Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of Information in the Heat Flow Clustering

Brian Weber

TL;DR

Addresses the lack of a universal cluster definition by proposing a stability-based clustering framework that uses a heat-flow analogy with a time-varying kernel to reveal persistent data structure. The method introduces a chronodendrogram and uses global stability scores $B(n)$ and local entropy-based measures $B_{s1}^{s2}$ to identify robust clusterings across time. Demonstrated on one-dimensional and two-dimensional datasets, the approach yields stable multi-scale clusters even under noise and kernel variance, with local analyses showing high persistence (for example $B_{0.0}^{0.0}$ values up to 0.795 for a cluster). The resulting automatic workflow does not require preselecting the number of clusters and is suitable for automating labeling tasks in noisy experimental data.

Abstract

Clustering methods must be tailored to the dataset it operates on, as there is no objective or universal definition of ``cluster,'' but nevertheless arbitrariness in the clustering method must be minimized. This paper develops a quantitative ``stability'' method of determining clusters, where stable or persistent clustering signals are used to indicate real structures have been identified in the underlying dataset. This method is based on modulating clustering methods by controlling a parameter -- through a thermodynamic analogy, the modulation parameter is considered ``time'' and the evolving clustering methodologies can be considered a ``heat flow.'' When the information entropy of the heat flow is stable over a wide range of times -- either globally or in the local sense which we define -- we interpret this stability as an indication that essential features of the data have been found, and create clusters on this basis.

Stability of Information in the Heat Flow Clustering

TL;DR

Addresses the lack of a universal cluster definition by proposing a stability-based clustering framework that uses a heat-flow analogy with a time-varying kernel to reveal persistent data structure. The method introduces a chronodendrogram and uses global stability scores and local entropy-based measures to identify robust clusterings across time. Demonstrated on one-dimensional and two-dimensional datasets, the approach yields stable multi-scale clusters even under noise and kernel variance, with local analyses showing high persistence (for example values up to 0.795 for a cluster). The resulting automatic workflow does not require preselecting the number of clusters and is suitable for automating labeling tasks in noisy experimental data.

Abstract

Clustering methods must be tailored to the dataset it operates on, as there is no objective or universal definition of ``cluster,'' but nevertheless arbitrariness in the clustering method must be minimized. This paper develops a quantitative ``stability'' method of determining clusters, where stable or persistent clustering signals are used to indicate real structures have been identified in the underlying dataset. This method is based on modulating clustering methods by controlling a parameter -- through a thermodynamic analogy, the modulation parameter is considered ``time'' and the evolving clustering methodologies can be considered a ``heat flow.'' When the information entropy of the heat flow is stable over a wide range of times -- either globally or in the local sense which we define -- we interpret this stability as an indication that essential features of the data have been found, and create clusters on this basis.
Paper Structure (5 sections, 21 equations, 7 figures)

This paper contains 5 sections, 21 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The chronodendrogram and information stability
  3. One dimensional clustering
  4. Higher dimensions
  5. Discussion

Figures (7)

  • Figure 1: Three time slices for the heat flow clustering of the dataset (\ref{['EqnToyModelData']}). Datapoints (circles along the lower axis) are depicted with maxima (triangles) and minima (squares) of the potentials. Cluster selections are indicated by dashed lines. Inset is the kernel choice. In the first subfigure we find five clusters, in the second three, and in the third two.
  • Figure 2: Above: The chronodendrogram for the dataset with five elements; linkage weights are indicated by thickness. Below: The two informational measures, $M_k$ and $S_k$, as a function of time.
  • Figure 3: (a) displays the three clusters with varying densities and numbers of points, (b) displays the full dataset with random noise added, and (c) shows the potential at $t=0.158$, the time indicated in Fig. \ref{['FigOneDComplicatedDendrogram']}.
  • Figure 4: Top: Chronodendrogram for the noisy dataset. Middle: The entropy and cluster number, showing stability at 3 clusters. Bottom: The local entropy scores for each of the three clusters observed at time $t=0.2206$.
  • Figure 5: Left: The data points, distributed randomly within three circles. Right: Clustering actually obtained at time $t_k=0.1833$.
  • ...and 2 more figures