Overview of ISOMAP (Isometric Mapping) and examples of algorithms and implementations

ISOMAP (Isometric Mapping) Overview

ISOMAP (Isometric Mapping) is a non-linear dimensionality reduction method, which is an algorithm for embedding high-dimensional data into low-dimensional space. It is particularly effective when the data has a ‘manifold structure’, such as a curved distribution, and was proposed by Tenenbaum, De Silva, and Langford in 2000.

The aim of ISOMAP is to obtain a low-dimensional representation of high-dimensional data and map it to a low-dimensional space while preserving the ‘geographical distance’ or ‘distance on manifold’ between the data, thus avoiding the curse of dimensionality and allowing the intrinsic structure of the data to be visualised and analysed It allows the essential structure of the data to be utilised for visualisation and analysis.

The steps of the algorithm are as follows.

1. Construction of a nearest neighbour graph: calculate the Euclidean distances between the data points and for each point find the nearest neighbour points (points at a distance within k or ε). This creates a graph ( G ), where the weights of the edges of this graph are the Euclidean distances.
2. Shortest path calculation: calculate the ‘geodesic distance’ between each point on the graph ( G ) and estimate the distance on the manifold between the data points using the shortest path algorithm (e.g. Dijkstra method, Floyd-Warchal method).
3. Low-dimensional embedding: embedding the shortest distance matrix in a low-dimensional space using Multidimensional Scaling (MDS) described in “Multidimensional Scaling (MDS)“, which is a method for placing points in a low-dimensional space based on a distance matrix.

ISOMAP has the advantages of being strong in the analysis of non-linear structures, where the data reside on a non-linear manifold and dimensionality reduction can be achieved while preserving the structure of the manifold, preserving geographical distances between the data, which enables data distributions that cannot be captured by a simple linear model to be represented, and low-dimensionality. space, which makes the data structure intuitive and easy to understand, and that the method is suitable for visualisation.

The limitations and challenges of ISOMAP include

• High computational cost: unsuitable for large datasets due to the time-consuming nature of nearest neighbour and shortest path calculations.
• Difficulty in parameter selection: the setting of the number of nearest neighbours ( k ) or the distance threshold ( epsilon ) has a significant impact on the results.
• Sensitive to noise: data with a lot of noise and outliers may not accurately capture the structure of the manifold.

A summary of the comparison between ISOMAP and other dimensionality reduction methods is given below.

ISOMAP is a fundamental method for manifold learning and non-linear dimensionality reduction and is one of the most important tools in data analysis.

Implementation example

Below is an example implementation of ISOMAP using Python. In this example, the Scikit-learn Isomap class is used to perform non-linear dimensionality reduction. The data used is the well-known ‘Swiss Roll’ data.

1. install the necessary libraries

pip install numpy matplotlib scikit-learn

2. implementation code

import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_swiss_roll
from sklearn.manifold import Isomap

# 1. generation of Swiss roll data
n_samples = 1000
noise = 0.05
X, color = make_swiss_roll(n_samples=n_samples, noise=noise, random_state=42)

# 3D plot of Swiss rolls.
fig = plt.figure(figsize=(10, 7))
ax = fig.add_subplot(111, projection='3d')
ax.scatter(X[:, 0], X[:, 1], X[:, 2], c=color, cmap=plt.cm.Spectral)
ax.set_title("Original Swiss Roll (3D)")
plt.show()

# 2. application of ISOMAP
n_neighbors = 10  # nearest neighbor number
n_components = 2  # Number of dimensions after dimensional reduction
isomap = Isomap(n_neighbors=n_neighbors, n_components=n_components)
X_reduced = isomap.fit_transform(X)

# 2D plot after dimensional reduction.
plt.figure(figsize=(10, 7))
plt.scatter(X_reduced[:, 0], X_reduced[:, 1], c=color, cmap=plt.cm.Spectral)
plt.title("ISOMAP Result (2D)")
plt.xlabel("Component 1")
plt.ylabel("Component 2")
plt.colorbar(label="Color")
plt.show()

