We orthogonalize the NSS model to condition and diagnose its ill-conditioned parameters.
problem The ill-conditioning of the NSS model's design matrix.
method Exact orthogonal reparametrization via QR decomposition.
result Orthogonalization isolates the conditioning structure and maintains fit uncertainty.
Two new algorithms reduce online kernel regression's computational cost while maintaining optimal regret bounds.
problem Trade-off between regret and computational cost in online kernel regression.
method AOGD-ALD and NONS-ALD algorithms dynamically maintain nearly orthogonal basis to approximate kernel mapping and control approximate error.
result Achieves nearly optimal regret bounds at sublinear computational complexity.
We prove effective uniformization for nearly round 2-spheres and investigate their stability.
problem Proving effective uniformization for nearly round 2-spheres and their stability.
method Utilizing an identity related to the third-order differential of the conformal factor, and an isometric embedding of a round sphere into Euclidean space using an orthogonal basis of the first eigenspace of the Laplacian operator.
result We provide a simplified proof of effective uniformization and its stability.
Study shows RFRR's effectiveness with nearly orthogonal data in overparameterized settings.
problem Understanding the effectiveness of random feature regression with nearly orthogonal data.
method Investigates RFRR with nearly orthogonal deterministic unit-length input data vectors in the overparameterized regime.
result Shows high-probability non-asymptotic concentration results for RFRR's training, cross-validation, and generalization errors.
Paper solves NGCA for discrete distributions using LLL method.
problem Learning hidden non-Gaussian components in discrete distributions.
method Utilizes LLL lattice basis reduction method.
result Sample and computationally efficient algorithm for NGCA in discrete distributions.
Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, I considered the definition of orthonormal basis in Minkowsk…
Gradient descent biases towards stable rank networks for nearly-orthogonal data.
problem Understanding implicit bias in non-smooth neural networks trained by gradient descent.
method Analysis of two-layer ReLU and leaky ReLU networks trained by gradient descent on nearly-orthogonal data.
result Gradient descent biases towards networks with stable rank and uniform margin for nearly-orthogonal data.
Study and classify totally geodesic submanifolds in nearly Kaehler flag manifold.
problem Classifying totally geodesic submanifolds in nearly Kaehler flag manifold.
method Developed structural approach to nearly Kaehler flag manifold, expressed curvature tensor in terms of nearly Kaehler structure and canonical complex structures.
result Classified almost complex totally geodesic submanifolds of nearly Kaehler flag manifold and its semi-Riemannian counterpart.
The study explores various localized bases and their duals for scattered data approximation.
problem Scattered data approximation using radial basis functions.
method Examines different localized bases including Lagrange, Newton, and multiresolution versions, and their duals.
result Localized orthogonal bases, such as the Newton basis, offer symmetric preconditioners and are feasible for scattered data approximation.
New algorithm identifies best arm in semiparametric bandits with near optimal efficiency.
problem Fixed-confidence Best Arm Identification in semiparametric bandits with unknown baseline shift.
method Phase-elimination algorithm based on orthogonalized regression design.
result Nearly optimal high-probability sample-complexity upper bound established.
Introduces tunable basis functions for Gaussian processes.
problem Reduces computational complexity in Gaussian process approximations.
method Introduces tunable, local, and bounded basis functions for kernel approximation.
result Demonstrates superior performance compared to state-of-the-art methods, especially with poorly chosen kernel functions.
A machine learning method selects optimal orthonormal bases for functional data analysis.
problem Lack of formal criteria for choosing initial orthonormal bases in functional data methods.
method Proposes a machine learning algorithm to learn and place knots for efficient orthogonal spline bases (splinets).
result Demonstrates efficiency, especially for sparse functional data and complex physical systems.
Characterizes group-equivariant neural networks for three groups.
problem Understanding equivariant neural networks for orthogonal, special orthogonal, and symplectic groups.
method Characterized all possible group-equivariant neural networks for three groups.
result Found spanning sets of matrices for learnable, linear equivariant layer functions.
A new method for sparse PCA using orthogonal rotations and soft-thresholding.
problem Sparse PCA with a new basis using orthogonal rotations.
method Initialize with leading principal components, apply kimesk orthogonal rotation, and soft-threshold the rotated components. result The proposed method is more stable and explains more variance compared to alternatives.
