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Why Data Scientists Can't Afford Too Many Dimensions and What They Can Do About It

An in-depth article about dimensionality reduction and its most popular methods Niklas Lang January 16, 202514 min read Photo by Paulina Gasteiger on Unsplash Photo by Paulina Gasteiger on Unsplash Dimensionality reduction is a central method in the field of Data Analysis and Machine Learning that makes it possible to reduce the number of dimensions in a data set while retaining as much of the information it contains as possible. This step is necessary to reduce the dimensionality of the dataset before training to save computing power and avoid the problem of overfitting. In this article, we take a detailed look at dimensionality reduction and its objectives. We also illustrate the most commonly used methods and highlight the challenges of dimensionality reduction. Learn this step by step with the interactive Machine Learning roadmap.

What is Dimensionality Reduction?

Dimensionality reduction comprises various methods that aim to reduce the number of characteristics and variables in a data set while preserving the information in it. In other words, fewer dimensions should enable a simplified representation of the data without losing patterns and structures within the data. This can significantly accelerate downstream analyses and also optimize machine learning models. In many applications, problems occur due to the high number of variables in a data set, which is also referred to as the curse of dimensionality. For example, too much dimensionality can lead to these problems:

Although large data sets with many characteristics are very informative and valuable, the high number of dimensions can also quickly lead to problems. Dimensionality reduction is a method that attempts to preserve the information content of the data set while reducing the number of dimensions.

What is the Curse of Dimensionality?

The Curse of Dimensionality occurs with high-dimensional data sets, i.e. those that have a large number of attributes or features. At first, many attributes are a good thing because they contain a lot of information and describe the data points well. For example, if we have a dataset about people, the attributes can be information such as hair color, height, weight, eye color, etc. In mathematical terms, however, each additional attribute means a new dimension in the space and therefore a significant increase in possibilities. This becomes clear from the following example, in which we want to find out which customers buy which products. In the first step, we only look at the age of the prospective customers and whether they have bought the product. We can still depict this relatively easily in a two-dimensional diagram. Diagram of Buyers divided by Age | Source: Author Diagram of Buyers divided by Age | Source: Author As soon as we add more information about the customer, things get a little more complex. The information on the customer's income would mean a new axis on which the numerical income is mapped. So the two-dimensional diagram becomes a three-dimensional one. The additional attribute "gender" would lead to a fourth dimension and so on. When working with data, it is desirable to have a lot of attributes and information in the data set to give the model many opportunities to recognize structures in the data. However, it can also lead to serious problems, as the name Curse of Dimensionality suggests. Data Sparsity The example shown illustrates a problem that occurs with many attributes. Due to the large number of dimensions, the so-called data space, i.e. the number of values a dataset can take on, also grows. This can lead to what is known as data sparsity meaning that the training data set used to train the model does not contain certain values at all or only very rarely. As a result, the model only delivers poor results for these marginal cases. Let's assume that we examine 1,000 customers in our example, as it would be too time-consuming to survey even more customers or this data is simply not available. All age groups from young to old may be well represented among these customers. However, if the additional dimension of income is added, it becomes less likely that the possible characteristics, such as "young" and "high income" or "old" and "medium income", will occur and be backed up with enough data points. Distance Concentration If you want to evaluate the similarity of different data sets in the field of machine learning, distance functions are often used for this. The most common clustering algorithms, such as k-means clustering, rely on calculating the distance between points and assigning them to a cluster depending on their size. In multidimensional spaces, however, it can quickly become the case that all points are at a similar distance from each other so it seems almost impossible to separate them. We are also familiar with this phenomenon from everyday life. If you take a photo of two objects, such as two trees, they can look very close to each other in the picture, as it is only a two-dimensional image. In real life, however, the trees may be several meters apart, which only becomes clear in three dimensions. All these problems, which can occur in connection with many dimensions, are summarized under the term Curse of Dimensionality.

What are the Goals of Dimensionality Reduction?

