Many problems in machine learning and statistics can be formulated as (generalized) eigenproblems. In terms of the associated optimization problem, computing linear eigenvectors amounts to finding critical points of a quadratic function subject to quadratic constraints. In this paper we show that a certain class of con…
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20 results for “Eigenproblem”
SPONGE clusters signed graphs by solving a generalized eigenproblem.
problem Clustering signed graphs where affinity can be positive or negative.
method Generalized eigenproblem inspired by social balance theory.
result The method provides theoretical guarantees and outperforms existing methods.
It has been recently shown that a large class of balanced graph cuts allows for an exact relaxation into a nonlinear eigenproblem. We review briefly some of these results and propose a family of algorithms to compute nonlinear eigenvectors which encompasses previous work as special cases. We provide a detailed analysis…
Researchers extend M-eigenvalues to Riemann curvature tensor, finding real values and invariants.
problem Extending M-eigenvalues to Riemann curvature tensor in general relativity.
method Introduced M-eigenproblem definition from minimization of associated function, proved existence and real values of M-eigenvalues.
result M-eigenvalues of Riemann curvature tensor are real and invariants, related to curvature invariants.
A new clustering method for simplicial complexes using homology.
problem Clustering simplicial complexes efficiently and accurately.
method Inspired by graph spectral clustering, the method uses sparse eigenproblems.
result Produces clusters sensitive to simplicial complex homology.
New moving average adapts weight dynamically based on polynomial and wavefunction.
problem Lagging traditional moving averages in adjusting to changes in data.
method Develops a moving average with weight as a polynomial of a wavefunction from an eigenproblem.
result Immediate 'switch' without lag, adapting to changes in data.
New method scales features for better clustering.
problem Irregular features disrupt classification.
method Spectral clustering with modified feature scales.
result Outperforms existing methods in experiments.
Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
Lower bounds for Dirac eigenvalues on manifolds with boundary.
problem Finding lower bounds for eigenvalues of the Dirac operator on manifolds with boundary.
method Using the relative Yamabe constant to derive a conformal lower bound.
result Equality in the lower bound holds if and only if the manifold is a hemisphere and the eigenfunction is a Killing spinor.
The execution flow drives market dynamics, validated on real data.
problem Understanding the fundamental driving force of market dynamics.
method Developed a numerical framework using the Radon-Nikodym derivative to calculate execution flow and determined thresholds and characteristic time scales.
result Execution flow is the fundamental driving force of market dynamics.
We discuss the portfolio optimization problem with the obligatory deposits constraint. Recently it has been shown that as a consequence of this nonlinear constraint, the solution consists of an exponentially large number of optimal portfolios, completely different from each other, and extremely sensitive to any changes…
Additive principal components (APCs for short) are a nonlinear generalization of linear principal components. We focus on smallest APCs to describe additive nonlinear constraints that are approximately satisfied by the data. Thus APCs fit data with implicit equations that treat the variables symmetrically, as opposed t…
New method improves subspace iteration for eigenvectors in machine learning.
problem Computing eigenvectors for large-scale problems in machine learning.
method Subspace iteration with norm convergence analysis.
result Deterministic bounds and practical stopping criterion for improved performance.
Develops numerical method for joint probability estimation from random processes.
problem Estimating joint probability distribution from random processes.
method Formulates and solves generalized eigenvalue problems for two random processes, then uses projections of eigenvectors to build a joint distribution estimator.
result Develops a new type of probability correlation, , for random processes.
Two adaptive kernel selection methods improve the accuracy of Kernelized Diffusion Maps.
problem Selecting an appropriate kernel for Kernelized Diffusion Maps.
method Two complementary approaches: variational outer loop and unsupervised cross-validation.
result Both methods improve the quality and stability of the recovered eigenfunctions.
New asymmetric kernel methods improve feature learning.
problem Improving feature learning with asymmetric kernels.
method Coupled covariance eigenproblem and Nyström method.
result Empirical evaluations show benefits of KSVD.
PCA++ improves robustness to background noise in contrastive learning.
problem Recovering shared signal subspaces from positive pairs in high-dimensional data with structured background noise.
method PCA++ uses hard uniformity-constrained contrastive learning to enforce identity covariance on projected features.
result PCA++ outperforms standard PCA and alignment-only PCA+ in simulations and real-world datasets.
A novel approach to interpolation, classification, and clustering using Radon-Nikodym derivatives.
problem Interpolation, classification, and clustering problems in data analysis.
method Radon-Nikodym approach with Lebesgue quadrature for optimal clustering.
result The approach changes both probabilities and the probability space with new observations.
Unified framework for scale-invariant representation learning using MAPCA.
problem Learning invariant representations in data.
method Metric-Aware Principal Component Analysis (MAPCA) based on generalized eigenproblem.
result MAPCA provides a unified geometric language for various self-supervised learning objectives.
Regularized spectral methods improve clustering in signed graphs, especially for sparse data.
problem Clustering signed graphs with positive and negative edges.
method Developed regularized versions of SPONGE and Signed Laplacian methods for clustering signed graphs, especially for sparse data.
result Theoretical guarantees and empirical performance improvements for clustering signed graphs, especially in sparse regimes.