Novel prior for orthogonal functions improves functional component estimation.
problem Improving orthogonality in functional principal component analysis.
method Sequential adaptive priors for orthogonal functions using hierarchical conditionally normal distributions.
result Proposed prior leads to nearly orthogonal posterior estimates.
Paper proposes a method to recover point configurations from noisy distance data.
problem Recovering point configurations from noisy distance data.
method Robust Euclidean Distance Geometry via Dual Basis (RoDEoDB) algorithm.
result Exact recovery guarantees for point configuration and Gram matrix under mild conditions.
Gradient-enhanced GSA uses Poincaré chaos expansions for accurate sensitivity analysis.
problem Accurately estimating Sobol' indices with limited data.
method Integrates sparse, gradient-enhanced regression with Poincaré chaos expansions for derivative-based sensitivity analysis.
result Accurately estimated Sobol' indices using limited data.
AGI modifies utility function to cooperate, conflicting with orthogonality thesis.
problem AGI modifying utility function to improve cooperation, conflicting with orthogonality thesis.
method Observing and influencing interactions, converging to similar utility functions.
result AGIs in competitive environments may optimize for favorable influence.
We propose a penalized orthogonal-components regression (POCRE) for large p small n data. Orthogonal components are sequentially constructed to maximize, upon standardization, their correlation to the response residuals. A new penalization framework, implemented via empirical Bayes thresholding, is presented to effecti…
Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, we considered the definition of orthonormal basis in Minkows…
New method combines spectral and sparse methods for Gaussian processes.
problem Efficiently fitting Gaussian processes to large datasets.
method Orthogonally decoupled variational Fourier features.
result Competitive performance on synthetic and real-world data.
A l1-norm penalized orthogonal forward regression (l1-POFR) algorithm is proposed based on the concept of leaveone- out mean square error (LOOMSE). Firstly, a new l1-norm penalized cost function is defined in the constructed orthogonal space, and each orthogonal basis is associated with an individually tunable regulari…
This paper deals with the problem of describing the vector spaces of divergence-free, natural tensors on a pseudo-Riemannian manifold that are second-order; i.e., that are defined using only second derivatives of the metric. The main result establishes isomorphisms between these spaces and certain spaces of tensors (at…
Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…
In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this…
Proposes FOAGP for efficient orthogonal effect decomposition of black-box computer experiments.
problem Challenges in sensitivity analysis of black-box computer experiments with complex, nonlinear functional outputs.
method Functional-output orthogonal additive Gaussian process (FOAGP) with conditional orthogonality constraint.
result Demonstrates effectiveness in orthogonal effect decomposition and variance decomposition through simulations and real-world application.
A method for interpreting SVMs using polynomial kernels, revealing model complexity.
problem Interpreting SVMs built with truncated orthogonal polynomial kernels.
method Orthogonal Representation Contribution Analysis (ORCA) with normalized Orthogonal Kernel Contribution (OKC) indices.
result The method reveals structural aspects of model complexity not captured by predictive accuracy.
New algorithms improve tensor CP decomposition under mild conditions.
problem Improving tensor CP decomposition with theoretical guarantees under mild incoherence conditions.
method Composite PCA and Concurrent Orthogonalization algorithms.
result Theoretical guarantees and practical superiority over existing methods.
ManifoldFlow relaxes fixed-spectrum Stiefel layers to learn a positive spectrum.
problem Fixed-spectrum Stiefel layers impose rigid spectral constraints.
method Introduces ManifoldFlow, a relaxation that learns a positive spectrum while keeping the basis on the Stiefel manifold.
result Learnable SPD spectrum improves performance in various settings.
A new method for learning manifolds efficiently using canonical basis functions.
problem Learning manifolds in high-dimensional data with efficient and distinct latent dimensions.
method Proposes a novel optimization objective to enforce a transformation matrix with a few prominent and non-degenerate basis functions.
result Demonstrates that minimizing the off-diagonal manifold metric elements ℓ1-norm results in a more efficient latent space representation. New framework models complex spatial data with basis functions and graphical vectors.
problem Modeling highly-multivariate spatial processes with varying resolutions.
method Extends graphical lasso to multivariate Gaussian processes with independent graphical vectors at different resolutions, using an orthogonal basis and fusion penalty.
result Linear complexity and parsimonious conditional independence structure in multilevel graphical model.
