The study assesses low-rank approximations in Gaussian Process regression.
problem Improving Gaussian Process regression efficiency with low-rank approximations.
method Analyzes two low-rank approximations: random Fourier features and Mercer expansion truncation.
result Bounds on the divergence and error between exact and approximate GP models.
The study assesses low-rank approximations in Gaussian Process regression.
problem Improving the efficiency of Gaussian Process regression while maintaining accuracy.
method Analyzes two low-rank approximations: random Fourier features and Mercer expansion truncation, and bounds the divergence and error between exact and approximate models.
result Theoretical bounds on the divergence and error between exact and approximate Gaussian Process models are provided.
A new PCA method for analyzing point processes.
problem Analyzing variability in replicated point processes.
method Functional Principal Component Analysis (fPCA) on cumulative mass functions.
result Established convergence and introduced principal measures.
Bayesian neural networks with Mercer priors for interpretable uncertainty quantification.
problem Uncertainty quantification in neural networks, especially for complex input-to-output mappings.
method Introducing Mercer priors for BNNs, which approximate a specified GP and are scalable.
result BNNs with Mercer priors can approximate the uncertainty of a specified GP, making them interpretable and scalable.
Survey of kernels, RKHS, and their applications in machine learning.
problem Understanding kernels and their applications in machine learning.
method Review of historical context, mathematical definitions, and practical applications of kernels.
result Comprehensive overview of kernels, RKHS, and their applications.
This paper presents a unified framework to tackle estimation problems in Digital Signal Processing (DSP) using Support Vector Machines (SVMs). The use of SVMs in estimation problems has been traditionally limited to its mere use as a black-box model. Noting such limitations in the literature, we take advantage of sever…
Paper identifies key function spaces for ReLU networks based on Fisher information.
problem Understanding the structure of Fisher information matrices in ReLU networks.
method Spectral decomposition of Fisher information matrices, focusing on the first three eigenspaces.
result The first three eigenspaces account for 97.7% of the trace of the Fisher information matrix, corresponding to spherical harmonic functions of order ≤2.
Study bounds on kernel function entropy for finite measures.
problem Investigate bounds on the ε-entropy of kernel classes.
method Sharp upper and lower bounds for p in [1, +∞] derived from eigenvalue behavior and Mercer series convergence.
result Proves tighter bounds for general kernels compared to previous work.
Develops a framework for learning nonlinear operators using Mercer kernels.
problem Learning nonlinear operators between infinite-dimensional spaces.
method Stochastic approximation framework with Mercer operator-valued kernels.
result Establishes dimension-free polynomial convergence rates for nonlinear operator learning.
Transformers are explained as infinite-dimensional kernel machines.
problem Understanding the mechanics of Transformers in AI.
method Characterized Transformers' attention mechanism as a kernel learning method on Banach spaces.
result Transformer's kernel has infinite feature dimension and can learn any binary non-Mercer reproducing kernel Banach space pair.
Lecture notes on kernel functions and Random Fourier Features.
problem Understanding and approximating kernel functions in machine learning.
method Mathematical background and proofs of concentration results.
result Estimation of error in Random Fourier Features approximation.
Kernel interpolation improved with continuous volume sampling.
problem Approximating functions from RKHS using weighted sums of kernel translates.
method Continuous volume sampling for choosing node locations.
result Proved almost optimal bounds for interpolation and quadrature under VS.
New method approximates MMD using pseudo-differential operators and singular values.
problem Approximating MMD with pseudo-differential operators and singular values.
method Corresponding pseudo-differential operators to Mercer kernels, approximating p(x,y) with its first r singular values. result The new MMD distance measures the difference of two distributions with respect to r∗ local moments, where r∗ depends on singular values decay rate. Uniform bounds for neural networks' generalization error in overparameterized settings.
problem Generalization error in overparameterized neural networks.
method Neural Tangent kernel theory and Mercer decomposition of the NT kernel in spherical harmonics.
result Uniform generalization bounds for overparameterized neural networks in RKHS.
Overlapping clustering problem is an important learning issue in which clusters are not mutually exclusive and each object may belongs simultaneously to several clusters. This paper presents a kernel based method that produces overlapping clusters on a high feature space using mercer kernel techniques to improve separa…
Devoted to multi-task learning and structured output learning, operator-valued kernels provide a flexible tool to build vector-valued functions in the context of Reproducing Kernel Hilbert Spaces. To scale up these methods, we extend the celebrated Random Fourier Feature methodology to get an approximation of operator-…
New learning rates derived for Tikhonov-regularized problems without kernel assumptions.
problem Learning rates for Tikhonov-regularized learning problems.
method Minimax adaptive rates derived using Fourier isocapacitary condition and interpolation theory.
result Derivation of minimax adaptive rates without requiring kernel assumptions.
Rescaling expansiveness proven for k*-expansive vector fields.
problem Proving rescaling expansiveness for k*-expansive vector fields.
method Introducing and exploring singular-expansive flows.
result Rescaling expansiveness established for k*-expansive vector fields.
We investigate a generic problem of learning pairwise exponential family graphical models with pairwise sufficient statistics defined by a global mapping function, e.g., Mercer kernels. This subclass of pairwise graphical models allow us to flexibly capture complex interactions among variables beyond pairwise product. …
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
Interest in multioutput kernel methods is increasing, whether under the guise of multitask learning, multisensor networks or structured output data. From the Gaussian process perspective a multioutput Mercer kernel is a covariance function over correlated output functions. One way of constructing such kernels is based …
As a robust nonlinear similarity measure in kernel space, correntropy has received increasing attention in domains of machine learning and signal processing. In particular, the maximum correntropy criterion (MCC) has recently been successfully applied in robust regression and filtering. The default kernel function in c…
A new approach to cost-sensitive multiclass classification prioritizes certain classes over others.
problem Cost-sensitive multiclass classification where some classes are more important than others.
method Apportioned margin framework that shifts the decision boundary to prioritize certain classes.
result The method improves the error rate for important classes while reducing overall error.
In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space Rd. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the sol…
Develops a martingale expansion for stochastic volatility models.
problem Approximating marginal distributions of stochastic volatility models.
method Martingale expansion framework for continuous stochastic volatility models.
result First-order perturbation expansions for small volatility-of-volatility and fast mean-reversion models.
Taylor expansions improve reinforcement learning policies.
problem Improving reinforcement learning policy optimization.
method Taylor expansion policy optimization.
result Taylor expansions enhance performance of distributed algorithms.
Analytic torsion expansions for symmetric and complex homogeneous spaces.
problem Calculating the full asymptotic expansion of analytic torsion for various spaces.
method Explicit calculation and comparison with existing results.
result Explicit full asymptotic expansions for symmetric and complex homogeneous spaces.
Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
Study of hypersurfaces with specific expansion properties.
problem Existence and properties of hypersurfaces with prescribed null expansion.
method Adapted Eichmair's Perron approach for existence.
result Topology theorem for hypersurfaces with prescribed null expansion.
A new hypergraph expansion method treats vertices and hyperedges equally, improving node classification.
problem Information loss in hypergraph expansions on either vertex or hyperedge level.
method Proposes a new hypergraph formulation named line expansion (LE) that treats vertices and hyperedges symmetrically.
result The proposed line expansion method outperforms state-of-the-art baselines on five hypergraph datasets.
Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-p…
Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
The paper calculates asymptotic expansions for specific types of oscillatory integrals.
problem Analyzing oscillatory integrals with complex phase functions.
method Using asymptotic expansions of simpler phase functions to derive results for more complex cases.
result Explicit computation of coefficients in asymptotic expansions for certain integrals.
Dropout increases the generalization of neural networks by expanding the weight space.
problem Understanding and improving the generalization of neural networks.
method Introducing weight expansion and showing that dropout leads to it.
result Dropout increases the generalization of neural networks by expanding the weight space.
This work analyzes how different layers in deep neural networks contribute to generalization error.
problem Understanding the role of each layer in deep neural networks for generalization.
method Spectral analysis, Neural Tangent Kernel, Hermite polynomials, Spherical Harmonics.
result Initial layers in deep neural networks have a larger bias towards high-frequency functions.
New derivation of knot invariants from universal invariant.
problem Deriving knot invariants from universal invariant.
method Using Hopf algebra D and a Mathematica implementation to compute ZD(K). result Derivation of large-color expansion from universal invariant.
This work explores functional expansions to handle path dependence in various fields.
problem Path dependence and infinite-dimensional problems in non-Markovian systems.
method Generalizes Wiener series and functional Taylor expansion to handle static and dynamic functionals.
result Elegant separation of functionals from future trajectories in dynamic cases.
In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus correctio…
Paper calculates third coefficient in Kaehler-Einstein metric expansion.
problem Understanding Kaehler-Einstein metrics and their epsilon functions.
method Computes the third coefficient in the TYCZ-expansion of the epsilon function.
result Discovers the vanishing of the third coefficient's significance.
We describe the first known mean-field study of landing probabilities for random walks on hypergraphs. In particular, we examine clique-expansion and tensor methods and evaluate their mean-field characteristics over a class of random hypergraph models for the purpose of seed-set community expansion. We describe paramet…
Study on heat trace expansion for thermoelastic Dirichlet-to-Neumann map.
problem Asymptotic expansion of heat trace for thermoelastic Dirichlet-to-Neumann map.
method Provided a method to obtain all coefficients of the asymptotic expansion.
result Explicitly gave the first two coefficients involving volume and total mean curvature of the boundary.
The paper proposes and proves asymptotic expansions for quantum invariants.
problem Quantum invariants and their expansions under varying metrics.
method Asymptotic expansion conjectures for relative Reshetikhin-Turaev, Turaev-Viro invariants and quantum 6j-symbols.
result Proved asymptotic expansions for special cases, showing geometric dependence on metrics.
New method models portfolios with leptokurtic risk factors using Gram-Charlier expansions.
problem Modeling portfolios with excess kurtosis.
method GC-like expansions of the hyperbolic-secant law to account for leptokurtosis.
result Portfolio distribution with risk factors modeled as GC-like expansions of the HS law.
Develops AMITE for analyzing neural network nonlinearities.
problem Addressing difficulties in verification, explainability, and security in neural network analysis.
method Analytically modified integral transform expansion (AMITE) for neural network nonlinearities.
result First to provide six mutually exclusive desired expansion properties.
We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.
Paper presents new expansions for option pricing with cash dividends.
problem No exact formula for European options with cash dividends.
method Uses Etore and Gobet's technique for piecewise lognormal process with jumps.
result Provides more robust first, second, and third-order expansions.
The validity of an approximation formula for European option prices under a general stochastic volatility model is proved in the light of the Edgeworth expansion for ergodic diffusions. The asymptotic expansion is around the Black-Scholes price and is uniform in bounded payoff func- tions. The result provides a validat…
The paper uses polyhedral expansions to capture the shape of compact metric spaces.
problem Capturing the shape of compact metric spaces using finite approximations.
method Inverse sequences of polyhedra based on finite approximations of a compact metric space.
result Proves the General Principle and computes inverse persistent homology groups.