Investigates a conjugate prior for Dirichlet distribution.
problem No specific problem stated; focuses on mathematical investigation.
method Investigates a conjugate class for the Dirichlet distribution within the exponential family.
result Identifies a conjugate prior for the Dirichlet distribution.
Paper constructs conjugate pairs for Bayesian nonparametric models with continuous likelihoods.
problem Limited conjugate pairs for Bayesian nonparametric models with continuous likelihoods.
method Develops a general construction for prior, likelihood, and posterior in conjugate pairs for processes with Levy measure densities from positive exponential families.
result Demonstrates conjugacy for processes with Levy measure densities from positive exponential families.
We present a general method for deriving collapsed variational inference algo- rithms for probabilistic models in the conjugate exponential family. Our method unifies many existing approaches to collapsed variational inference. Our collapsed variational inference leads to a new lower bound on the marginal likelihood. W…
Extends likelihood ratio exponential families to analyze various optimization methods.
problem Analyzing optimization methods like rate-distortion and information bottleneck.
method Linking geometric mixture paths to exponential families and using hypothesis testing.
result Provides a common mathematical framework for understanding these methods.
We characterize conjugate nonparametric Bayesian models as projective limits of conjugate, finite-dimensional Bayesian models. In particular, we identify a large class of nonparametric models representable as infinite-dimensional analogues of exponential family distributions and their canonical conjugate priors. This c…
The study identifies conjugate and cut points in ideal fluid motion configurations.
problem Understanding stability and re-convergence of fluid configurations.
method Existence and non-existence of conjugate points in specific fluid configurations, using geometric and physical analysis.
result Existence of conjugate points in Kolmogorov flows and non-existence in Arnold steady states.
The MAP estimate's log-likelihood sub-optimality is hard to bound in general.
problem Bounding the expected log-likelihood sub-optimality of MAP for exponential families.
method Interpreting MAP as stochastic mirror descent and analyzing convergence rates.
result Current convergence results do not apply to standard examples of exponential families.
Study the geometry of hydrodynamics equations using diffeomorphism groups.
problem Investigate the Euler equations and surface quasi-geostrophic equation family.
method Realize equations as geodesic equations on diffeomorphism groups and analyze Riemannian exponential maps.
result Show precise conditions for non-linear Fredholm maps of index 0.
Locally, isoperimetric problems on Riemannian surfaces are sub-Riemannian problems in dimension 3. The particular case of Dido problems corresponds to a class of singular contact sub-Riemannian metrics : metrics which have the charateristic vertor field as symmetry. We give a classification of the generic conjugate loc…
In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponent…
Improves variational inference for sparse models using mixtures of exponential families.
problem Intractability of posterior distributions in Bayesian sparse models.
method Flexible mean field variational inference using mixtures of non-overlapping exponential families.
result Mixtures of exponential families with non-overlapping support form an exponential family, enabling analytical updates.
The paper explains how the conjugate locus of a point on a surface can gain or lose cusps as the point moves.
problem Understanding the behavior of the conjugate locus as a point moves on a surface.
method Explains bifurcations in terms of vanishing higher derivatives of the exponential map and classifies them using scalar invariants.
result Simple equations for higher derivatives of the exponential map are derived, and the bifurcations of cusps are classified based on the local structure of the conjugate locus.
Privacy-preserving Bayesian inference framework for sensitive data.
problem Protecting sensitive information in Bayesian data analysis.
method Differential privacy framework for Variational Bayes, tailored to CE and non-CE models.
result Effective privatization of VB for CE models and improved privacy for non-CE models.
New method uses exponential family priors to handle shuffled data problems.
problem Handling mismatch errors in record linkage of two data files.
method Flexible exponential family prior on the permutation group for regularization.
result The proposed method outperforms competing methods in synthetic and real data.
Improved SVI with adjustable annealing for better optimization.
problem Improving optimization in stochastic variational inference.
method Tuneable stochastic annealing in SVI with adjustable batch size.
result Approximation to maximum entropy stochastic gradient at desired variance level.
BayesPy is an open-source Python software package for performing variational Bayesian inference. It is based on the variational message passing framework and supports conjugate exponential family models. By removing the tedious task of implementing the variational Bayesian update equations, the user can construct model…
Study on conjugate points in a C∗-algebra's Grassmann manifold.
problem Characterizing conjugate points in a C∗-algebra's Grassmann manifold. method Analyzes connections and geodesics in the Riemannian metric induced by the Killing form.
result Points that are tangent conjugate in the classical setting may not be conjugate in a C∗-algebra's Grassmann manifold. The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| < K(|f|+|g|), where |.| denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set. As a consequence, the conjugacy problem for mapping class groups is exponentiall…
We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate po…
A new method reduces variance in black-box variational inference.
problem High variance in black-box variational inference.
method Importance sampling from an overdispersed distribution.
result Effective reduction in variance with negligible overhead.
Study on sub-Riemannian manifolds, focusing on conjugate points and caustic stability.
problem Analyzing sub-Riemannian manifolds and their conjugate points.
method Computed sub-Riemannian Hamiltonian flow, approximated conjugate locus, introduced geometric invariant.
result Contact distributions exhibit unique behavior, different from 3D case.