Dimensionality reduction primarily pursues three primary goals: Improving model performance, visualizing data, and increasing processing speed. We will examine these in more detail in the following section. Improving Model Performance One of the main goals of dimensionality reduction is to improve model performance. By reducing the number of variables in a dataset, a less complex model can be used, which in turn reduces the risk of overfitting. Models that have a large number of parameters and are therefore highly complex tend to overfit the training dataset and the noise in the data. As a result, the model delivers poorer results with new data that does not contain this noise, while the accuracy of the training data set is very good. This phenomenon is known as overfitting. During dimensionality reduction, unimportant or redundant features are removed from the data set, which reduces the risk of overfitting. As a result, the model delivers better quality for new, unseen data. Visualization of Data If you want to visualize data sets with many features, you face the challenge of mapping all this information in a two- or at most three-dimensional space. Any dimensionality beyond this is no longer directly tangible for us humans, but it is easiest to assign a separate dimension to each feature in the data set. Therefore, with high-dimensional data sets, we are often faced with the problem that we cannot simply visualize the data to gain an initial understanding of the peculiarities of the data and, for example, to recognize whether there are outliers. Dimensionality reduction helps to reduce the number of dimensions to such an extent that visualization in two- or three-dimensional space is possible. This makes it easier to better understand the relationships between the variables and the data structures. Increasing the Processing Speed Computing time and the necessary resources play a major role in the implementation of projects, especially for machine learning and deep learning algorithms. Often, only limited resources are available, which should be used optimally. By removing redundant features from the data set at an early stage, you not only save time and computing power during data preparation but also when training the model, without having to accept lower performance. In addition, dimensionality reduction makes it possible to use simpler models that not only require less power during initial training but can also perform calculations faster later during operation. This is an important factor, especially for real-time calculations. Overall, dimensionality reduction is an important method for improving data analysis and building more robust machine-learning models. It is also an important step in the visualization of data.

Which Methods are used for Dimensionality Reduction?

In practice, various methods for dimensionality reduction have become established, three of which are explained in more detail below. Depending on the application and the structure of the data, these methods already cover a broad spectrum and can be used for most practical problems.

Principal Component Analysis

Principal component analysis (PCA) assumes that several variables in a data set possibly measure the same thing, i.e. are correlated. These different dimensions can be mathematically combined into so-called principal components without compromising the significance of the data set. The shoe size and height of a person, for example, are often correlated and can therefore be replaced by a common dimension to reduce the number of input variables. Principal component analysis describes a method for mathematically calculating these components. The following two key concepts are central to this: The covariance matrix is a matrix that specifies the pairwise covariances between two different dimensions of the data space. It is a square matrix, i.e. it has as many rows as columns. For any two dimensions, the covariance is calculated as follows: Here n stands for the number of data points in the data set, _X i is the value of the dimension X of the i -th data point and X̅ is the mean value of the dimension X for all n data points. As can be seen from the formula, the covariances between two dimensions do not depend on the order of the dimensions, so the following applies COV(X,Y) = COV(Y,X). These values result in the following covariance matrix C for the two dimensions X and Y : The covariance of two identical dimensions is simply the variance of the dimension itself, i.e: The covariance matrix is the first important step in the principal component analysis. Once this matrix has been created, the eigenvalues and eigenvectors can be calculated from it. Mathematically, the following equation is solved for the eigenvalues: Here λ is the desired eigenvalue and I is the unit matrix of the same size as the covariance matrix C. When this equation is solved, one or more eigenvalues of a matrix are obtained. They represent the linear transformation of the matrix in the direction of the associated eigenvector. An associated eigenvector can therefore also be calculated for each eigenvalue, for which the slightly modified equation must be solved: Where v is the desired eigenvector, according to which the equation must be solved accordingly. In the case of the covariance matrix, the eigenvalue corresponds to the variance of the eigenvector, which in turn represents a principal component. Each eigenvector is therefore a mixture of different dimensions of the data set, the principal components. The corresponding eigenvalue therefore indicates how much variance of the data set is explained by the eigenvector. The higher this value, the more important the principal component is, as it contains a large proportion of the information in the data set. Therefore, after calculating the eigenvalues, they are sorted by size and the eigenvalues with the highest values are selected. The corresponding eigenvectors are then calculated and used as principal components. This results in a dimension reduction, as only the principal components are used to train the model instead of the individual features of the data set.

t-Distributed Stochastic Neighbor Embedding (t-SNE)

t-Distributed Stochastic Neighbor Embedding, or t-SNE for short, approaches the problem of dimensionality reduction differently by attempting to create a new, lower-dimensional space that adopts the distances of the data points from the higher-dimensional space as far as possible. The basic idea of this becomes clear in the following example. It is not easy to transfer data sets from a high dimensionality to a low dimensionality while retaining as much information as possible from the data set. The following figure shows a simple, two-dimensional data set with a total of 50 data points. Three different clusters can be identified, which are also well separated from each other. The yellow cluster is furthest away from the other two clusters, while the purple and blue data points are closer to each other. Two-dimensional data for t-SNE | Source: Author Two-dimensional data for t-SNE | Source: Author The aim now is to convert this two-dimensional data set into a lower dimension, i.e. into one dimension. The simplest approach for this would be to represent the data either only by its X or Y coordinate. Dimensionality reduction by X- or Y-coordinate | Source: Author Dimensionality reduction by X- or Y-coordinate | Source: Author However, it is clear that this simple transformation has lost much of the information in the data set and gives a different picture than the original two-dimensional data. If only the X coordinates are used, it looks as if the yellow and purple clusters overlap and that all three clusters are roughly equidistant from each other. If, on the other hand, only the Y coordinates are used for dimensionality reduction, the yellow cluster is much better separated from the other clusters, but it looks as if the purple and blue clusters overlap. The basic idea of t-SNE is that the distances from the high dimensionality are transferred to the low dimensionality as far as possible. To do this, it uses a stochastic approach and converts the distances between points into a probability that indicates how likely it is that two random points are next to each other. More precisely, it is a conditional probability that indicates how likely it is that one point would choose the other point as a neighbor. Hence the name "Stochastic Neighbor Embedding". Dimensionality reduction according to t-SNE | Source: Author Dimensionality reduction according to t-SNE | Source: Author As you can see, this approach leads to a much better result, in which the three different clusters can be clearly distinguished from one another. It is also clear that the yellow data points are significantly further away from the other data points and the blue and purple clusters are somewhat closer together. To better understand the details of this approach, you are welcome to read our detailed article on this topic.

Mastering t-SNE: A Comprehensive Guide to Understanding and Implementation in Python

Linear Discriminant Analysis

Linear Discriminant Analysis (LDA for short) aims to identify and maximize the separability between classes by projecting the data onto a smaller dimension. In contrast to other methods, it places particular emphasis on maximizing the separability between classes and is therefore particularly important for classification tasks. Two central key figures are calculated:

Simply put, these key figures are used to form matrices and calculate their eigenvalues. The eigenvectors for the largest eigenvalues are in turn the dimensions in the new feature space, which has fewer dimensions than the original data set. All data points can then be projected onto the eigenvectors, which reduces the dimensions. Linear Discriminant Analysis is particularly suitable for applications in which the classes are already known and the data is also clearly labeled. It uses this class information from supervised learning to find a low-dimensional space that separates the classes as well as possible. A disadvantage of LDA is that the maximum reduction to a dimensional space is limited and depends on the number of classes. A data set with (n) classes can therefore be reduced to a maximum of (n-1) dimensions. In concrete terms, this means, for example, that a data set with three different classes can be reduced to a two-dimensional space. This approach of dimensionality reduction is also particularly suitable for large data sets, as the computing effort is only moderate and scales well with the amount of data so that the computing effort is kept within limits even with large amounts of data.

What are the Challenges of Dimensionality Reduction?

Dimensionality reduction is an important step in data pre-processing to increase the generalizability of machine learning models and the general model performance and possibly save computing power. However, the methods also bring with them some challenges that should be considered before use. These include

Dimensionality reduction has many advantages and is an integral part of data pre-processing in many applications. However, the disadvantages and challenges mentioned must also be taken into account to train an efficient model.

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