The Curse of Dimensionality in Machine Learning!

Introduction

In this article, we tackle the Curse of Dimensionality in machine learning, examining its origins and impact on algorithm performance. We discuss practical strategies, including dimensionality reduction and feature selection, to mitigate its effects, paving the path for more effective data-driven insights.

Learning Objectives:

• Understand the concept of the curse of dimensionality in machine learning
• Recognize the problems caused by high-dimensional data
• Learn strategies to mitigate the effects of the curse of dimensionality
• Explore how deep learning approaches handle high-dimensional data

This article was published as a part of the Data Science Blogathon

What is the curse of dimensionality?

It refers to the phenomena of strange/weird things happening as we try to analyze the data in high-dimensional spaces. Let us understand this peculiarity with an example, suppose we are building several machine learning models to analyze the performance of a Formula One (F1) driver. Consider the following cases:

i) Model_1 consists of only two features say the circuit name and the country name.

ii) Model_2 consists of 4 features say weather and max speed of the car including the above two.

iii) Model_3 consists of 8 features say driver’s experience, number of wins, car condition, and driver’s physical fitness including all the above features.

iv) Model_4 consists of 16 features say driver’s age, latitude, longitude, driver’s height, hair color, car color, the car company, and driver’s marital status including all the above features.

v) Model_5 consists of 32 features.

vi) Model_6 consists of 64 features.

vii) Model_7 consists of 128 features.

viii) Model_8 consists of 256 features.

ix) Model_9 consists of 512 features.

x) Model_10 consists of 1024 features.

Assuming the training data remains constant, it is observed that on increasing the number of features the accuracy tends to increase until a certain threshold value and after that, it starts to decrease. From the above example the accuracy of Model_1 < accuracy of Model_2 < accuracy of Model_3 but if we try to extrapolate this trend it doesn’t hold true for all the models having more than 8 features. Now you might wonder if we are providing some extra information for the model to learn why is it so that the performance starts to degrade. My friends welcome to the curse of dimensionality!

If we think logically some of the features provided to Model_4 don’t actually contribute anything towards analyzing the performance of the F1 driver. For example, the driver’s height, hair color, car color, car company, and the driver’s marital status is giving useless information for the model to learn, hence the model gets confused with all this extra information, and the accuracy starts to go down.

The curse of curse of dimensionality in data science was first termed by Richard E. Bellman when considering problems in dynamic programming.

Curse of dimensionality in various domains

There are several domains where we can see the effect of this phenomenon. Machine Learning is one such domain. Other domains include numerical analysis, sampling, combinatorics, data mining, and databases. As it is clear from the title we will see its effect only in Machine Learning.

What problems does curse of dimensionality cause?

The curse of curse of dimensionality in data science refers to the difficulties that arise when analyzing or modeling data with many dimensions. These problems can be summarized in the following points:

• Data Sparsity: Data points become increasingly spread out, making it hard to find patterns or relationships.
• Computational Complexity: The computational burden of algorithms increases exponentially.
• Overfitting: Models become more likely to memorize the training data without generalizing well.
• Distortion of Distance Metrics: Traditional distance metrics become less reliable in measuring proximity.
• Visualization Challenges: Projecting high-dimensional data onto lower dimensions leads to loss of information.
• Data Preprocessing: Identifying relevant features and reducing curse of dimensionality in data science is crucial for effective analysis.
• Algorithmic Efficiency: Algorithms need to be scalable and efficient to handle the complexity of high-dimensional spaces.
• Domain-Specific Challenges: Each domain faces unique challenges in high-dimensional spaces, requiring tailored approaches.
• Interpretability Issues: Understanding the decision-making process of high-dimensional models becomes increasingly difficult.
• Data Storage Requirements: Efficient data storage and retrieval strategies are essential for managing large volumes of high-dimensional data

How to overcome its effect

This was a general overview of the curse of dimensionality. Now we will go slightly technical to understand it completely. In ML, it can be defined as follows: as the number of features or dimensions ‘d’ grows, the amount of data we require to generalize accurately grows exponentially. As the dimensions increase the data becomes sparse and as the data becomes sparse it becomes hard to generalize the model. In order to better generalize the model, more training data is required.

1. Hughes phenomenon

Again let’s take an example under this phenomenon. Assume all the features in a dataset are binary. If the dimensionality is 3 i.e. there are 3 features then the total number of data points will be equal to 23 = 8. If the dimensionality is 10 i.e. there are 10 features then the total number of data points will be equal to 210 = 1024. It is clear that as dimensionality increases the number of data points also increases exponentially which implies high dimensional data in machine learning is directly proportional to the number of data points required for training a machine learning model.

There is a very interesting phenomenon called the Hughes phenomenon which states that for a fixed size dataset the performance of a machine learning model decreases as the machine learning curse of dimensionality increases.

2. Distance functions (especially Euclidean distance)

Let’s think of a 1D world where n points are spread randomly between 0 and 1, we have a point xi.

From the above two figures, it is clear that the Euclidean distance between pair of points is very close to 0.

Now let me define two terms,

Dist_min (xi) = min{euc-dist(xi, xj} where xi is not equal to xj.

Dist_max (xi) = max{euc-dist(xi, xj} where xi is not equal to xj.

For 1D, 2D and 3D,

{[dist-max(xi) – dist-min(xi)] / dist-min(xi)} > 0

Taking the limit as d -> infinity, {[dist-max(xi) – dist-min(xi)] / dist-min(xi)} tends towards 0. Now you might wonder what happens if this ratio tends to 0.

