Reviews state-of-the-art methods in dimensionality reduction techniques, written in a clear but precise mathematical language

Presents application of the methods to the representation of expert-designed fault indicators for smart buildings, I-V curves for photovoltaic systems and acoustic signals for Li-ion batteries

Numerous appendices provide mathematical background to facilitate the understanding of the main text

1 Data science context

1.1 Data in a metric space

1.1.1 Measuring dissimilarities and similarities

1.1.2 Neighbourhood ranks

1.1.3 Embedding space notations

1.1.5 Sequence data

1.1.6 Network data

1.2 Automated tasks

1.2.1 Underlying distribution

1.2.2 Category identification

1.2.3 Data manifold analysis

1.2.4 Model learning

1.2.5 Regression

1.3.1 Human in the loop using graphic variables

1.3.2 Spatialization and Gestalt principles

1.3.3 Scatter plots

1.3.4 Parallel coordinates

1.3.5 Colour coding

1.3.6 Multiple coordinated views and visual interaction

2 Intrinsic dimensionality

2.1 Curse of dimensionality

2.1.2 Norm concentration

2.2 ID estimation

2.2.1 Covariance-based approaches

2.2.2 Fractal approaches

2.2.3 Towards local estimation

2.3 TIDLE

2.3.1 Gaussian mixture modelling

2.3.2 Test of TIDLE on a two clusters case

3 Map evaluation

3.1 Objective and practical indicators

3.1.1 Subjectivity of indicators .

3.1.2 User studies on specific tasks

3.2 Unsupervised global evaluation

3.2.1 Types of distortions

3.2.3 Reasons of distortions ubiquity

3.2.4 Scalar indicators

3.2.5 Aggregation

3.2.6 Diagrams

3.3 Class-aware indicators

3.3.1 Class separation and aggregation

3.3.3 Class cohesion and distinction

3.3.4 The case of one cluster per class

4 Map interpretation

4.1 Axes recovery

4.1.1 Linear case: biplots

4.1.2 Non-linear case

4.2 Local evaluation

4.2.2 One to many relations with focus point .

4.2.3 Many to many relations

4.3 MING

4.3.1 Uniform formulation of rank-based indicator

4.3.2 MING graphs

4.3.3 MING analysis for a toy dataset

4.3.4 Impact of MING parameters

4.3.5 Visual clutter

4.3.6 Oil flow

4.3.7 COIL-20 dataset

4.3.8 MING perspectives

5 Unsupervised DR

5.1 Spectral projections

5.1.1 Principal Component Analysis

5.1.3 Kernel methods: Isompap, KPCA, LE

5.2 Non-linear MDS

5.2.2 Non-metric MultiDimensional Scaling

5.3 Neighbourhood Embedding

5.3.1 General principle: SNE

5.3.2 Scale setting

5.3.3 Divergence choice: NeRV and JSE

5.3.4 Symmetrization

5.3.5 Solving the crowding problem: tSNE

5.3.6 Kernel choice

5.3.7 Adaptive Student Kernel Imbedding

5.4.1 Force directed graph layout: Elastic Embedding

5.4.2 Probabilistic graph layout: LargeVis

5.4.3 Topological method UMAP

5.5 Artificial neural networks

5.5.2 IVIS

6 Supervised DR

6.1 Types of supervision

6.1.1 Full supervision

6.1.2 Weak supervision

6.1.3 Semi-supervision

6.2 Parametric with class purity

6.2.1 Linear Discriminant Analysis

6.3 Metric learning

6.3.1 Mahalanobis distances

6.3.2 Riemannian metric

6.3.3 Direct distances transformation

6.3.4 Similarities learning

6.3.5 Metric learning limitations

6.4 Class adaptive scale

6.5 Classimap

6.6 CGNE

6.6.1 ClassNeRV stress

6.6.2 Flexibility of the supervision

6.6.3 Ablation study

6.6.4 Isolet 5 case study

6.6.5 Robustness to class misinformation

6.6.6 Extension to the type 2 mixture: ClassJSE

6.6.7 Extension to semi-supervision and weak-supervision

6.6.8 Extension to soft labels

7 Mapping construction

7.1 Optimization

7.1.1 Global and local optima

7.1.2 Descent algorithms

7.1.3 Initialization

7.1.5 Force-directed placement interpretation

7.2 Acceleration strategies

7.2.2 Binary search trees

7.2.3 Repulsive forces

7.2.4 Landmarks approximation

7.3 Out of sample extension

7.3.1 Applications

7.3.2 Parametric case

7.3.3 Non-parametric stress with neural network model

7.3.4 Non-parametric case

8 Applications

8.1 Smart buildings commissioning

8.1.1 System and rules

8.1.2 Mapping

8.2.1 I¿V curves

8.2.2 Comparing normalized I¿V curves

8.2.3 Colour description of the chemical compositions

8.3 Batteries

8.3.1 Case 1 1

8.3.2 Case 2 2

9 Conclusions

Nomenclature

A Some technical results

A.1 Equivalence between triangle inequality and convexity of balls for

a pseudo-norm

A.2 From Pareto to exponential distribution

A.3 Spiral and Swiss roll

B Kullback¿Leibler divergence

B.1 Generalized Kullback¿Leibler divergence

B.1.1 Perplexity with hard neighbourhoods

B.2 Link between soft and hard recall and precision

Details of calculations

C.1 General gradient of stress function

C.2 Neighbourhood embedding

C.2.2 Mixtures

C.2.3 Belonging rates

C.2.5 Scale setting by perplexity

C.2.6 Force interpretation

D Spectral projections algebra

D.1 PCA as matrix factorization and SVD resolution

D.2 Link with linear projection

D.3 Sparse expression

D.4 PCA and centering: from affine to linear

D.6 From distances to Gram matrix

D.6.1 Probabilistic interpretation and maximum likelihood

D.7 Nyström approximation

References

Index 7