The paper introduces structured variational families to improve scalability in black-box variational inference.
problem Scalability issues in black-box variational inference, especially for large datasets and hierarchical models.
method Developed structured variational families that achieve better iteration complexity of O(N) compared to full-rank families.
result Structured variational families can achieve better scaling with respect to dataset size N, improving iteration complexity from O(N^2) to O(N).
Study of double complexes on Iwasawa manifold yields 3 isomorphism types.
problem Understanding the structure of double complexes on the Iwasawa manifold.
method Used Stelzig and Qi-Khovanov's structure theorem for double complexes.
result Identified and described exactly 3 isomorphism types of double complexes.
Constructs independent bases for cubic curve families using Hessian structures.
problem Finding independent bases for cubic curve families.
method Uses a Hessian structure to define a cost function for constructing bases.
result Constructs valuatively independent bases for H0(X,Lk). The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that th…
Study geometric structures on LVM threefolds, focusing on resonant structures.
problem Understanding deformations of geometric structures on LVM threefolds.
method Using the Ehresmann-Thurston principle and Kuranishi family construction.
result Construction of a family containing all LVM threefolds and complete at every point.
The paper classifies natural almost Hermitian structures on Lie groups with minimal conformal leaves.
problem Classifying natural almost Hermitian structures on Lie groups with minimal conformal leaves.
method Analyzing Lie groups with a 2-dimensional conformal foliation and classifying structures based on Lie algebra properties.
result 16 multi-dimensional almost Kähler families, 18 integrable families, and 11 Kähler families were constructed.
Constructs Lorentzian manifolds from Riemannian conformal structures.
problem Creating Lorentzian manifolds from Riemannian conformal structures.
method Starting from a Riemannian conformal structure, a family of Lorentzian manifolds is constructed using a metric in the conformal class and a 1-parameter family of tensor fields.
result Every Mobius structure on a Riemannian conformal structure arises from this construction.
We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform…
Hopf manifolds can be given lcK structures, shown by constructing a family.
problem Constructing locally conformally Kaehler structures for Hopf manifolds.
method Analytic family construction and application of Ornea-Verbitsky's theorem.
result Hopf manifolds can be endowed with lcK structures.
We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products, with an extra natural condition (namely, rotating structure vector field). We prove that, provided the surface is not tot…
The study explores generalized divergences and exponential families with a focus on sufficient conditions and laws of large numbers.
problem Generalization of Kullback-Leibler divergence and exponential families.
method Investigation of (h,τ)-divergence and (h,τ)-exponential families, definition of (h,τ)-dependence, proof of law of large numbers. result Sufficient condition for (h,τ)-divergence to induce Hessian structure on (h,τ)-exponential family, proof of law of large numbers. The paper explores families of almost complex structures and transverse (p,p)-forms.
problem Understanding and producing families of almost p-Kähler structures.
method Producing families of almost p-Kähler structures (Jt,Ωt) on $\C^3$, $\C^4$, and T6. result The almost complex structures Jt cannot be locally compatible with any symplectic form for teq0. One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces J1M, equipped with the standard contact structure, is through the Morse theoretic technique of generating families. This paper extends the invariant of generating family cohomology by giving it a product μ2. To define the product, moduli…
We construct some families of complex structures on compact manifolds by means of normal almost contact structures (nacs) so that each complex manifold in the family has a non-singular holomorphic flow. These families include as particular cases the Hopf and Calabi-Eckmann manifolds and the complex structures on the pr…
Complex structures found on product of Sasakian manifolds.
problem Exploring complex structures on the product of two Sasakian manifolds.
method Examined a family of complex structures on the product of two compact Sasakian manifolds, computed Dolbeault cohomology groups.
result None of the complex structures in the family admit compatible locally conformally Kähler metrics if both Sasakian manifolds are of dimension greater than 1.
The study proves rigidity for mixed Hodge structures and applies to curve families.
problem Rigidity of period maps for mixed Hodge structures.
method Holomorphic bisectional curvature approach.
result Establishes rigidity in various cases, including curve families.
We investigate the curvature properties of a two-parameter family of Hermitian structures on the product of two Sasakian manifolds, as well as intermediate relations. We give a necessary and sufficient condition for a Hermitian structure belonging to the family to be Einstein and provide concrete examples.
Paper explores duality in DPPs using embedding structure analysis.
problem Understanding the geometric structure of determinantal point processes.
method Analyzes the exponential family embedding of DPPs and uses the e-embedding curvature tensor.
result Discovers the duality between marginal and L-ensemble kernels.
Classifies tight contact structures on surgeries of the Whitehead link.
problem Classifying tight contact structures on surgeries of the Whitehead link.
method Analyzes various surgeries on the Whitehead link to classify tight contact structures.
result Determines tight contact structures, Stein fillability, and virtually overtwisted properties.
The abstract discusses families of holomorphic maps to Oka manifolds with approximation theorems.
problem Approximating \(J_b\)-holomorphic maps to Oka manifolds.
method Constructing continuous or smooth families of \(J_b\)-holomorphic maps to Oka manifolds with approximation on compact Runge sets.
result Runge and Mergelyan approximation theorems and Weierstrass interpolation theorem for families of open Riemann surfaces.
Researchers compute mod 2 Seiberg-Witten invariants for spin structures and families.
problem Computing mod 2 Seiberg-Witten invariants for spin structures and families.
method Using Pin(2)-symmetry and localisation in equivariant cohomology, the researchers enhance and compute the invariants.
result The mod 2 Seiberg-Witten invariants are computed for spin structures and families, confirming the simple type conjecture mod 2.
