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Algebraic Multigrid Acceleration for Efficient Label Spreading
Antonia van Betteray, Jonathan Klees, Miriam Schäfers, Matthias Rottmann
TL;DR
Large-scale, high-dimensional label spreading is constrained by the cost and memory demands of graph construction and solving for label estimates. AMELS combines efficient sparse graph construction with AMG solvers, and evaluations report lower computational cost while preserving accuracy and improving robustness to hyperparameter choices. This makes label spreading applicable to large-scale image datasets and supports semiautomated annotation at lower cost.
Problem
Label spreading is limited on large-scale, high-dimensional datasets by the computational and memory costs of graph construction and solving the associated label estimates.
Method
AMELS combines sparse k-NN graphs built from meaningful image embeddings with AMG methods that approximate the linear system underlying label spreading.
Results
AMELS significantly reduces label-spreading computational cost while preserving classification accuracy and improving robustness to hyperparameter choices.
Takeaways & Limitations
AMELS makes label spreading applicable to large-scale image datasets and enables semiautomated annotation to reduce annotation cost.
Abstract
from arXiv · showhide
Modern machine learning models rely on large amounts of labeled data. However, manual annotation of large-scale datasets is expensive and time-consuming. Label spreading is a semi-supervised learning technique that addresses this challenge by propagating information from a few labeled examples to a larger pool of unlabeled data. Despite its effectiveness, its application to large-scale, high-dimensional datasets is limited by computational costs and memory constraints. To address these limitations, we propose Algebraic Multigrid Acceleration for Efficient Label Spreading (AMELS), an efficient label spreading framework that improves scalability by fast construction of neighborhood graphs and the incorporation of algebraic multigrid solvers. The latter is an iterative solver that replaces the ordinary random walk iteration typically performed in label spreading. Due to the multilevel nature of algebraic multigrid solvers, AMELS spreads given label information across a graph of any size in a single multigrid cycle. We demonstrate that AMELS achieves significant runtime reductions compared to existing implementations while also being more robust to hyperparameter choices in terms of both runtime and classification accuracy. Our framework therefore enables efficient label spreading on large-scale image datasets and produces accurate labels even when only a few labeled samples are available.
I. INTRODUCTION
AMELS targets label spreading’s graph-construction and solution-computation bottlenecks for large-scale, high-dimensional image datasets. It combines efficient neighborhood graphs, image embeddings, and AMG solvers, with evaluations reporting lower computational cost, preserved accuracy, and greater hyperparameter robustness.
- Label spreading becomes difficult at scale because graph construction and matrix inversion require quadratic and cubic computational costs, respectively.The associated memory requirement for matrix inversion is quadratic in the number of samples.
- AMELS uses semantically meaningful low-dimensional image embeddings and efficient nearest-neighbor search to construct sparse k-NN graphs.The framework leverages vision-language models and state-of-the-art nearest-neighbor methods for high-dimensional image data.
- AMELS applies algebraic multigrid solvers to approximate the label-spreading linear system while capturing local and distant graph connections hierarchically.Coarse levels accelerate propagation over large distances, and AMG methods can robustly handle ill-conditioned graph Laplacians associated with diffusion strengths near 1.
- AMELS reduces label-spreading computational cost, with lower runtime and fewer iterations than existing approaches on standard image-classification benchmarks.The evaluation also reports accurate labeling of large-scale image datasets at a fraction of the annotation cost.
- AMELS preserves classification accuracy while achieving efficiency gains over existing approaches.
- AMELS is more robust to hyperparameter choices in both runtime and performance.
II. RELATED WORK
Label spreading propagates labels through similarity graphs, but conventional implementations remain costly at large scale because graph construction and solution computation are major bottlenecks.
- Label Propagation and Label Spreading: Label propagation uses hard clamping, whereas label spreading uses normalized graph-Laplacian regularization with soft clamping for improved stability and robustness.
- Iterative Solution: The conventional iterative solver computes a truncated Neumann series, with each iteration adding a term until convergence.
- Computational Challenges: Large graphs make the conventional iterative approach computationally expensive, motivating more efficient label-spreading methods.
- Efficient Graph Construction: k-NN graphs reduce edges from O(n^2) to O(kn), lowering memory and matrix-vector-operation costs, although nearest-neighbor search remains challenging at scale.
