The report analyzes infinite-dimensional output space regression.
problem Learning theory in vector-valued RKHS regression.
method Integral operator technique with spectral theory for non-compact operators.
result Results with minimal assumptions using Chebyshev's inequality.
Reduced-rank method improves least-squares regression under output regularity.
problem Least-squares regression with infinite dimensional outputs.
method Reduced-rank method for solving least-squares problems with output regularity assumptions.
result Learning bounds and improved statistical performance compared to full-rank method.
Develops a new method for functional regression that works with non-Gaussian data.
problem Limited models for regression in function spaces with Gaussian process priors.
method Introduces Neural Operator Flows (OpFlow) for non-Gaussian function spaces.
result OpFlow enables robust and accurate uncertainty quantification for functional regression.
The paper reformulates regression in infinite dimensions as an inverse problem, showing it's equivalent to compact inverse problems.
problem Learning a linear operator between Hilbert spaces from empirical observations.
method Reformulates regression as an inverse problem, proving equivalence to compact inverse problems under specific conditions.
result The inverse problem is equivalent to compact inverse problems in terms of spectral properties and regularisation theory.
Universal algorithm learns unknown distribution for various decision-making problems.
problem Various statistical measures in contextual sequential decision-making.
method Infinite-dimensional functional regression oracle for cumulative distribution functions.
result Utility regret rate bounded by polynomial decay of eigenvalue sequence.
Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with i…
Online learning of linear operators between infinite-dimensional spaces is possible but with limitations.
problem Learning linear operators between infinite-dimensional Hilbert spaces in an online setting.
method Online learning approach for linear operators with bounded p-Schatten norm, proving impossibility for operator norm. result Separation between online learnability and uniform convergence for bounded linear operators.
Proposes a lasso variant of MARS for nonparametric regression.
problem Nonparametric regression with MARS.
method Least squares estimation over convex function combinations with a complexity constraint.
result Achieves logarithmic convergence rate in dimensionality.
We study the learnability of a class of compact operators known as Schatten--von Neumann operators. These operators between infinite-dimensional function spaces play a central role in a variety of applications in learning theory and inverse problems. We address the question of sample complexity of learning Schatten-von…
Paper analyzes error bounds for learning with vector-valued RF, improving existing analyses.
problem Learning with vector-valued random features in infinite-dimensional settings.
method Direct analysis of risk functional, avoiding random matrix theory.
result Strong consistency and minimax optimal convergence rates established.
We study the stability properties of nonlinear multi-task regression in reproducing Hilbert spaces with operator-valued kernels. Such kernels, a.k.a. multi-task kernels, are appropriate for learning prob- lems with nonscalar outputs like multi-task learning and structured out- put prediction. We show that multi-task ke…
New algorithms improve machine learning performance with explicit regret bounds.
problem Improving machine learning performance with explicit regret bounds.
method Projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance.
result Established a priori regret bounds with explicit λ-dependence.
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…
Develops methods for selecting and estimating smooth functional coefficients in high-dimensional multivariate functional data.
problem Functional predictor selection and estimation of smooth functional coefficients in high-dimensional multivariate functional data.
method Functional group-sparse regression methods in a generic Hilbert space of infinite dimension.
result Consistency of estimation and selection (oracle property) under infinite-dimensional Hilbert spaces.
Study confirms learning rates for vector-valued spectral algorithms, proving consistency.
problem Theoretical confirmation of learning rates for vector-valued spectral algorithms.
method Rigorous analysis of learning rates for various vector-valued spectral algorithms, including kernel ridge regression and gradient descent.
result Upper and lower bounds on learning rates for vector-valued spectral algorithms, proving minimax optimality in various scenarios.
Randomized algorithm solves vector-valued regression problems with low-rank operators.
problem Vector-valued regression problems involving infinite-dimensional spaces.
method Randomized Reduced Rank Regression (R4) using Gaussian sketching for optimization.
result R4 estimators are efficient and accurate, with empirical risk close to optimal.
Data splitting enhances model performance in overparametrized ridgeless regression.
problem Computational inefficiency in training models with large datasets.
method Data splitting as a regularization technique in overparametrized ridgeless regression.
result Data splitting improves statistical performance and computational complexity.
Optimal multiscale learning of linear operators
problem Statistical and computational limits of learning bounded linear operators between Sobolev spaces
method Reformulate as an infinite-dimensional matrix regression problem with heterogeneous multiscale structure
result Establish minimax rates and construct a finite-resolution blockwise least-squares estimator attaining these rates
Study high-dimensional Bayesian linear regression using variational inference.
problem High-dimensional Bayesian linear regression with product priors.
method Non-linear large deviations theory and variational inference.
result Unique optimizer in variational problem governs posterior distribution under separation condition.
Optimal rates for vector-valued regression on various norms.
problem Optimal rates for vector-valued ridge regression on continuous norms.
method Combining standard capacity assumptions with tensor product constructions of vector-valued interpolation spaces.
result Optimal rates for vector-valued ridge regression, independent of output space dimension.
Functional input neural networks approximate continuous functions on weighted spaces.
problem Approximating continuous functions on infinite-dimensional weighted spaces.
method Additive family mapping, non-linear activation, linear readouts, Stone-Weierstrass theorem.
result Global universal approximation of continuous functions on weighted spaces.
Estimates path-valued data using signature metrics and local kernels.
problem Nonparametric regression and classification for path-valued data.
method Combines signature transform and local kernel regression.
result Establishes convergence bounds and demonstrates competitive accuracy.
New approach to quantify posterior concentration rates using Wasserstein dynamics.
problem Quantifying the speed of posterior distribution concentration in Bayesian statistics.
method Combining local Lipschitz-continuity with dynamic formulation of Wasserstein distance.
result Optimal posterior contraction rates in finite and infinite-dimensional models.
New algorithms solve large-scale convex regression problems.
problem Large-scale convex regression with subgradient regularization.
method Active set type algorithm on dual QP, approximate optimization, randomized augmentation.
result Solves problems with n=10^5 and d=10 in minutes.
There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for …
Kernel ridge regression for causal inference with missing data.
problem Estimating treatment effects with missing data in selected samples.
method Kernel ridge regression estimators for nonparametric dose response curves and semiparametric treatment effects.
result Uniform consistency and finite sample rates for continuous treatment, root-n consistency for discrete treatment.
Most Finsler metrics have infinite-dimensional holonomy groups.
problem Understanding the holonomy groups of Finsler metrics.
method Analyzing the set of Finsler metrics on a manifold.
result An open dense subset of Finsler metrics have infinite-dimensional holonomy groups.
We find a deterministic equivalent for random feature regression's test error, independent of feature map dimension.
problem Understanding the generalization performance of random feature ridge regression.
method We derive a deterministic equivalent for the test error of RFRR under a concentration property, showing it can be approximated by a closed-form expression dependent on feature map eigenvalues.
result Our approximation guarantee is non-asymptotic, multiplicative, and independent of the feature map dimension, providing a tight result for the smallest number of features achieving optimal minimax error rate.
Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.
problem Efficient inference in infinite-dimensional diffusion models.
method Derives PF-ODE in infinite-dimensional function spaces.
result Reduces function evaluations while maintaining sample quality.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal rates are constructed …
Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
problem Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
method Analyzing and expanding the notion of non-commutative cross-ratios, proving their smoothness.
result Smoothness of non-commutative cross-ratios.
Proves uniqueness of embedding complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifold into infinite-dimensional projective space.
method Fine estimates of asymptotics of a balanced embedding.
result Uniqueness of embedding proven.
In high-dimensional data, structured noise caused by observed and unobserved factors affecting multiple target variables simultaneously, imposes a serious challenge for modeling, by masking the often weak signal. Therefore, (1) explaining away the structured noise in multiple-output regression is of paramount importanc…
The paper extends Cartan development to infinite dimensional Lie groups.
problem Generalizing Cartan development to infinite dimensional Lie groups.
method Generalization of Cartan development to infinite dimensional manifolds and Lie groups.
result The tangent mapping of a Cartan development is another Cartan development.
A new model approximates complex functions in parameter space.
problem Complex and nonlinear functional regression problems.
method Mapping-to-Parameter function model with B-spline free knot placement.
result Robust knot placement algorithms improve model performance.
The paper explores infinite-dimensional nonholonomic and vakonomic systems.
problem Understanding dynamics of infinite-dimensional systems with constraints.
method Visualizing and revisiting classical and new examples of nonholonomic and vakonomic systems.
result Infinite-dimensional systems exhibit both nonholonomic and vakonomic dynamics.
We construct a infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spa…
Paper investigates optimal transport map estimation in infinite-dimensional spaces.
problem Estimating optimal transport maps in infinite-dimensional spaces is challenging.
method Characterizes γ-smoothness for optimal transport maps and develops a polynomial-rate estimator. result Shows polynomial-order minimax risk for optimal transport map estimation.
Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of infinite-dimensional operator-valued kernels which are suitable for extending func…
ReTaSA tackles continuous target shift in regression problems.
problem Continuous target shift in regression settings.
method Nonparametric regularized approach to estimate importance weight function.
result The method provides theoretical justification for the estimated importance weight function.
Introduces infinite-dimensional differential geometry using Bastiani calculus.
problem Calculus breakdown in infinite-dimensional settings.
method Uses Bastiani calculus for directional derivatives.
result Develops and connects infinite-dimensional Lie groups and weak Riemannian geometry.
We extend rectified flow to infinite-dimensional Hilbert space.
problem Extending rectified flow to infinite-dimensional spaces.
method Established a rigorous functional formulation using the superposition principle for continuity equations.
result Demonstrated superior performance compared to existing models.
Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.
problem No specific problem stated; focuses on mathematical construction.
method Symplectic and Kaehler quotient construction.
result Infinite-dimensional Siegel disc constructed as symplectic and Kaehler quotient.
New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.
problem Non-trivial Kaehler-Ricci solitons in infinite dimensional complex space forms.
method Exhibited families of non trivial radial Kaehler-Ricci solitons in infinite dimensional complex space forms.
result Non-trivial Kaehler-Ricci solitons exist in infinite dimensional complex space forms, contradicting previous finite dimensional results.
This research develops approximation theory for OOMs of infinite-dimensional processes.
problem Developing an approximation theory for OOMs of infinite-dimensional processes.
method Establishing an inner product structure and proving continuity of observable operators.
result A fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space is described.
The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares ∑j(Yj−μ(tj))2+λ∫ab[μ"(t)]2dt, where the data are tj,Yj, j=1,...,n. The minimization is taken over an infinite-dimensional function space, the space of all functions wi…
We define submersions f between manifolds M and N modelled on locally convex spaces. If the range N is finite-dimensional or a Banach manifold, then these coincide with the naive notion of a submersion. We study pre-images of submanifolds under submersions and pre-images under mappings whose differentials have dense im…