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A new model for natural groupings in high-dimensional data

Mireille Boutin, Evzenie Coupkova

TL;DR

The paper addresses why high-dimensional data often exhibit clear groupings after random 1D projections even when no distinct clusters exist in the original space. It introduces a proof-of-concept density model built from a stretched hypercube (a stretched multivariate Bernoulli/Dirac-delta skeleton) that yields bimodal projections along many directions while remaining sparsely clustered in the full space. Through a two-stage construction and a generalization to noisy, affine-transformed forms, it demonstrates that random projections can reveal multiple, overlapping groupings corresponding to different Bernoulli processes, not a single clustering. Numerical experiments on synthetic image datasets and a real 8×8 digit dataset validate that 1D projections can produce multiple, sometimes hierarchical, binary groupings, highlighting the need to distinguish between groupings observed in projections and true clusters in the high-dimensional space. This framework offers a lens to interpret projection-based clustering results and informs caution in clustering interpretations and preprocessing choices.

Abstract

Clustering aims to divide a set of points into groups. The current paradigm assumes that the grouping is well-defined (unique) given the probability model from which the data is drawn. Yet, recent experiments have uncovered several high-dimensional datasets that form different binary groupings after projecting the data to randomly chosen one-dimensional subspaces. This paper describes a probability model for the data that could explain this phenomenon. It is a simple model to serve as a proof of concept for understanding the geometry of high-dimensional data. We start by building a rescaled multivariate Bernouilli model (stretched hypercube) so to create several overlapping grouping structures in the data. The size of each scaling parameter is related to the likelihood of uncovering the corresponding grouping by random 1D projection. Clusters in the original space are then created by adding noise to this cluster-free model. In high dimension, these clusters would hardly be observable given a sample set from the distribution because of the curse of dimensionality, but the binary groupings are clear. Our construction makes it clear that one needs to make a distinction between "groupings" and "clusters" in the original space. It also highlights the need to interpret any clustering found in projected data as merely one among potentially many other groupings in a dataset.

A new model for natural groupings in high-dimensional data

TL;DR

The paper addresses why high-dimensional data often exhibit clear groupings after random 1D projections even when no distinct clusters exist in the original space. It introduces a proof-of-concept density model built from a stretched hypercube (a stretched multivariate Bernoulli/Dirac-delta skeleton) that yields bimodal projections along many directions while remaining sparsely clustered in the full space. Through a two-stage construction and a generalization to noisy, affine-transformed forms, it demonstrates that random projections can reveal multiple, overlapping groupings corresponding to different Bernoulli processes, not a single clustering. Numerical experiments on synthetic image datasets and a real 8×8 digit dataset validate that 1D projections can produce multiple, sometimes hierarchical, binary groupings, highlighting the need to distinguish between groupings observed in projections and true clusters in the high-dimensional space. This framework offers a lens to interpret projection-based clustering results and informs caution in clustering interpretations and preprocessing choices.

Abstract

Clustering aims to divide a set of points into groups. The current paradigm assumes that the grouping is well-defined (unique) given the probability model from which the data is drawn. Yet, recent experiments have uncovered several high-dimensional datasets that form different binary groupings after projecting the data to randomly chosen one-dimensional subspaces. This paper describes a probability model for the data that could explain this phenomenon. It is a simple model to serve as a proof of concept for understanding the geometry of high-dimensional data. We start by building a rescaled multivariate Bernouilli model (stretched hypercube) so to create several overlapping grouping structures in the data. The size of each scaling parameter is related to the likelihood of uncovering the corresponding grouping by random 1D projection. Clusters in the original space are then created by adding noise to this cluster-free model. In high dimension, these clusters would hardly be observable given a sample set from the distribution because of the curse of dimensionality, but the binary groupings are clear. Our construction makes it clear that one needs to make a distinction between "groupings" and "clusters" in the original space. It also highlights the need to interpret any clustering found in projected data as merely one among potentially many other groupings in a dataset.

Paper Structure

This paper contains 13 sections, 2 theorems, 13 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Model Construction Overview
  3. A highly likely clusterable Model
  4. A model with no clusters but clusterable in many different ways
  5. A model with no cluster but likely clusterable
  6. Stretched hypercube model construction
  7. Bounds on the probability of a separation along the largest separation direction
  8. Note about Data Rescaling
  9. General Model
  10. Numerical Experiments
  11. Synthetic image datasets
  12. Real image dataset
  13. Conclusion and Future Work

Key Result

Lemma 1

Suppose $\{D_k\}$ is a sequence with $D_k\to\infty$ and suppose $\{a_k\}_{k=1}^{\infty}$ is a sequence such that $a_k/\sqrt{D_k}\to a_0$. As $k$ approaches infinity, the random variable $a_k\mathbf{v}\cdot\mathbf{e}$ converges in distribution to a normally distributed random variable,

Figures (11)

  • Figure 1: The probability of finding a good separation by random projection, for various values of $r$ and $D$. These curves were empirically computed by simulation, with 1,000,000 trials for each point.
  • Figure 2: The computed bound on the probability of a separation along the longest axis.
  • Figure 3: Four of the vertices of a parallelotope according to Formula \ref{['vertices']}. The background $t$ is a light grey image, the number of elements in the dictionary is $k=10$ and the scaling coefficients (edge lengths) are $a_i=(1.5)^{i-1}$, i=1,…,10.
  • Figure 4: Dictionary of images $\{D_i\}_{i=1}^{10}$ used to generate the vertices of the parallelotope illustrated in Figure \ref{['fig:vertices']}.
  • Figure 5: Histogram of image distribution after 10 different projections: $200$ vertices were generated following Formula \ref{['vertices']} in a similar manner as the ones in Figure \ref{['fig:vertices']}. The images were then projected onto the lines spanned by the dictionary images, resulting in 10 perfect (different) clusterings.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof : Outline of proof