- Efficient Solvers: Efficient implementations also use iterative solvers such as conjugate gradient and graph constructions based on anchor points, against which AMELS is compared.
- Label Spreading Formulation: Label spreading predicts labels for n−a unlabeled samples using a similarity graph built over n data points.
- Label Spreading Formulation: The diffusion parameter α controls information exchange, while label spreading solves c linear systems sharing the SPD matrix A := I − αS.
2) Iterative Solvers and Krylov Subspace Methods.:
Krylov methods solve large sparse systems through low-memory iterative approximations, but their convergence depends on the system’s spectral properties and conditioning.
- Krylov Methods: Krylov subspace methods are attractive for large sparse systems because of their scalability and low memory requirements.
- Krylov Methods: Krylov methods construct iterates in expanding subspaces and use smoothing to damp high-frequency error components.
- Krylov Methods: CG minimizes error in the A-norm for SPD matrices, while GMRES minimizes the Euclidean residual norm over an affine Krylov subspace.
- Convergence Limitations: Convergence can deteriorate for ill-conditioned systems, where a small residual does not necessarily imply a small error.
- Convergence Limitations: Eigenvalues near zero can separate residual size from error size in SPD systems.
3) Algebraic Multigrid (AMG) Methods.:
AMG improves iterative linear-system solution by combining preconditioning, local smoothing, and coarse-grid corrections across a hierarchy of systems.
- AMG Preconditioning: Preconditioning replaces the original system with an equivalent one using M^−1 ≈ A^−1 to improve spectral properties and convergence.
- AMG Preconditioning: AMG approximates the preconditioner action through a hierarchy that combines local smoothing with coarse-grid corrections.
- AMG Hierarchy: AMG constructs coarser systems during setup, recursively defining operators from the fine system A0 = A.
- AMG Hierarchy: Coarsening aggregates strongly connected variables into coarser representations using prolongation and restriction operators between levels.
- AMG V-cycle: A V-cycle smooths on fine levels, restricts residuals to coarser levels, solves directly at the coarsest level, then interpolates corrections and post-smooths.
- GMRES with AMG: GMRES applies AMG preconditioning while minimizing residuals over an affine Krylov subspace.
IV. METHOD
AMELS addresses label spreading’s two key computational challenges: graph construction and determining the label spreading solution.
- AMELS is a framework designed to address computational challenges in graph construction and label spreading solution estimation.
- The framework combines efficient neighborhood-graph construction with an algebraic multigrid-based approach to approximate label spreading.
- AMELS targets scalability by addressing both stages that limit label spreading on large datasets.
A. Graph Construction
The method constructs sparse similarity graphs from embedding-space neighborhoods and uses AMG to approximate label spreading efficiently across graph scales.
- Graph Construction: AMELS builds an affinity matrix from embedding-space similarities and sparsifies it with a symmetric k-NN connectivity rule.The supplied passage describes a Gaussian-kernel affinity matrix and edges added when at least one node belongs to the other’s k-NN.
- Efficient Solution Approximation via AMG: AMG exploits hierarchical graph structure and preconditioning to approximate the label spreading system efficiently.
- Efficient Solution Approximation via AMG: When α approaches 1, the system becomes increasingly ill-conditioned and residual-based stopping criteria can underestimate true error.
- Efficient Solution Approximation via AMG: Because all class score vectors share the coefficient matrix I − αS, the AMG hierarchy is constructed once and reused across right-hand sides.
- Efficient Solution Approximation via AMG: Coarse-grid correction propagates label information globally across the graph within a single iteration.
V. NUMERICAL RESULTS
The experiments assess AMELS’s efficiency, hyperparameter dependence, and classification accuracy against several label spreading baselines under matched residual tolerances.
- Research Questions: The evaluation asks whether AMELS improves computational efficiency relative to existing baselines.
- Research Questions: The study examines how key hyperparameters influence AMELS runtime compared with other label spreading implementations.
- Research Questions: The experiments test whether greater runtime efficiency affects label spreading classification accuracy.
- Experimental Setup: AMELS uses FGMRES with an AMG preconditioner and conducts one V-cycle iteration with an initial Krylov subspace dimension t = 20.