3. code description

1. Generating Swiss roll data.
   • The make_swiss_roll function is used to create three-dimensional non-linear data.
   • Swiss roll is a typical non-linear data set with a manifold structure.
2. Data visualisation.
   • Check the structure of the Swiss roll in a 3D plot.
3. Applying Isomap.
   • Perform dimensionality reduction using the Isomap class.
   • Set n_neighbours (number of nearest neighbours) and n_components (number of dimensions after dimensionality reduction).
4. Visualisation after dimensionality reduction
   • Display the mapped results in a 2D plot in low-dimensional (2D) space.

4. run results

1. Original Swiss roll (3D plot): the 3D structure of the Swiss roll can be seen.
2. Dimensionality reduction with ISOMAP (2D plot): the Swiss rolls are expanded into a 2D plane while preserving the geographical distance of the original data.

5. adjustment of parameters

• n_neighbours: increasing the number of nearest neighbour points preserves a more extensive structure, but local structure may be lost.
• n_components: by changing the number of dimensions after dimensional reduction, it can be embedded in other dimensions, such as 1D or 3D.

Application examples

ISOMAP (Isometric Mapping) is a method dedicated to non-linear dimensionality reduction and is specifically used in the following areas.

1. image processing

Dimensionality reduction and clustering of face images:
– Abstract: Converts high-dimensional face image data (e.g. 64 x 64 pixels = 4096 dimensions) into a low-dimensional space and analyses the similarities between the images.
– Case study: face recognition system, reducing the dimensionality of image data and extracting features efficiently. Embedding is robust to differences in lighting conditions and facial expressions.
– Advantages: preserves geographical distance, so similar faces are placed closer together, which is useful for clustering and retrieval.

2. natural language processing (NLP)

Similarity analysis between documents:
– Abstract: Analyses the non-linear structure between documents by dimensionality reduction of document and word embeddings (e.g. feature vectors from Word2Vec or TF-IDF).
– Case study: 2D or 3D mapping and visualisation of textual data. Used for clustering related documents and topic modelling.
– Advantages: easier to capture non-linear relationships between high-dimensional word vectors.

3. medical data analysis

Gene expression data visualisation:.
– Abstract: dimensionality reduction of gene expression data (thousands of dimensions) to analyse disease-specific patterns and clusters.
– Case study: analysis of gene expression data to identify cancer subtypes. Support for early diagnosis of Parkinson’s disease and Alzheimer’s disease.
– Advantages: visualisation of non-linear manifold structures hidden in high-dimensional data, useful for disease grouping.

4. robotics.

Dimensionality reduction of the robot’s motion space:
– Abstract: Efficiently analyses movement patterns by mapping robot joint angles and sensor values (high-dimensional data) into low-dimensional space.
– Case study: analysis of robot arm movement optimisation and obstacle avoidance. Visualisation of movement trajectories of autonomous mobile robots.
– Advantages: simplifies movement data while preserving the manifold structure and is useful for building control models.

5. biometrics

Fingerprint and iris authentication data analysis:
– Abstract: Dimensionality reduction of high-dimensional biometric data to improve authentication accuracy.
– Case study: feature extraction and mapping of fingerprint patterns. Clustering of iris scan data and personal identification.
– Advantages: reduced influence of noise and improved computational efficiency.

6. geographic information systems (GIS)

Dimension reduction of topographical data:
– Abstract: analysis of topographical and remote sensing data by converting them into low-dimensional space.
– Case study: satellite image data analysis for urban planning and forest monitoring. Identification of disaster risk areas.
– Advantages: efficient analysis of topography and land use patterns while maintaining geographical distances.

7. financial data analysis

Portfolio dimension reduction:
– Abstract: visualisation of risk-return relationships by representing the characteristics of financial instruments in high-dimensional space and embedding them in a lower dimension with ISOMAP.
– Case study: analysis of similarities between instruments in asset diversification (risk diversification). Credit scoring and anomaly detection.
– Advantages: better capture of correlations between diverse financial instruments and enhanced risk management.