The paper establishes a correspondence between quandle and biquandle colorings.
problem Understanding the relationship between quandle and biquandle colorings.
method Defining a functor and showing a one-to-one correspondence between colorings.
result The set of Alexander biquandles colorings is isomorphic to that of Alexander quandles colorings.
The intimate relationship between coherent states and geodesics is pointed out. For homogenous manifolds on which the exponential from the Lie algebra to the Lie group equals the geodesic exponential, and in particular for symmetric spaces, it is proved that the cut locus of the point 0 is equal to the set of coheren…
We provide a generalization of Bianchi's triply conjugate systems containing a family of deformations of 2-dimensional quadrics together with its Bäcklund transformation to higher dimensions.
Fast approximate inference for non-Gaussian data.
problem Efficient inference for non-Gaussian data.
method Laplace Matching for fast approximate inference in latent Gaussian models.
result Achieves high approximation quality with low computational cost.
Unified method for deriving formulas on almost complex manifolds.
problem Deriving formulas for derivations conjugated by exponential functions.
method Unified method using exponential functions.
result Corrected mistakes in previous versions.
Study Chabauty limits of subgroups of SL(n,Qp), focusing on parahoric and conjugate subgroups.
problem Classify and understand Chabauty limits of subgroups of SL(n,Qp). method Use various Levi decompositions and Bruhat–Tits buildings to classify limits.
result Found infinitely many non-conjugate Chabauty limits for n≥7. A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
In this paper we study the reduction curves of a braid, and how they can be used to decompose the braid into simpler ones in a precise way, which does not correspond exactly to the decomposition given by Thurston theory. Then we study how a cyclic sliding (which is a particular kind of conjugation) affects the normal f…
Paper introduces kernel deformed exponential families for sparse continuous attention.
problem Creating efficient attention mechanisms for sparse data.
method Developed kernel deformed exponential families, theoretically and experimentally.
result Kernel deformed exponential families can attend to multiple compact regions of data.
The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkab…
Classifies good involutions in conjugation subquandles and racks.
problem Classifying quandles with good involutions for applications in surface-knot theory.
method Study of subquandles of conjugation quandles, including core quandles; analysis of good involutions of faithful racks.
result Sharp bounds on the number of good involutions of racks in these families.
Paper develops EP algorithm for t-exponential family using q-algebra.
problem Efficient learning algorithm for t-exponential family distributions.
method Borrowing q-algebra from statistical physics, develop EP algorithm.
result Demonstrates performance of EP algorithm on Bayes point machine and Student-t process classification.
Proves properties of sub-Riemannian exponential map, showing it's not injective.
problem Regularity and continuity of sub-Riemannian exponential map.
method Used sub-Riemannian Jacobi fields and Maslov index of Jacobi curves.
result Exponential map of 3D Heisenberg group is not injective near conjugate vectors.
Bayesian hierarchical clustering (BHC) is an agglomerative clustering method, where a probabilistic model is defined and its marginal likelihoods are evaluated to decide which clusters to merge. While BHC provides a few advantages over traditional distance-based agglomerative clustering algorithms, successive evaluatio…
Upper bound on Stiefel manifold's injectivity radius found.
problem Finding the maximum distance within which the Stiefel manifold remains injective.
method Exhibited conjugate points and calculated the minimum of geodesic lengths.
result Upper bound on Stiefel manifold's injectivity radius is conjectured to be equal to the injectivity radius.
Abstract manifolds split with no conjugate points using special functions.
problem Splitting compact manifolds without conjugate points.
method Developed functions that vanish under certain conditions on central Busemann functions.
result Central Busemann functions split linearly under specific conditions.
Bayesian model updates data streams with hierarchical priors.
problem Continuous model updating and adapt to changes in data distribution.
method Non-conjugate hierarchical priors and variational inference.
result Validated on real data sets, demonstrating adaptability.
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. Generalizes moment-matching for exponential families with conditioning or hidden data.
problem Generalizing moment-matching conditions for exponential families with conditioning or hidden data.
method First-principles explanation and self-contained derivation of generalized moment-matching conditions.
result Derives generalized moment-matching conditions for conditional exponential families and hidden data.
Correspondence found between exponential families and affine Grassmannians.
problem Understanding the relationship between exponential families and geometric structures.
method Established a one-to-one correspondence between exponential families and affine Grassmannians.
result Found a correspondence between minimal exponential families and affine Grassmannians.
Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.
Analytic pairs in 3D space constructed via power series.
problem Constructing analytic pairs of conjugate functions in 3D.
method Exploiting an ansatz to create power series expansions.
result Found entire solutions not harmonic and a 2-parameter family.
We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in C3 of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\…
The goal of this thesis is to study the singularities of the exponential map of Riemannian and Finsler manifolds (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton), to improve existing results about its structure, to look at it in new ways, and t…
Constructing exponential families from statistical manifolds.
problem The central problem of constructing exponential families from statistical manifolds.
method Constructive approach proving every compact statistical manifold admits a foliation of Hessian manifolds.
result Compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy.
We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks …