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What is Dimensionality Reduction?

Dimensionality reduction is a technique used in Data Science and Machine Learning to simplify complex datasets. It involves reducing the number of features or variables in a dataset while retaining as much of the relevant information as possible. Dimension reduction is essential in solving many real-world problems, including image recognition, natural language processing, and recommendation systems.

In this article, we will discuss what dimension reduction is, why it is important, and the various techniques used in dimensionality reduction.

What is Dimensionality Reduction?

Dimension reduction is the process of reducing the number of features or variables in a dataset while retaining the essential information. In other words, it is the process of converting a large dataset into a smaller one without losing significant information.

Why do you need the Dimensionality Reduction?

Various algorithms, such as linear regression, have problems if the data set has variables that are correlated with each other, i.e. depend on each other. To avoid this problem, it can make sense to remove the variables from the data set that correlate with another variable. At the same time, however, the data should not lose its original information content or should retain as much information as possible.

Another application we have is in cluster analysis, such as k-means clustering, where we need to define the number of clusters in advance. Reducing the dimensionality of the data set helps us to get a first impression of the information and to be able to estimate, for example, which are the most important variables and how many clusters the data set could have. For example, if we manage to reduce the data set to three dimensions, we can visualize the data points in a diagram. From this, it may then already be possible to read off the number of clusters.

Das Bild zeigt den Ablauf beim k-Means Clustering.
k-Means Clustering Algorithm

In addition, large data sets with many variables also offer the danger that the model overfits. Simply explained, this means that the model adapts too much to the training data during training and thus only delivers poor results for new, unseen data. Therefore, for neural networks, for example, it can make sense to first train the model with the most important variables and then add new variables bit by bit, which may further increase the performance of the model without overfitting.

What are the advantages of Dimension Reduction?

Dimensionality reduction offers several advantages, including:

  1. Reduced computational complexity: Dimension reduction simplifies complex datasets, making them easier to analyze and reducing computational complexity.
  2. Better accuracy: Reducing the number of features can lead to better accuracy in Machine Learning models, as it reduces the risk of overfitting.
  3. Improved interpretability: Simplified datasets are more interpretable, making it easier to understand the relationship between variables and make informed decisions.
  4. Reduced storage requirements: Dimension reduction reduces the size of datasets, reducing storage requirements and enabling faster processing.
  5. Better visualization: Simplified datasets are easier to visualize, enabling better data exploration and analysis.

What are the techniques used in Dimensionality Reduction?

There are two main techniques used in dimensionality reduction: feature selection and feature extraction.

  1. Feature Selection: Feature selection is the process of selecting a subset of the most relevant features from a dataset. This technique involves evaluating each feature’s importance and selecting only the most important ones for analysis. Feature selection can be done manually or using automated techniques such as correlation analysis, mutual information, and regression models.
  2. Feature Extraction: Feature extraction is the process of transforming a dataset into a lower-dimensional space. This technique involves creating new features that capture the essential information in the original dataset. There are two types of feature extraction techniques: linear and nonlinear. Feature extraction also involves creating new features which are a combination of the previous ones. In most cases, this also leads to the features being harder to interpret than before.

Linear techniques involve projecting the data onto a lower-dimensional space using linear transformations such as principal component analysis (PCA) and singular value decomposition (SVD). Nonlinear techniques involve projecting the data onto a lower-dimensional space using nonlinear transformations such as t-distributed stochastic neighbor embedding (t-SNE) and autoencoders.

Feature Selection vs. Feature Extraction

Feature selection and feature extraction are two common techniques used in machine learning to reduce the dimensionality of datasets by selecting or creating a subset of the most relevant features.

Both feature selection and feature extraction have their advantages and disadvantages. Feature selection is generally faster and easier to implement than feature extraction, and can often result in a more interpretable model. However, it may not always be effective in capturing complex nonlinear relationships between the features and the target variable. Feature extraction, on the other hand, can be more effective in capturing these relationships, but may result in a less interpretable model and can be computationally expensive.

In practice, the choice between feature selection and feature extraction depends on the specific problem at hand and the nature of the data. It is often a good idea to try both techniques and compare the results to see which approach works best for the particular problem.

How does the Principal Component Analysis work?

The core idea of Principal Component Analysis is that possibly several variables in a data set measure the same thing, i.e. are correlated. Thus, the different dimensions can be combined into fewer so-called principal components without compromising the validity of the data set. Body size, for example, has a high correlation with shoe size, since in many cases tall people also have a larger shoe size and vice versa. So if we remove shoe size as a variable from our data set, the information content does not decrease.

In statistics, the information content of a data set is determined by the variance. This indicates how far the data points are from the center. The smaller the variance, the closer the data points are to their mean value and vice versa. A small variance thus indicates that the mean value is already a good estimate for the data set.

In the first step, PCA tries to find the variable that maximizes the explained variance of the data set. Then, step by step, more variables are added to explain the remaining part of the variance, because the variance, i.e. the deviation from the mean, contains the most information. This should be preserved if we want to train a model based on it.

In the first step, Principal Component Analysis tries to find a line that minimizes the distance between it and the data points as much as possible. This procedure is the same as in linear regression. The line is therefore a summed combination of all individual features of the data set and forms the first principal component.

Das Bild zeigt ein zweidimensionales Diagramm mit verschiedenen orangenen Punkten und einer blauen Linie, die durch die Punktewolke verläuft. Dies ist die Gerade der Linearen Regression.
First Principal Component | Source: Author

An attempt is then made to create a second line that is orthogonal, i.e. perpendicular, to the first principal component and again minimizes the distance to the data points. The lines must be orthogonal to each other because the principal components should not be correlated with each other and because a perpendicular line is also very likely to explain variance that is not contained in the first component.

How does t-distributed stochastic neighbor embedding work?

The approach of tSNE is relatively simple in theory. Assuming we have a high-dimensional data set, we define a distance measure between the data points. This can be known distance measures, but it can also be custom functions that are defined. In many cases, this involves normalizing the distance so that it is the differences in the data points that matter, rather than the actual distance in space.

The tSNE algorithm then tries to find a low-dimensional space in which these distances are preserved as well as possible. For this purpose, it uses the so-called gradient method to improve the results step by step.

What are the applications of Dimensionality Reduction?

Dimensionality reduction has several applications in Data Science and Machine Learning, including:

  1. Image recognition: Images can have high dimensions making them too expensive to process due to their storage. By reducing the dimensions the processing is faster and cheaper
  2. Natural language processing: Large texts also have many dimensions and probably repetitive information that can be compressed by reducing the number of dimensions at hand.
  3. Recommendation systems: Dimensionality reduction is used in recommendation systems to reduce the dimensionality of high-dimensional user-item interaction datasets.
  4. Signal processing: Dimensionality reduction is used in signal processing to reduce the dimensionality of high-dimensional signal datasets.

This is what you should take with you

  • Dimensionality reduction is the process of reducing the number of features or variables in a dataset while retaining the essential information.
  • It is essential in solving many real-world problems, including image recognition, natural language processing, and recommendation systems.
  • Dimensionality reduction offers several advantages, including reduced computational complexity, better accuracy, improved interpretability, reduced storage requirements, and better visualization.
  • There are two main techniques used in dimensionality reduction: feature selection and feature extraction.
  • By understanding the different techniques used in dimensionality reduction, businesses and researchers can leverage this technique to simplify complex datasets and extract relevant insights.
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You can find practical examples of how to do dimensionality reduction in Scikit-Learn here.

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