8. speech processing

Dimensionality reduction of speech features:
– Abstract: lowers the dimensionality of high-dimensional features such as Mel-frequency cepstrum coefficients (MFCC) in speech data.
– Case study: speech clustering in speech recognition systems. Emotion recognition and speaker recognition.
– Advantages: Streamlining of features while preserving the non-linear relationships in speech.

9. bioinformatics

Protein structure analysis:
– Abstract: mapping of protein structural data (e.g. α-helix and β-sheet arrangements) in low-dimensional space.
– Case study: similarity comparisons between proteins. Structural prediction modelling.
– Advantages: visualisation of data while maintaining structural similarity.

ISOMAP is specialised in dimensionality reduction and visualisation of multidimensional data with non-linear structure and is effective in various fields. Depending on the characteristics and objectives of the data to be applied, it can be compared and selected with other dimensionality reduction methods (e.g. PCA, t-SNE, UMAP) for even more effective data analysis.

reference book

The following is a list of reference books that can be used to learn about the theory, implementation and applications related to Isometric Mapping (ISOMAP) and dimensionality reduction.

1. fundamentals and applications of dimensionality reduction
Pattern Recognition and Machine Learning
– Author: Christopher M. Bishop
– Publisher: Springer
– Abstract: Provides an in-depth look at the basic theory of machine learning, including dimension reduction, and discusses ISOMAP and other dimensionality reduction methods (e.g. PCA, LLE, t-SNE).
– Recommendations: useful for beginners and intermediate users, with plenty of theoretical background and application examples.

2. data visualisation and dimensionality reduction
Nonlinear Dimensionality Reduction
– Author(s): John A. Lee, Michel Verleysen
– Publisher: Springer
– Abstract: A specialist book dedicated to non-linear dimensionality reduction methods (e.g. ISOMAP, LLE, t-SNE). Covers the theoretical background and practical applications of manifold learning.
– Recommendations: detailed description of ISOMAP from theory to implementation, ideal for those wishing to deepen their specialist understanding.

3. machine learning in general and dimensionality reduction
Machine Learning: A Probabilistic Perspective
– Author: Kevin P. Murphy
– Publisher: MIT Press
– Abstract: Provides a comprehensive overview of machine learning algorithms in general. Dimensionality reduction methods are also included.
– Recommendations: Practical coverage of the fundamentals and applications of machine learning techniques, including dimensionality reduction.

4. practical dimension reduction and visualisation
“Python Data Science Handbook: Essential Tools for Working with Data–
– Author: Jake VanderPlas
– Publisher: O’Reilly Media
– Abstract: A practical book on data science using Python. It is rich in examples of implementations of dimensionality reduction methods (e.g. PCA, t-SNE, ISOMAP, etc.).
– Recommendations: implementation-oriented, with ISOMAP code examples to learn from. Can be used immediately for actual data analysis.

5. manifold learning and applications
Manifold Learning Theory and Applications
– Author(s): Yunqian Ma, Yun Fu
– Publisher: CRC Press
– Abstract: Covers the theory of manifold learning, details of methods such as ISOMAP and application examples.
– SUGGESTIONS: specialist content, particularly suitable for those interested in manifold learning.

6. applications and data analysis
Data Mining: concepts and techniques
– Author(s): Jiawei Han, Micheline Kamber, Jian Pei
– Publisher: Morgan Kaufmann
– Abstract: Describes methods for data mining in general. Includes applications of dimensionality reduction and ISOMAP.
– Recommendations: rich in application examples, allows the reader to learn ISOMAP from a data mining perspective.

7. practice based on Scikit-learn
Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow
– Author: Aurélien Géron
– Publisher: O’Reilly Media
– Abstract: A practical book on machine learning with Scikit-learn and TensorFlow. It also details examples of implementing dimensionality reduction methods.
– Recommendations: ideal for beginners who want to learn how to implement dimensionality reduction, including ISOMAP.