arXiv research

This paper studies the Fisher-Rao geometry on the parameter space of beta distributions. We derive the geodesic equations and the sectional curvature, and prove that it is negative. This leads to uniqueness for the Riemannian centroid in that space. We use this Riemannian structure to study canonical moments, an intrin…

Introduces a new canonical connection for Riemannian manifolds and proves Frobenius theorem geometrically.

In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions on a 5-dimensional manifold. We solve the analogous problems for rank 2 distributions on an n-dimensional manifold for arbitrary n greater than 5. Our method is a kind of symplectification…

Study on diffeologies on locally convex spaces and smooth multiplication of distributions.

Constructs canonical frames for specific distributions, proving maximality and describing germs.

A new method called MCLMC avoids dissipation in sampling from canonical distributions.

We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second ord…

Proposes GCCA for detecting latent relations in multiview data with sparse structures.

As was shown recently by P. Nurowski, to any rank 2 maximally nonholonomic vector distribution on a 5-dimensional manifold M one can assign the canonical conformal structure of signature (3,2). His construction is based on the properties of the special 12-dimensional coframe bundle over M, which was distinguished by E.…

To certain types of generic distributions (subbundles in a tangent bundle) one can associate canonical Cartan connections. Many of these constructions fall into the class of parabolic geometries. The aim of this article is to show how strong restrictions on the possibles sizes of automorphism groups of such distributio…

We construct a canonical frame for an arbitrary Gl(2)-structure thus solving the equivalence problem for Gl(2)-structures. Our treatment includes also a problem of contact equivalence of ordinary differential equations and applies to certain classes of vector distributions. Additionally we characterise Gl(2)-structures…

We regard a contact metric manifold whose Reeb vector field belongs to the $(κ,μ)$-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric $(κ,μ)$-spaces in terms of a canonical connection which can be naturally defined on them.

We aim to analyze the relation between two random vectors that may potentially have both different number of attributes as well as realizations, and which may even not have a joint distribution. This problem arises in many practical domains, including biology and architecture. Existing techniques assume the vectors to …

We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded En…

We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistr…

Study cone structures on contact manifolds to understand their geometric properties.

Paper optimizes classification of distributions using Wasserstein metric.

The paper explores geometric and algebraic structures on Lie groups.

We consider a problem of equivalence of generic pairs $(X,V)$ on a manifold $M$, where $V$ is a distribution of rank $m$ and $X$ is a distribution of rank one. We construct a canonical bundle with a canonical frame. We prove that two pairs are equivalent if and only if the corresponding frames are diffeomorphic. As a p…

The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.

In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in $\mathbb R^5$. We solve the analogous problem for germs of generic rank 2 distributions in ${\mathbb R}^n$ for n>5. We use a completely different approach based on the symplectification o…

Efficiently transforms samples from various statistical models.

Canonical transformation plays a fundamental role in simplifying and solving classical Hamiltonian systems. We construct flexible and powerful canonical transformations as generative models using symplectic neural networks. The model transforms physical variables towards a latent representation with an independent harm…

Simplified proofs and new distributions on anti-quasi-Sasakian manifolds.

I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…

We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in se…

Study Hochschild cohomology of dg manifolds linked to integrable distributions.

Robust test for distributions under Hellinger distance, simpler than optimal tests.

We study the sample complexity of canonical correlation analysis (CCA), \ie, the number of samples needed to estimate the population canonical correlation and directions up to arbitrarily small error. With mild assumptions on the data distribution, we show that in order to achieve $ε$-suboptimality in a properly define…

Solves parameter non-identifiability in Bayesian LTI system identification.

We use variational Gaussian approximations to analyze parametric models with unknown data-generating distributions.

A central tenet of probabilistic programming is that a model is specified exactly once in a canonical representation which is usable by inference algorithms. We describe JointDistributions, a family of declarative representations of directed graphical models in TensorFlow Probability.

Infinite-dimensional diffusion models tackle generative tasks for complex data.

We present RandomizedCCA, a randomized algorithm for computing canonical analysis, suitable for large datasets stored either out of core or on a distributed file system. Accurate results can be obtained in as few as two data passes, which is relevant for distributed processing frameworks in which iteration is expensive…

The mean field methods, which entail approximating intractable probability distributions variationally with distributions from a tractable family, enjoy high efficiency, guaranteed convergence, and provide lower bounds on the true likelihood. But due to requirement for model-specific derivation of the optimization equa…

Each lens space has a canonical contact structure which lifts to the distribution of complex lines on the three-sphere. In this paper, we show that a symplectic homology cobordism between two lens spaces, which is given with the canonical contact structure on the boundary, must be diffeomorphic to the product of a lens…

The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.

Canonical twistor fibrations lead to Pfaffian systems by means of their superhorizontal distribution. The aim of this note is to identify explicitly the Pfaffian systems of five or less variables that arise in this way in terms of the classification given by A.Awane and M.Goze.

We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution ("lagrangians") and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a conseque…

We construct canonical frames and find all maximally symmetric models for a natural generic class of corank 2 distributions on manifolds of odd dimension greater or equal to 7. This class of distributions is characterized by the following two conditions: the pencil of 2-forms associated with the corresponding Pfaffian …

We study the behaviour of differential forms in a manifold having at least one of their maximal isotropic local distributions endowed with the special algebraic property of being decomposable. We show that they can be represented as the sum of a form with constant coefficients and one that vanishes whenever contracted …

Two probability distributions $μ$ and $ν$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are al…

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. I…

New approach constructs symplectic structure on pseudo-Riemannian geodesics.

Motivated by the geometric theory of differential equations and the variational approach to the equivalence problem for geometric structures on manifolds, we consider the problem of equivalence for distributions with fixed submanifolds of flags on each fiber. We call them flag structures. The construction of the canoni…

We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Koda…

We provide a new and simple system of equations for the normal sub-Riemannian geodesics. These use a partial connection that we show is canonically available, given a choice of complement to the distribution. We also describe conditions which, if satisfied, mean that even this choice of complement is determined canonic…

In this article we construct a canonical Kähler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical Kähler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical Kähler-Einstein current the c…