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 logistic regression model is known to converge to a Poisson point process model if the binary response tends to infinitely imbalanced. In this paper, it is shown that this phenomenon is universal in a wide class of link functions on binomial regression. The proof relies on the extreme value theory. For the logit, p…
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. Exponential family distributions are highly useful in machine learning since their calculation can be performed efficiently through natural parameters. The exponential family has recently been extended to the t-exponential family, which contains Student-t distributions as family members and thus allows us to handle noi…
Paper extends sparse alternatives to softmax for continuous domains, enabling efficient attention mechanisms.
problem Efficiently assigning zero probability to irrelevant categories in continuous domains.
method Extend alpha-entmax to continuous domains, introducing continuous-domain attention mechanisms.
result Continuous attention allows attending to time intervals and compact regions, improving text classification, machine translation, and visual question answering.
Introduces Legendre bundle for dually flat manifolds and quantum field theories.
problem Understanding duality in geometric structures and quantum field theories.
method Introduces Legendre bundle and para-Kähler structure.
result Exponential families and Hessian QFTs are realizations of the Legendre bundle.
Study infinitesimal deformations of Lie algebroid pairs.
problem Infinitesimal deformations of Lie algebroid pairs.
method Investigate isomorphism classes of infinitesimal deformations of (L,A) modulo automorphisms from exponentials of derivations of L and those from the exponentials of inner derivations of L. result Find the associated governing L∞-algebras in the sense of extended deformation theory. We introduce a new distance metric for non-linear embeddings of Tempered Exponential Measures.
problem Non-linear embeddings of Tempered Exponential Measures (TEMs).
method Parameterization of finite discrete TEMs via Legendre functions, introducing tempered Hilbert co-simplex distance.
result Established a generalization of the Hilbert log cross-ratio simplex distance to a tempered Hilbert co-simplex distance.
We recall the construction of non-formal deformation quantization of the Poincare Group ISO(1,1) on its coadjoint orbit and exhibit the associated non-formal star-exponentials.
We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the phys…
New theorem connects minimal and maximal surfaces, affecting graphness.
problem Understanding graphness of minimal surfaces in different spaces.
method Introducing a new deformation family and proving Krust-type theorems.
result Graphness of minimal surfaces in isotropic 3-space affects deformed surfaces.
The paper introduces natural α-embeddings for item representations.
problem Overcoming computational limitations in item encoding.
method Interpreting item embeddings in an Information Geometric framework using α-geometry of exponential families.
result A family of natural α-embeddings in the tangent space of the probability simplex.
New method calibrates reference distributions for bounded support.
problem Lack of principled method for bounded-support statistical reference distributions.
method Formulated maximum entropy on projective space of nonnegative measures.
result Prescribed acceptance region uniquely determines deformation parameter.
Introduces q-paths for generalizing geometric annealing paths in machine learning.
problem Limited applicability of existing path methods in machine learning.
method Develops a family of paths derived from a generalized mean, including geometric and arithmetic mixtures.
result Empirical gains in Bayesian inference and generative model evaluation.
We prove the existence of a new 2-parameter family oΔ of embedded triply periodic minimal surfaces of genus 3. The new surfaces share many properties with classical orthorhombic deformations of Schwarz' D surface, but also exotic in many ways. In particular, they do not belong to Meeks' five-dimensional family. Never…
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.
This paper develops sparse alternatives to continuous distributions, including new types of Gaussians and attention mechanisms.
problem Creating flexible continuous distributions with varying support for machine learning applications.
method Defining Ω-regularized prediction maps and Fenchel-Young losses for arbitrary domains, and deriving new types of Gaussians and attention mechanisms. result Sparse alternatives to continuous distributions, including deformed exponential families and β-Gaussians, are introduced. New MD algorithms using Tempesta logarithms for machine learning.
problem Optimization in machine learning with tailored hyperparameters.
method Developed Mirror Descent algorithms using Tempesta multi-parametric logarithms.
result Wide and flexible family of Mirror Descent and mirror-less updates.
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.
Study deformations of Calabi-Yau foliations using Kuranishi spaces.
problem Deforming Calabi-Yau foliations and understanding their properties.
method Analysis of three types of deformations (unfoldings, holomorphic, transversally holomorphic) using Kuranishi spaces.
result Smoothness of Kf and product structure of Kh. 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.
In this paper, we prove several formulas related to Hodge theory, and using them to prove the deformations of a compact H-twisted generalized Calabi-Yau manifold are unobstructed and L2 convergence in a neighborhood in another power series . And if we assume that the deformation is smooth in a fixed neighborhood, …
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 …
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.
Study on deforming complex manifolds and Higgs bundles.
problem Deforming holomorphic-Higgs pairs on complex manifolds.
method Introduced a DGLA and derived the Maurer-Cartan equation to govern the deformation.
result Proved the local completeness of the Kuranishi family of the deformed holomorphic-Higgs pair.
Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We gi…
Nowozin \textit{et al} showed last year how to extend the GAN \textit{principle} to all f-divergences. The approach is elegant but falls short of a full description of the supervised game, and says little about the key player, the generator: for example, what does the generator actually converge to if solving the GAN…
Thompson Sampling has been demonstrated in many complex bandit models, however the theoretical guarantees available for the parametric multi-armed bandit are still limited to the Bernoulli case. Here we extend them by proving asymptotic optimality of the algorithm using the Jeffreys prior for 1-dimensional exponential …
Constructs families of Toeplitz operators for symplectic fibrations.
problem Quantization of symplectic fibrations.
method Smooth families of Szegö projections and Toeplitz operators.
result Deformation quantization of prequantizable symplectic fibrations.
New Thompson sampling algorithm reduces regret for exponential family bandits.
problem Minimizing regret in multi-armed bandit problems with exponential family rewards.
method Proposes ExpTS and ExpTS+ algorithms using novel sampling distributions. result Minimizes both finite-time and asymptotic regret for exponential family rewards.
Moment polytope of toric exponential families is a projection of a simplex.
problem Understanding the geometry of exponential families in finite sample spaces.
method Toric torification and projection of higher-dimensional simplices.
result Moment polytope is a projection of a higher-dimensional simplex.
We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012b) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if and only if the opti…
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 $\…
We propose a novel approach for density estimation with exponential families for the case when the true density may not fall within the chosen family. Our approach augments the sufficient statistics with features designed to accumulate probability mass in the neighborhood of the observed points, resulting in a non-para…
The versatility of exponential families, along with their attendant convexity properties, make them a popular and effective statistical model. A central issue is learning these models in high-dimensions, such as when there is some sparsity pattern of the optimal parameter. This work characterizes a certain strong conve…
New insights into natural exponential families improve regret bounds for bandit problems.
problem Improving regret bounds for bandit problems with subexponential tails.
method Proving self-concordance for natural exponential families and applying to bandits.
result Optimistic algorithms for generalized linear bandits have second-order regret bounds that are free of an exponential dependence on problem parameters.
Constructs universal local deformations for curves and differential forms.
problem Local deformations of curves and differential forms under preservation of periods.
method Develops Kuranishi families for pairs of curves and meromorphic 1-forms, focusing on hyperelliptic cases.
result First paper in a series developing a deformation theory for spectral curve data of integrable systems.
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.
Unified diffusive bounds for non-linear parabolic equations.
problem Proving diffusive upper bounds for parabolic equations.
method Simple exponential deformation argument.
result Unified diffusive upper bounds for a wide class of non-linear parabolic equations.
The paper introduces a new method for graph embedding using exponential family distributions.
problem Representing networks in a low dimensional latent space for various applications.
method Introduces the exponential family graph embedding model, generalizing random walk-based techniques to exponential family conditional distributions.
result The proposed techniques outperform existing methods in link prediction and node classification tasks.
Efficient method for learning continuous exponential families beyond Gaussian.
problem Learning continuous exponential families with unbounded support.
method Interaction Screening approach for scalable learning of continuous graphical models.
result Our estimator maintains similar accuracy and sample complexity scalings compared to alternative approaches, while improving run-time.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
EFDA extends LDA to non-Gaussian models using exponential families.
problem Classifying non-Gaussian data with LDA's limitations.
method EFDA uses exponential families to derive closed-form estimators for natural parameters and a linear decision rule.
result EFDA matches LDA's accuracy while reducing ECE by 2-6x, proving asymptotic calibration and efficiency.
We realize the Reeb foliation of S^3 as a family of Legendrian submanifolds of the unit S^5 \subset C^3, Moreover we construct a deformation of the standard contact S^3 in S^5, via a family of contact submanifolds, into this realization.
New invariant metrics preserved under deformed Markov embeddings.
problem Preserving invariance in probability measure spaces under deformed embeddings.
method Deforming Markov embeddings while maintaining sufficiency, proving existence and uniqueness of invariant families.
result Existence and uniqueness of invariant families of tensor fields under deformed embeddings.
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 examines the stability of a specific flow on complex manifolds.
problem Stability of line bundle mean curvature flow on complex manifolds.
method Analyzes the convergence of the line bundle mean curvature flow to a deformed Hermitian-Yang-Mills metric.
result The flow converges exponentially to the deformed Hermitian-Yang-Mills metric in the C∞ sense. The paper solves a geometric problem related to K3 surfaces and complex-hyperkähler metrics.
problem Geometric meaning of small deformations of twistor cycles in K3 period domain.
method Construction of a moduli space for families of marked K3 surfaces and use of Penrose's Non-linear Graviton construction.
result Small deformations of twistor cycles induce complex-hyperkähler metrics on K3 surface families.