- Baselines: Baselines include the scikit-learn power method, sparse direct solving, and CG implementations from SciPy and AmgX.
- Experimental Setup: All methods solve the linear system to the same relative residual tolerance for comparable accuracy.
3) Datasets.:
AMELS is evaluated on image datasets spanning different scales and complexities, with runtime–accuracy and scalability comparisons conducted under fixed experimental settings.
- Datasets: The evaluation uses EMNIST-Digits with 280,000 images, CIFAR-10 with 50,000 images, and Tiny ImageNet with 100,000 images.The datasets contain handwritten digits, 10-class images, and 200-class images, respectively.
- Experimental Setup: Experiments use an NVIDIA Quadro P6000 GPU, two 20-core CPUs, and 502 GB of RAM.
- Parameter Configurations: The fixed configuration uses k = 20 neighbors, α = 0.99, 100 randomly drawn labels, and residual tolerance 10^-3.
- Classification Accuracy versus Runtime: AMELS consistently achieves high classification accuracy with minimal runtime, particularly on EMNIST-digits and CIFAR-10.
2) Scalability Across Dataset Sizes.:
AMELS scales label spreading more effectively than conventional and competing iterative implementations as dataset size and hyperparameters vary. It reduces runtime and iterations while maintaining or improving classification accuracy.
- Dataset-size scalability: AMELS scales better than direct solving, the power method, and CG as dataset size increases, requiring particularly few iterations.On a 1,000,000-point augmented EMNIST-digits dataset, direct solving and the power method become costly at moderate sizes, while AMELS remains more runtime-efficient than CG.
- Diffusion strength: AMELS runtime is only minimally affected by spreading intensity α, unlike the sharply worsening power method and slower SciPy CG solver as α approaches 1.AMG can approximate weak long-range graph interactions through its multilevel structure.
- Graph connectivity: AMELS and baseline methods show limited runtime sensitivity to the neighbor count k, allowing moderately dense graphs without significant runtime increases.Increasing k raises both graph-construction and linear-system-solving costs because it produces denser matrices.
- Classification accuracy: AMELS is more robust to α in classification accuracy, with less degradation at small α and best performance near α = 1.The accuracy advantage over competing implementations is especially pronounced for small neighbor counts.
- Overall comparison: AMELS improves label-spreading efficiency without sacrificing classification performance and is more robust to hyperparameter choices.The reported gains cover both runtime and accuracy across spreading intensity and graph-connectivity studies.
APPENDIX
The appendix describes CG and GMRES as Krylov-subspace methods for solving the linear systems arising in label spreading. CG minimizes error in the A-norm, whereas GMRES minimizes the Euclidean residual norm.
- Conjugate gradient: CG generates iterates in an affine Krylov subspace and minimizes the error in the A-norm.The method applies under the stated symmetric positive-definite conditions on A.
- GMRES: GMRES generates iterates in an affine Krylov subspace by minimizing the Euclidean norm of the residual.Its Krylov basis is typically constructed with the Arnoldi iteration.
- GMRES: The GMRES iterate is obtained from a small least-squares problem involving the corresponding Hessenberg matrix.The least-squares solution determines the coefficients used for the iterate.
B. Spectral Properties of the Graph Laplacian
For spreading intensity α between 0 and 1, the graph Laplacian I − αS is positive definite. Its condition number increases as α grows, indicating worsening conditioning near large diffusion strengths.
- Eigenvalue bounds: The graph Laplacian I − αS is positive definite because S is stochastic and its eigenvalues lie between −1 and 1.The eigenvalues of I − αS lie in [1 − α, 1 + α] for α ∈ (0, 1).
- Eigenvalue bounds: The largest eigenvalue of S is 1, with 1/√n as an associated eigenvector because W1 = D1.The degree-matrix identity establishes the stochastic normalization used in the spectral argument.
- Condition number: The condition number of I − αS is determined by λ_min = 1 − α and λ_max = 1 + α.These extremal eigenvalues imply increasing ill-conditioning as α approaches 1.
- Condition number: Figure A.1 displays the condition number of I − αS as the spreading intensity α varies.The figure supports examining how conditioning changes across diffusion strengths.