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Approximating Metric Magnitude of Point Sets

Rayna Andreeva, James Ward, Primoz Skraba, Jie Gao, Rik Sarkar

TL;DR

The paper tackles the heavy computational cost of computing metric magnitude Mag$(X,d)$ for point sets, which traditionally requires inverting the $n\times n$ similarity matrix $\zeta$. It introduces scalable approximations: a convex optimization formulation, the Iterative Normalization algorithm, and greedy/subset approaches including a Discrete Center Hierarchy for fast, scale-aware estimation, along with analyses of submodularity. It also extends magnitude to practical ML applications, showing magnitude-based neural network regularization and a magnitude-driven clustering criterion, and demonstrates that longer training trajectories strengthen the correlation between magnitude-derived measures and generalization. The results show substantial speedups and scalability, enabling broader use of magnitude in ML, optimization, and data analysis, with evidence of improved generalization and clustering quality in experiments.

Abstract

Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization algorithms. But its usability is limited due to the computational cost when the dataset is large or when the computation must be carried out repeatedly (e.g. in model training). In this paper, we study the magnitude computation problem, and show efficient ways of approximating it. We show that it can be cast as a convex optimization problem, but not as a submodular optimization. The paper describes two new algorithms - an iterative approximation algorithm that converges fast and is accurate, and a subset selection method that makes the computation even faster. It has been previously proposed that magnitude of model sequences generated during stochastic gradient descent is correlated to generalization gap. Extension of this result using our more scalable algorithms shows that longer sequences in fact bear higher correlations. We also describe new applications of magnitude in machine learning - as an effective regularizer for neural network training, and as a novel clustering criterion.

Approximating Metric Magnitude of Point Sets

TL;DR

The paper tackles the heavy computational cost of computing metric magnitude Mag for point sets, which traditionally requires inverting the similarity matrix . It introduces scalable approximations: a convex optimization formulation, the Iterative Normalization algorithm, and greedy/subset approaches including a Discrete Center Hierarchy for fast, scale-aware estimation, along with analyses of submodularity. It also extends magnitude to practical ML applications, showing magnitude-based neural network regularization and a magnitude-driven clustering criterion, and demonstrates that longer training trajectories strengthen the correlation between magnitude-derived measures and generalization. The results show substantial speedups and scalability, enabling broader use of magnitude in ML, optimization, and data analysis, with evidence of improved generalization and clustering quality in experiments.

Abstract

Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization algorithms. But its usability is limited due to the computational cost when the dataset is large or when the computation must be carried out repeatedly (e.g. in model training). In this paper, we study the magnitude computation problem, and show efficient ways of approximating it. We show that it can be cast as a convex optimization problem, but not as a submodular optimization. The paper describes two new algorithms - an iterative approximation algorithm that converges fast and is accurate, and a subset selection method that makes the computation even faster. It has been previously proposed that magnitude of model sequences generated during stochastic gradient descent is correlated to generalization gap. Extension of this result using our more scalable algorithms shows that longer sequences in fact bear higher correlations. We also describe new applications of magnitude in machine learning - as an effective regularizer for neural network training, and as a novel clustering criterion.
Paper Structure (37 sections, 6 theorems, 50 equations, 13 figures, 2 tables, 4 algorithms)

This paper contains 37 sections, 6 theorems, 50 equations, 13 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Technical Background
  3. Metric Magnitude
  4. Scaling and the Magnitude Function
  5. Submodular Functions and Maximization Algorithm
  6. Approximation Algorithms
  7. Convex optimization formulation and gradient descent
  8. Iterative Normalization Algorithm
  9. Approximation via greedy subset selection
  10. Greedy set selection algorithm
  11. Discrete Center Hierarchy Algorithm
  12. Applications in Machine Learning
  13. Neural Network Regularization
  14. Clustering
  15. Experiments
  16. ...and 22 more sections

Key Result

Theorem 2

Let $X = \{te_1,-te_1,...,te_D,-te_D\}$ be a set of points in $\mathbb{R}^D$ as described above. Then in the limit:

Figures (13)

  • Figure 1: Consider the magnitude function of a 3-point space, visualized above at different scales. (a) For a small value of the scale parameter (e.g. $t=0.0001$), all the three points are very close to each other and appears as a single unit. This space has magnitude close to $1$. (b) At $t=0.01$ the distance between the two points on the right is still small and they are clustered together, and the third point is farther away. This space has Magnitude close to $2$ (c) When $t$ is large, all the three points are distinct and far apart, and Magnitude is 3.
  • Figure 2: Greedy algorithm approximates magnitude with small number of points. Plot (a) shows magnitude approximation of a Gaussian blobs, 3 centers, with 500 points. Plot (b) shows Gaussian blobs with 3 clusters and $10^4$ points.
  • Figure 3: Discrete centers are close to Greedy Maximization at a fraction of the computational cost and better than random. In plot (a) we have the Iris dataset, in plot (b) the Breast cancer dataset, in plot (c) the Wine dataset. In the remaining plots, we see subsamples of size 500 for popular image datasets: (d) MNIST, (e) CIFAR10 and (f) CIFAR100.
  • Figure 4: Iterative algorithms comparison Comparison of Inversion, Iterative Normalization and GD (a) Mean and standard deviation over 10 different runs, with 50 iterations of both iterative algorithms. (b) Number of iterations for convergence of Iterative Normalization for a randomly generated sample of 10000 points. (c) Iterative Normalization vs GD. Iterative Normalization converges fast, GD takes a longer number of iterations. 100 runs. Comparison on larger point sets in supplementary materials.
  • Figure 5: Subset selection algorithms comparison (a) Time taken for Inversion, Greedy Maximization and Discrete Centers to execute. (b) zoom on the performance of Inversion and Discrete Centers, and note that Discrete Centers performs better as the number of points increases. Comparison on larger datasets in supplementary materials.
  • ...and 8 more figures

Theorems & Definitions (18)

  • Definition 1: Weighting $w$
  • Definition 2: Metric Magnitude $\mathrm{Mag}(X,d)$
  • Example 1
  • Definition 3: scaling and $tX$
  • Definition 4: Magnitude function
  • Definition 5: Submodular Function
  • Theorem 2
  • Theorem 3
  • proof
  • Definition 6
  • ...and 8 more