This paper provides a dictionary of closed-form kernel mean embeddings.
problem Challenges in deriving closed-form kernel mean embeddings.
method Comprehensive dictionary and practical tools for deriving new embeddings.
result Provides a Python library with minimal implementations of embeddings.
This note optimizes distributions using kernel mean embeddings with a new parameterization.
problem Optimizing distributions using kernel mean embeddings is challenging due to the difficulty of characterizing probability distribution vectors.
method Proposes a new parameterization of positive functions using kernel sums-of-squares to fit distributions in the MMD geometry.
result Distributions with kernel sum-of-squares densities are dense in the MMD geometry, allowing optimization in the finite-sample setting.
Bayesian deconditional embeddings solve complex function recovery.
problem Recovering original functions from conditional mean observations.
method Formalizes deconditional kernel mean embeddings as Bayesian inference, connects to task-transformed Gaussian processes.
result Establishes deconditional kernel means as posterior predictive mean, providing Bayesian interpretations and uncertainty.
We offer a new, rigorous approach to conditional mean embeddings without operator constraints.
problem Lack of rigorous, operator-free approach to conditional mean embeddings.
method Measure-theoretic approach to conditional mean embeddings.
result Natural regression interpretation and universal consistency of empirical estimates.
Kernel mean estimation for functions of random variables provides consistent estimators.
problem Estimating functions of random variables using kernel mean embeddings.
method Kernel mean embeddings for continuous functions of random variables.
result Consistent estimators of mean embeddings of functions of random variables.
MONK improves mean embedding estimation by reducing outlier sensitivity.
problem Outliers severely affect classical mean embedding estimators.
method Uses median-of-means principle to design robust estimators.
result Optimal sub-Gaussian deviation bounds with resistance to outliers.
Efficiently approximates kernel mean embeddings using Nyström method.
problem Computational cost of kernel mean embeddings in large-scale settings.
method Nyström method for approximating a small random subset of the dataset.
result Upper bound on approximation error with sufficient subsample size conditions.
Bayesian model improves kernel learning for probability measures.
problem Challenges in kernel learning, especially for characteristic kernels.
method Bayesian model combining Gaussian process prior and conjugate likelihood.
result Closed form posterior over mean embedding with uncertainty.
New KQEs improve probability metrics without mean function constraints.
problem Improving probability metrics without relying on mean function representations.
method Kernel quantile embeddings (KQEs) to construct new distances.
result KQEs offer a competitive alternative to MMD with near-linear cost.
Authors construct hypertori with constant negative mean curvature in a sphere.
problem Constructing constant mean curvature hypertori in a sphere.
method Constructing two different constant mean curvature (2n−1)-dimensional hypertori in a 2n-dimensional sphere. result Two different constant mean curvature (2n−1)-dimensional hypertori with negative mean curvature in a 2n-dimensional sphere. Develops a rigorous theory for conditional mean embeddings.
problem Efficient conditioning of probability distributions in RKHSs.
method Mathematical theory for both centred and uncentred covariance operators.
result Significantly weakens conditions for applicability of CMEs.
We introduce the moduli space of spectral curves of constant mean curvature (\cmc\hspace{-5pt}) cylinders of finite type in the round unit 3-sphere. The subset of spectral curves of mean-convex Alexandrov embedded cylinders is explicitly determined using a combination of integrable systems and geometric analysis techni…
We consider mean-convex Alexandrov embedded surfaces in the round unit 3-sphere, and show under which conditions it is possible to continuously deform these preserving mean-convex Alexandrov embeddedness.
New algorithm for RL using mean embeddings of return distributions.
problem Improving reinforcement learning algorithms for dynamic programming.
method Mean embeddings of return distributions, novel algorithms for RL.
result Asymptotic convergence and improved performance in deep RL.
New learning rates for embeddings in RKHSs, even when the target is not Hilbert-Schmidt.
problem Applying conditional mean embeddings to complex ML/RL settings with infinite-dimensional RKHSs.
method Developed novel learning rates using interpolation theory for RKHSs, derived explicit adaptive rates for sample estimator.
result Achieved uniform convergence rates in the output RKHS for certain parameter regimes.
