prohibitive in some real-world applications. By converting dense models into sparse ones, pruning appears to be a promising solution to reducing the computation/memory cost. This paper studies classificat…

Study heat kernel on quaternionic contact manifolds, finding linear dependence of coefficients on curvature.

Paper proves conditions for 3D submanifolds to embed in 4D space.

Study properties of sets with constant normal in Carnot groups.

In this paper we establish the basic tools to develop the "Calculus" associated with group-valued continuously Pansu differentiable mappings. We develop the technical machinery on which all of our results rely. In particular, the linearization of addends appearing in the Baker-Campbell-Hausdorff formula is one of the m…

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

The Fisher-Rao geometry is applied to elliptical distributions for optimization and classification.

We applied Generative Adversarial Networks (GANs) to learn a model of DOOM levels from human-designed content. Initially, we analysed the levels and extracted several topological features. Then, for each level, we extracted a set of images identifying the occupied area, the height map, the walls, and the position of ga…

Dirac structures are geometric objects that generalize both Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems. In this paper, we show that the evolution equa- tions for nonequilibrium thermodynamics admit an intrinsic formulation in …

In this paper we generalize special geometry to arbitrary signatures in target space. We formulate the definitions in a precise mathematical setting and give a translation to the coordinate formalism used in physics. For the projective case, we first discuss in detail projective Kaehler manifolds, appearing in N=1 supe…

Bayesian model predicts mid-price dynamics in financial markets.

Survey of intrinsically linked or knotted graphs.

The paper studies properties of intrinsically Lipschitz constants in metric spaces.

We prove an analogue of the classical Steiner formula for the $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Stein…

Learning can be seen as approximating an unknown function by interpolating the training data. Kriging offers a solution to this problem based on the prior specification of a kernel. We explore a numerical approximation approach to kernel selection/construction based on the simple premise that a kernel must be good if t…

We introduce new sufficient conditions for intrinsic knotting and linking. A graph on n vertices with at least 4n-9 edges is intrinsically linked. A graph on n vertices with at least 5n-14 edges is intrinsically knotted. We also classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respe…

A geometric account explains why 'The Dress' is ambiguous, predicting observable signatures in image processing.

New graph shows edge deletion/contraction doesn't always result in intrinsically linked graphs.

New method estimates intrinsic dimensionality using angles, not distances.

Identifying meaningful signal buried in noise is a problem of interest arising in diverse scenarios of data-driven modeling. We present here a theoretical framework for exploiting intrinsic geometry in data that resists noise corruption, and might be identifiable under severe obfuscation. Our approach is based on uncov…

A directed graph $G$ is $\textit{intrinsically linked}$ if every embedding of that graph contains a non-split link $L$, where each component of $L$ is a consistently oriented cycle in $G$. A $\textit{tournament}$ is a directed graph where each pair of vertices is connected by exactly one directed edge. We consider intr…