From the above figures, we can see how those peaks are getting formed as the dimensions are increasing. At the heart of KNN, it works well if the pair of points are closer together in a cluster but at higher dimensions, we can see the pair of points that are very close to each other reduces and we have lot many pair of points having distance 5-10 and 15-20 when d=100 and it only increases on increasing the dimensions. So we know for sure KNN will break apart in such conditions.

Let me Break it Down for you Even Further:

{[dist-max(xi) – dist-min(xi)] / dist-min(xi)}

The above ratio will only become 0 when the numerator becomes 0 i.e. dist-max and dist-min are equal, which means in high dimensional data in machine learning spaces every pair of points are equally distant from every other pair of points. For example, the distance between xi and xj is almost equal to the distance between xi and xk. This is true for every pair of points.

In high machine learning curse of dimensionality data in machine learning spaces, whenever the distance of any pair of points is the same as any other pair of points, any machine learning model like KNN which depends a lot on Euclidean distance, makes no more sense logically. Hence KNN doesn’t work well when the dimensionality increases. Even though this was theoretically proven for n random points, it has been observed experimentally also that KNN doesn’t work well in higher dimensional spaces. So what is the solution?

The solution is very simple. Use cosine-similarity instead of Euclidean distance as it is impacted less in higher dimensional spaces. That’s why especially in-text problems where we use a bag of words, TF-IDF, word-to-vec, etc., cosine similarity is preferred because of high dimensional data in machine learning space.

It is important to note that all these observations were made assuming the spread of points is uniform and random. So the very next thing that comes into mind is what if the spread of points are not uniform and random. We can think of this from a different angle i.e.

• When dimensionality is high and points are dense, the impact of dimensionality is high.
• When dimensionality is high and points are sparse, the impact of dimensionality is low.

3. Overfitting and Underfitting

There is a relationship between ‘d’ and overfitting which is as follows:

‘d’ is directly proportional to overfitting i.e. as the machine learning curse of dimensionality increases the chances of overfitting also increases.

Let’s discuss the solutions to tackle this problem.

a) Model-dependent approach: Whenever we have a large number of features, we can always perform forward feature selection to determine the most relevant features for the prediction.

b) Unlike the above solution which is classification-oriented, we can also perform dimensionality reduction techniques like PCA and t-SNE which do not use the class labels to determine the most relevant features for the prediction.

So it is important to keep in mind whenever you download a new dataset that has a large number of features, you can reduce it by some of the techniques like PCA, t-SNE, or forward selection in order to ensure your model is not affected by the curse of dimensionality.

How does deep learning tackle the curse of dimensionality?

Deep learning aids in handling high-dimensional data through various mechanisms:

Identifying Key Features: It discerns the crucial aspects of the data, filtering out less significant elements.

Constructing a Comprehensive View: Deep learning dissects the data into simpler components and then assembles them to grasp the broader context. This process resembles assembling a complex structure from individual building blocks, gradually forming a coherent whole.

Preventing Information Overload: Deep learning employs techniques to avert confusion caused by an abundance of data. These strategies maintain focus during the learning process and mitigate errors.

Dimensionality Reduction: Certain deep learning approaches condense the data into a lower-dimensional space while retaining essential information. This analogy mirrors framing a large picture in a smaller frame without sacrificing its core details.

Selective Utilization: At times, deep learning prioritizes the most pertinent data points while disregarding extraneous information, streamlining the analysis process.

Uncovering Latent Patterns: Deep learning unveils concealed patterns within the data, even amidst its vastness. This capability is akin to discerning recognizable shapes amidst a cluster of clouds.

In essence, deep learning streamlines the comprehension of extensive datasets by deconstructing them, prioritizing critical components, and revealing hidden structures. This proficiency renders it invaluable in various domains, including the analysis of complex phenomena and the optimization of machine learning algorithms, such as linear discriminant analysis, in lower-dimensional spaces.

Principal Component Analysis in Machine Learning

Principal Component Analysis (PCA) is a powerful technique used in machine learning for transforming high-dimensional data into a more manageable form. It works by extracting important features, known as principal components, which capture the maximum variance in the data. These components are linear combinations of the original features and provide a new coordinate system for the data. By doing so, PCA enables a deep neural network to focus on the most relevant aspects of the data, thereby improving its performance.

Moreover, PCA facilitates distraction-free reading by simplifying complex data while retaining essential information for analysis. However, it’s important to note that PCA assumes linear relationships between variables, which means it may not perform optimally with nonlinear data. Nonetheless, it remains a valuable tool for visualizing data and speeding up algorithms by reducing input dimensions. The steps involved in PCA include data standardization, computation of the covariance matrix, eigenvalue decomposition, selection of principal components based on eigenvalues, and projection of data onto these components. Overall, PCA serves as a fundamental technique for dimensionality reduction and feature extraction in machine learning.

Conclusion

In summary, the Curse of Dimensionality in Machine Learning highlights challenges when dealing with high-dimensional data. It affects diverse domains, increasing computational demands and reducing model performance. Overcoming it involves feature selection, dimensionality reduction, and careful algorithm choices. Understanding and addressing these aspects are crucial for efficient and accurate machine learning models in various applications.

Key Takeaways:

• Increasing dimensions can lead to decreased model performance beyond a certain threshold
• The curse of dimensionality affects various aspects, including data sparsity and distance metrics
• Dimensionality reduction techniques like PCA are crucial for managing high-dimensional data
• Deep learning offers unique approaches to tackle the challenges of high-dimensional spaces

Frequently Asked Questions

Swapnil Vishwakarma 28 Jun, 2024