New bound for neural networks with full-rank weights, independent of network width.
problem Understanding generalization of neural networks with full-rank weight matrices.
method Using Koopman operators to derive a tighter generalization bound for full-rank weight matrices.
result The bound is tighter than existing norm-based bounds when condition numbers are small.
When estimating finite mixture models, it is common to make assumptions on the mixture components, such as parametric assumptions. In this work, we make no distributional assumptions on the mixture components and instead assume that observations from the mixture model are grouped, such that observations in the same gro…
New methods improve stability of Sinkhorn algorithm in machine learning.
problem Stability of Sinkhorn semigroups in high-dimensional settings.
method Semigroup analysis based on contraction coefficients and Lyapunov-type operator-theoretic techniques.
result Unified and simplified arguments in Sinkhorn algorithm stability.
A large number of algorithms in machine learning, from principal component analysis (PCA), and its non-linear (kernel) extensions, to more recent spectral embedding and support estimation methods, rely on estimating a linear subspace from samples. In this paper we introduce a general formulation of this problem and der…
Paper develops a novel approach for optimal control using kernel methods.
problem Optimal control of nonlinear stochastic systems.
method Infinitesimal generator approach in reproducing kernel Hilbert spaces.
result Data-driven solution to optimal control problems.
In this paper, we get a Kastler-Kalau-Walze type theorem associated to nonminimal de Rham-Hodge operators on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of four dimensional compact manifolds with flat boundary.
We prove a Kastler-Kalau-Walze type theorem for the Dirac operator and the signature operator for 3,4-dimensional manifolds with boundary. As a corollary, we give two kinds of operator theoretic explanations of the gravitational action in the case of 4-dimensional manifolds with flat boundary.
We investigate in detail the connection between harmonic maps from Riemann surfaces into the unitary group $\U(n)$ and their Grassmannian models: these are families of shift-invariant subspaces of $L^2(S^1,\C^n)$. With the help of operator-theoretic methods we derive a criterion for finiteness of the uniton number whic…
In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-fo…
In this article, we introduce the notion of cycling operations of arbitrary order in Garside groups, which is a full generalization of the cycling and decycling operations. Theoretically, this notion together with other related concepts provides a context in which various definitions and arguments concerning Garside gr…
Unified framework explains why overfitting is benign in interpolating learning.
problem Understanding why overfitting is benign in highly overparameterized models.
method Spectral-transport stability framework.
result Sharp benign-overfitting criterion and explicit phase-transition rates.
Proves inequality for special 3D shapes, generalizing to non-symmetric ones.
problem Proving a mathematical inequality for specific 3D shapes.
method Operator theoretic approach combined with spherical function decomposition.
result Generalized inequality for non-symmetric bodies of revolution.
In this paper, we prove a Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional spin manifolds with boundary associated with the conformal Robertson-Walker metric. And we give two kinds of operator theoretic explanations of the gravitational action for boundary in the case of 4-dimensional manifolds with…
New framework detects directional influence in multivariate time series.
problem Detecting directional influence in multivariate time series.
method Order-constrained spectral non-invariance.
result Unique diagnostic functional for directional influence.
Kernel-based Bayesian filter for nonlinear systems using infinite-dimensional operators.
problem Modeling and predicting nonlinear dynamical systems.
method Functional Bayesian perspective, reproducing kernel Hilbert space, Gaussian kernel.
result Effective approximation and accurate results for nonlinear systems.
New framework uses elliptic operators to study projective maps.
problem Understanding projective structures on Riemannian manifolds.
method Develops two elliptic operators of second and fourth order.
result Establishes a natural correspondence between analytical and geometric properties.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. A framework uses free probability to analyze Transformer models.
problem Understanding the dynamics and complexity of Transformer-based language models.
method Formal operator-theoretic analysis using free probability theory.
result Entropy-based generalization bounds derived under freeness assumptions.
Operator-theoretic analysis of nonlinear dynamical systems has attracted much attention in a variety of engineering and scientific fields, endowed with practical estimation methods using data such as dynamic mode decomposition. In this paper, we address a lifted representation of nonlinear dynamical systems with random…
New framework allows selective removal of stale data in option calibration.
problem Inability to remove old data from calibrated option pricing models without full retraining.
method Introduces operator-theoretic Gauss-Newton framework for selective forgetting.
result Provides stability guarantees and perturbation bounds for selective data removal.
Unified framework for analyzing graph neural operators converging to graph limits.
problem Analyzing convergence of graph neural operators to graph limits.
method Develops a unified spectral framework for graph neural operators under various graphon assumptions.
result Unified framework enables direct comparison of convergence rates and tradeoffs.
Paper tackles conditional expectation estimation using compactification operators.
problem Estimating conditional expectations from product of two random variables.
method Operator theoretic approach using kernel integral operators in reproducing kernel Hilbert space.
result Solutions allow numerical approximation and convergence of data-driven implementations.
NCP uses neural networks to efficiently learn conditional distributions.
problem Learning conditional distributions for statistical inference.
method Neural Conditional Probability (NCP) approach.
result NCP efficiently handles complex probability distributions and matches leading methods.
Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue,…
Geometric formalism views optimization algorithms as discrete connections, revealing their algebraic curvature and flatness properties.
problem Understanding and optimizing the behavior of iterative optimization algorithms.
method Introducing a geometric and operator-theoretic formalism where optimization algorithms are encoded by coupled channels (drift and diffusion) whose algebraic curvature measures the deviation from ideal reversibility.
result Flat connections correspond to methods whose updates commute up to higher order, achieving minimal numerical dissipation and preserving stability.
Kernel Dynamic Mode Decomposition reconstructs dynamical systems using Laplacian kernel.
problem Reconstructing spatial-temporal dynamics of complex systems.
method Kernel Dynamic Mode Decomposition with Laplacian kernel.
result Laplacian kernel allows for the closability of Koopman operators in RKHS, enabling reconstruction.
The paper tackles data-driven optimal control of unknown nonlinear systems using RKHS.
problem Unknown nonlinear dynamics and stage cost functions.
method Embed state densities into RKHS, learn Markov operators, solve Hamilton-Jacobi-Bellman recursions.
result Solves a wide range of nonlinear control problems, including depth regulation.
Develops a new framework for temporal anchoring in deep embedding spaces.
problem Temporal anchoring in deep embedding spaces, especially drift and convergence issues.
method Operator-theoretic framework with drift maps and event-indexed blocks, proving convergence theorems and equivalence theorems.
result Proves convergence theorems and equivalence theorems for the proposed framework.
New method learns policies from offline data using operator models.
problem Limited understanding of approximation errors in offline reinforcement learning.
method Linking reinforcement learning to Hamilton-Jacobi-Bellman equation, proposing operator-theoretic algorithm.
result Global convergence of the value function and finite-sample guarantees derived.
Develops a smooth operator framework for analyzing neural network representations.
problem Analyzing the geometry of feedforward neural network representations.
method Introduces a smooth operator-theoretic approach based on diffusion Markov operators derived from feature clouds.
result Establishes a stable operator-geometric framework for tracking training, width, and perturbation stability.
The paper proposes a new method for creating interpretable models using convex optimization.
problem Creating models that are both accurate and interpretable for decision-making.
method Formulates convex learning problems that combine interpretability with accuracy, using operator theory and parametric nonlinear models.
result Shows how to create efficient surrogate models that are both accurate and interpretable.
New stability theory for Sinkhorn semigroups with explicit decay rates.
problem Stability and convergence of Sinkhorn iterations for various divergences.
method Operator-theoretic framework based on Lyapunov techniques.
result Explicit exponential decay rates for Sinkhorn iterates.
DOODL learns shared spectral dynamics across related dynamical systems.
problem Learning independent dynamical operators for each system limits discovery of shared structure.
method DOODL learns a dictionary of characteristic spectral dynamics on a manifold of related systems.
result DOODL achieves errors one to two orders of magnitude lower than independent operator estimation methods.