In a previous paper, the authors together with L. Vrancken initiated the study of 3-dimensional CR submanifolds of the nearly K\" ahler homogeneous S3×S3. As is shown by Butruille this is one of only four homogeneous 6-dimensional nearly Kähler manifolds. Besides its almost complex structu…
This paper develops efficient surrogate models for optimization of complex dynamical systems.
problem Computational expense in solving complex dynamical systems through numerical simulation.
method Combination of proper orthogonal decomposition and radial basis functions for constructing low-dimensional surrogate models.
result Surrogate models reduce computational time for optimization problems while maintaining accuracy.
Emergency Department (ED) crowding is a worldwide issue that affects the efficiency of hospital management and the quality of patient care. This occurs when the request for an admit ward-bed to receive a patient is delayed until an admission decision is made by a doctor. To reduce the overcrowding and waiting time of E…
Study on holomorphic curves in 6-sphere with boundary conditions.
problem Characterizing holomorphic curves in nearly-Kähler 6-manifolds with boundary conditions.
method Complex-geometric methods, including second variation formula for area.
result Obtained rigidity results for reflection-invariant holomorphic curves and topological lower bounds for Morse index.
Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element η\in Λ^3L\subset Λ^3V. If ηis of type (3,0)+(0,3) with respect to a skew-symmetric endomorphis…
The sum-product or belief propagation (BP) algorithm is a widely used message-passing technique for computing approximate marginals in graphical models. We introduce a new technique, called stochastic orthogonal series message-passing (SOSMP), for computing the BP fixed point in models with continuous random variables.…
Deterministic bounds for tensor singular values and vectors, differing from matrix cases.
problem Spectral learning of higher-order orthogonally decomposable tensors.
method Deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors.
result Perturbation affects each essential singular value/vector in isolation, independent of multiplicity and distance from other singular values.
Orthogonal deep models defend against black-box attacks by ensuring internal representations are nearly orthogonal.
problem Vulnerability of deep learning models to black-box adversarial attacks.
method Introduce a gradient regularization scheme to encourage deep models' internal representations to be orthogonal to another model's.
result Orthogonal deep models significantly boost robustness against transferable black-box adversarial attacks.
Sparse principal component analysis (sparse PCA) aims at finding a sparse basis to improve the interpretability over the dense basis of PCA, meanwhile the sparse basis should cover the data subspace as much as possible. In contrast to most of existing work which deal with the problem by adding some sparsity penalties o…
Enhances Gaussian processes with spherical features for better scalability and flexibility.
problem Lack of representation learning in Gaussian processes compared to deep neural networks.
method Introduces spherical inter-domain features to improve GP approximation and scalability.
result The method alleviates limitations and improves scalability compared to alternative strategies.
In this paper, we study the problem of compressed sensing using binary measurement matrices and ℓ1-norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditi…
NS-RGS improves orthogonal group synchronization with faster convergence.
problem Orthogonal group synchronization from pairwise measurements.
method Newton-Schulz iteration for Riemannian gradient optimization.
result NS-RGS achieves linear convergence and near-optimal accuracy.
We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice b…
Paper finds conditions for benign overfitting in neural networks.
problem Benign overfitting in leaky ReLU two-layer neural networks.
method Established directional convergence and classification error bounds.
result Benign overfitting occurs with high probability on mixture data.
Adapts POD basis for parametric ROMs using pGP.
problem Updating POD basis for accurate system behavior over parameter space.
method Formulates problem as supervised statistical learning, uses pGP to learn mapping between parameter space and Grassmann manifold.
result Proposes pGP for optimal estimation of POD basis parameters and quantifies uncertainty.
Archetype and archetypoid analysis can be extended to functional data. Each function is represented as a mixture of actual observations (functional archetypoids) or functional archetypes, which are a mixture of observations in the data set. Well-known Canadian temperature data are used to illustrate the analysis develo…
This paper simplifies conditional Sobol' indices calculation using PCE bases.
problem Computational inefficiency and lack of consistency in evaluating conditional Sobol' indices.
method Analytical extraction of conditional Sobol' indices via basis decomposition of PCE expansions.
result Derives closed-form expressions for conditional Sobol' indices.