Constructs special surfaces in hyperbolic space with constant mean curvature.
problem Creating surfaces with specific geometric properties in hyperbolic space.
method Using the DPW method to construct surfaces with constant mean curvature.
result Constructs surfaces with constant mean curvature in hyperbolic space.
Diffusion means converge to extrinsic means for long times on spheres.
problem Understanding the long-time behavior of diffusion means on manifolds.
method Introduced diffusion means as a parameterized family of location statistics on manifolds, and analyzed their convergence to extrinsic means for long times.
result For real projective spaces and connected compact symmetric spaces, the long-time limit of diffusion means is conjectured to be the extrinsic mean in the isometric embedding.
We prove a chord arc bound for disks embedded in R3 with constant mean curvature. This bound does not depend on the value of the mean curvature. It is inspired by and generalizes the work of Colding and Minicozzi in [2] for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamen…
We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.
Study on embedding sphere subbundles with prescribed mean curvature in Riemannian vector bundles.
problem Embedding sphere subbundles with prescribed mean curvature in Riemannian vector bundles.
method Analyzes embeddings of sphere subbundles into Riemannian vector bundles with a prescribed mean curvature.
result Conditions for the existence of such embeddings are derived.
New surfaces in 3D space with constant curvature.
problem Constructing surfaces with constant curvature in a specific space.
method Created surfaces with constant mean curvature H in H^2xR.
result Successfully constructed non-properly embedded surfaces.
Novel approach to OT using kernel mean embeddings controls overfitting and achieves dimension-free sample complexity.
problem Consistently estimate optimal transport plan from samples.
method Pose OT as learning kernel mean embedding, employ MMD regularization.
result ε-optimal recovery of transport plan and map with dimension-free sample complexity.
New compact mean convex hypersurfaces found for positive λ.
problem Finding compact embedded hypersurfaces for positive λ.
method Constructing compact mean convex hypersurfaces diffeomorphic to spheres.
result No compact convex embedded λ-hypersurfaces except a round sphere for λ > 0.
Paper improves MMD estimation for analytical mean embeddings.
problem Improving MMD estimation for distributions with analytical mean embeddings.
method Proposes a tighter concentration result for MMD estimation under semi-explicit settings and extends to unbounded kernels.
result Demonstrates efficiency in real-world applications like index replication and calibration.
The study finds conditions for embedding sphere subbundles with specific mean curvatures.
problem Embedding sphere subbundles with prescribed mean curvatures in Riemannian vector bundles.
method Analyzes embeddings of sphere subbundles into Riemannian vector bundles with prescribed mean curvatures.
result Conditions for embedding sphere subbundles with specific mean curvatures are identified.
Ancient pancakes solve mean curvature flow problem.
problem Mean curvature flow problem
method Constructing an embedded ancient solution as a stack of pancakes
result Embedded ancient solution to mean curvature flow
Estimates curvature and radius for constant mean curvature surfaces.
problem Understanding the geometry of surfaces with constant mean curvature.
method Intrinsic curvature and radius estimates for compact disks, applied to complete surfaces.
result Global geometry of constant mean curvature surfaces studied.
New recursive algorithm estimates conditional kernel mean embeddings in Hilbert space.
problem Estimating conditional distributions in RKHS for supervised learning.
method Recursive algorithm in L2 space for conditional kernel mean map. result Strong L2 consistency of recursive estimator proved. A new method estimates multi-dimensional value distributions using Hilbert space embeddings.
problem Estimating value distributions in complex, multi-dimensional reinforcement learning settings.
method Hilbert space mappings and kernel mean embeddings to estimate the kernel mean embedding of multi-dimensional value distributions.
result Uniform convergence guarantees and robust off-policy evaluation demonstrated in simulations.