Consider two families of closed oriented curves in a d-manifold. At each point of intersecction of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an i-dimensional family and a j-dimensional family will produce …
Geometric structures on 5-manifolds from surface group representations of G2'.
problem Constructing geometric structures on 5-manifolds from G2'-surface group representations.
method Using Higgs bundles and partial flag manifolds of G2' to construct geometric structures.
result Developing maps of geometric structures are the domain of discontinuity.
We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable 1-forms on $R^3
Constructs continuous families of minimal surfaces and holomorphic immersions.
problem Creating continuous families of minimal surfaces and holomorphic immersions.
method Continuous family of complex structures and conformal minimal immersions.
result Continuous families of proper Jb-conformal minimal immersions and holomorphic null immersions. We put in a general framework the situations in which a Riemannian manifold admits a family of compatible complex structures, including hyperkahler metrics and the Spin-rotations of arxiv:1302.2846. We determine the (polystable) holomorphic bundles which are rotable, i.e., they remain holomorphic when we change a compl…
On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of…
New maximal families of compatible Poisson structures derived from geodesically equivalent metrics.
problem Constructing maximal families of compatible Poisson structures.
method Connecting geodesically equivalent metrics and compatible Poisson structures of hydrodynamic type.
result Maximal families of compatible Poisson structures of dimension (n+1)(n+2)/2 are constructed. New separation concepts for Anosov representations help bound Thurston asymmetric metric.
problem Understanding diverging families of Anosov representations.
method Introducing separation concepts and analyzing combinatorial invariants.
result Critical exponent asymptotic to a graph invariant.
The space of marked n distinct points on the complex projective line up to projective transformations will be called a configuration space in this paper. There are two families of complex hyperbolic structures on the configuration space constructed by Deligne-Mostow and Thurston. We first confirm that these families ar…
Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
Stochastic variational inference offers an attractive option as a default method for differentiable probabilistic programming. However, the performance of the variational approach depends on the choice of an appropriate variational family. Here, we introduce automatic structured variational inference (ASVI), a fully au…
This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying struc…
Orbits of families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows for a global description of a smooth geometric structure on a family of manifolds in terms of a single object defined on the corresponding family of vector fields. Stratified spaces, Poisson spaces and almost compl…
Monopole invariant studies contact structures on 3-manifolds.
problem Topology of contact structures on 3-manifolds.
method Generalized Kronheimer--Mrowka--Ozsváth--Szabó contact invariant for families of contact structures.
result Obstructs the existence of sections and detects exotic loops of contact structures.
The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yi…
We obtain new families of (1,2)-symplectic invariant metrics on the full complex flag manifolds F(n). For n > 4, we characterize n-3 different n-dimensional families of (1,2)-symplectic invariant metrics on F(n). Any of these families corresponds to a different class of non-integrable invariant almost complex structure…
Study complex structure deformations on Lie algebras and Dolbeault cohomology.
problem Deformations of complex structures on Lie algebras and their associated Dolbeault cohomology.
method Construct a complete deformation of complex structures similar to the Kuranishi family, showing extension isomorphism validity.
result Analytic open subset of deformations where Dolbeault cohomology can be computed by left invariant tensor fields.
We consider relations between two families of flat manifolds with holonomy group (Z_2)^k of diagonal type. The family RBM of real Bott manifolds and the family GHW of generalized Hantzsche-Wendt manifolds. In particular, we prove that the intersection GHW∩RBM is not empty. We also …
Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.
problem Proving the existence of bounding cochains for unobstructed Lagrangians.
method Introducing non-archimedean analytic structure and using family Floer techniques.
result All Lagrangians in a connected family are unobstructed if one is.
In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…
We prove the existence of a one-parameter family of nearly parallel G2-structures on the manifold S3×R4, which are mutually non isomorphic and invariant under the cohomogeneity one action of the group SU(2)3. This family connects the two locally homogeneous nearly parallel G2-structures which…
We characterize unimodular solvable Lie algebras with Vaisman structures in terms of Kähler flat Lie algebras equipped with a suitable derivation. Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sa…
New tree-structured Markov fields with Poisson marginals for counting variables.
problem Counting variables with complex dependencies.
method Tree-structured Markov random fields with Poisson marginals.
result Straightforward sampling and joint probability calculations.
Study examines heart and football-shaped metrics, verifying geometric structure.
problem Analyzing reducible spherical conical metrics and their geometric properties.
method Examined 1-parameter heart shape and 3-parameter football shape families, verified structure theorem, used explicit metric and geodesic calculations.
result Naturally arise from Abelian differentials of the third kind, offer new evidence for spherical geometry and complex analytic structure interaction.
New polynomials defined for quandle structures, enhancing graph invariants.
problem Enhancing the counting invariant for spatial graphs and handlebody-links.
method Introducing quandle polynomials and G-family polynomials for quandles, defining enhancements for invariants.
result New enhancements of the G-family counting invariant for trivalent spatial graphs and handlebody-links.
Found a new compact G2-structure on a 7-manifold.
problem Identifying compact G2-structures on manifolds.
method Lattice in solvable Lie group, Laplacian coflow analysis.
result Obtained a 4-parameter subfamily of expanding solitons.
A complex orthogonal (geometric) structure on a complex manifold is a geometric structure locally modelled on a non-degenerate quadric. One of the first examples of such a structure on a compact manifold of dimension three was constructed by Guillot. In this paper, we show that the same manifold carries a family of uni…