Develops a new deep learning formulation using Mori-Zwanzig formalism.
problem Improves deep learning by introducing a new concept of memory.
method Uses Mori-Zwanzig formalism to propagate quantities of interest through neural networks.
result Rigorously transforms deep networks into shallow ones using decay property of memory operator.
Improved spectral convergence bounds for diffusion maps on tori.
problem Weak theoretical error bounds for diffusion maps.
method Spatial Hardy space estimates, PDE spectral stability, Sinkhorn weights.
result Matched pointwise error bounds for spectral data and operator convergence.
MGDL refines deep neural networks by training grades sequentially, improving stability.
problem Training deep neural networks is challenging due to nonconvex optimization landscapes.
method MGDL trains deep networks grade by grade, freezing previously learned grades and training new ones to fit residuals.
result MGDL guarantees vanishing error in a fixed-width multigrade ReLU architecture.
New method uses weighted SDEs to improve sampling from complex distributions.
problem Sampling from highly non-log-concave distributions.
method Introduces weighted stochastic differential equations to augment diffusion-based samplers.
result Demonstrates improved exploration of nonconvex or multimodal landscapes.
The paper analyzes variance reduction in stochastic gradient Langevin dynamics.
problem Reducing the variance of stochastic gradient estimators in Langevin dynamics.
method Central limit theorem and Poisson equation analysis for variance characterization.
result Anti-symmetric perturbations can reduce the variance of non-reversible Langevin dynamics.
Paper develops a minimax optimal test for goodness-of-fit using kernel Stein discrepancy.
problem Developing a robust goodness-of-fit test for general domains.
method Kernel Stein Discrepancy (KSD) with spectral regularization and adaptive testing.
result Proposed regularized test achieves minimax optimality up to a logarithmic factor.
New methods for calculating curvature in graph theory.
problem Calculating curvature in graphs and random walks.
method Analyzing continuous and discrete-time Ollivier-Ricci curvatures of weighted graphs.
result Generalized existence and properties of Ollivier-Ricci curvature for various random walks.
Bayesian method with Gaussian process priors achieves optimal convergence rates for regression function and its derivatives.
problem Estimating the regression function and its derivatives in nonparametric regression.
method Bayesian approach with Gaussian process priors, focusing on convergence rates and plug-in property.
result Equivalence of convergence rates of posterior distributions and Bayes estimators for regression function and its derivatives.
Efficient private matrix analysis algorithms for recent variants.
problem Private analysis of recent matrix updates.
method Identifying sufficient conditions on positive semidefinite matrices.
result First efficient differentially private algorithms for various matrix analysis tasks.
This paper introduces compositional data analysis for financial ratios, improving industry-level analysis.
problem Statistical issues with standard financial ratios at industry level.
method Compositional data analysis techniques for financial ratios.
result Improved analysis of financial ratios using compositional data methods.
In this dissertation, the main goal is visualisation of financial time series. We expect that visualisation of financial time series will be a useful auxiliary for technical analysis. Firstly, we review the technical analysis methods and test our trading rules, which are built by the essential concepts of technical ana…
Paper combines geometry and time-series analysis for spatiotemporal data.
problem Multivariate time-series data from multiple sensors.
method Combines manifold learning, Riemannian geometry, and spectral analysis.
result Proposes Riemannian multi-resolution analysis (RMRA) for dynamic mode extraction.
In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis …
This paper investigates to identify the requirement and the development of machine learning-based mobile big data analysis through discussing the insights of challenges in the mobile big data (MBD). Furthermore, it reviews the state-of-the-art applications of data analysis in the area of MBD. Firstly, we introduce the …
Interactive DR framework for comparing datasets.
problem Limited flexibility in existing DR methods for comparative analysis.
method Unified linear comparative analysis (ULCA) with interactive optimization and visualization.
result ULCA and optimization algorithm improve comparative analysis efficiency and flexibility.