A new method learns hyperparameters for conditional kernel mean embeddings using Rademacher complexity bounds.
problem Hyperparameter tuning for conditional kernel mean embeddings is challenging and computationally expensive.
method Proposes a hyperparameter learning framework based on Rademacher complexity bounds for scalable kernel hyperparameter tuning.
result Demonstrates improved performance over competing methods and can incorporate deep neural network weights.
Dynamic embeddings capture evolving word meanings over time.
problem Capturing how word meanings change over time in historical texts.
method Developed dynamic Bernoulli embeddings based on exponential family embeddings.
result Dynamic embeddings provide better fits and reveal interesting language change patterns.
We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.
The paper shows translating solitons in R4 have SO(2) symmetry.
problem Understanding the symmetry of translating solitons in R4. method Analyzing the blow-up limits of embedded, mean convex mean curvature flow.
result Translating solitons in R4 have SO(2) symmetry. Paper learns identity-sensitive word embeddings from text corpora.
problem Lack of context-aware word embeddings.
method Constructs a heterogeneous network of words and identities, then embeds into a low-dimensional space.
result Identity-sensitive word embeddings capture different meanings of words.
Word2vec improved but lacks multi-meaning words; ConEc creates new embeddings.
problem Lack of meaningful embeddings for words with multiple meanings and OOV words.
method Context encoders (ConEc) extend word2vec by multiplying embeddings with context vectors.
result ConEc creates embeddings for OOV words and words with multiple meanings based on local contexts.
J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{\mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding.
Smooths out complex shapes into simpler forms.
problem Transforming complex shapes into simpler, smooth forms.
method Perturbing minimizing hypercones and viscosity mean convex cones into smooth, properly embedded hypersurfaces.
result Properly embedded smooth minimizing hypersurfaces and self-expanders are achieved.
New algorithm tackles multi-agent reinforcement learning issues.
problem Multi-agent reinforcement learning suffers from the curse of many agents.
method Proposes MF-FQI algorithm based on mean embeddings of distributions.
result Establishes a non-asymptotic analysis for MF-FQI algorithm.
New method constructs surfaces with constant mean curvature.
problem Constructing surfaces with specific curvature properties.
method Combining DPW method and opening nodes.
result Embedded surfaces with positive constant mean curvature.
We show that one-sided Alexandrov embedded constant mean curvature cylinders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are rotational.
In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; the…
Word embeddings improve classification of research stages.
problem Classifying research stages using conventional methods.
method Used pre-trained and custom word embeddings for classification.
result Custom embeddings outperform general embeddings for research classification.
The paper studies how surfaces move by mean curvature flow and what happens at singular points.
problem Understanding the behavior of surfaces moving by mean curvature flow at singular points.
method Proves that tangent flows at singular times are smooth shrinkers, with a new local Gauss-Bonnet formula.
result Smooth shrinkers without branch points if the initial surface is embedded in 3-manifold.
Theorem proves congruence for compact submanifolds in a sphere.
problem Understanding submanifolds in a sphere with specific embedding properties.
method Used a Reilly type formula for space forms.
result Proved a congruence theorem for compact embedded hypersurfaces.
We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.
KELFI improves inference accuracy in likelihood-free settings with limited simulations.
problem Intractable likelihood evaluations in likelihood-free inference.
method Kernel embedding likelihood-free inference (KELFI) learns model hyperparameters to balance accuracy and efficiency.
result Improved accuracy and efficiency on challenging inference problems in ecology.
The paper studies how certain surfaces shrink to points under a fractional mean curvature flow, revealing singularities.
problem Understanding the behavior of surfaces evolving by fractional mean curvature flow, especially near singularities.
method Fractional mean curvature flow, focusing on embedded surfaces in Rn. result For n>2, there exists an embedded surface that develops a singularity before shrinking to a point. Maximal mean curvature limits surface area in higher dimensions.
problem Maximal mean curvature limits surface area in higher dimensions.
method Smooth embeddings of the ball with arbitrary small volume for given maximal mean curvature.
result For a given maximal mean curvature, smooth embeddings of the ball with arbitrary small volume are provided.