Distances between samples in high dimensions. Curse of dimensions?

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In summary, the curse of dimensionality refers to the challenges of analyzing and interpreting high-dimensional data due to the exponential increase in data required and the less meaningful distance between points. Various approaches, such as dimensionality reduction and feature selection, have been proposed to address this issue. References such as "The Curse of Dimensionality: A Review" and "Pattern Recognition and Machine Learning" provide more information on this topic.
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Gerenuk
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Hello,

I remember that I heard something about the curse of dimensions when discrimitating between samples with high dimensional parameters. Therefore a sample might have the property vector (x1,x2,...,xn) and one could define a measure of distance (euclidean, manhattan, ...) to judge how close two samples are in their qualities.

The problem that arises in high dimension is, that effectively all random samples will have a very similar distance between a pair! Somehow that was also obvious from looking at volume of high dimensional cube and spheres.

Does anyone know good references to read up about this problem or does anyone know about solutions to this dilemma?
 
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Hello,

You are correct, there is a concept known as the "curse of dimensionality" that refers to the difficulty of analyzing and interpreting data in high-dimensional spaces. This phenomenon was first described by Richard Bellman in 1961 and has since been studied extensively in various fields of science and engineering.

The main issue with high-dimensional data is that as the number of dimensions increases, the amount of data required to adequately represent the space also increases exponentially. This means that in order to accurately capture the relationships and patterns within the data, a large amount of data is needed. This can be a problem when working with real-world data, as collecting and processing such large amounts of data can be time-consuming and resource-intensive.

In addition, as you mentioned, the distance between points in high-dimensional spaces becomes less meaningful. This is because in high-dimensional spaces, the distance between any two points tends to be similar, making it difficult to distinguish between different samples. This can lead to misleading results and make it challenging to identify meaningful patterns and relationships within the data.

There are various approaches that have been proposed to address the curse of dimensionality, such as dimensionality reduction techniques, feature selection methods, and data preprocessing techniques. These methods aim to reduce the dimensionality of the data while retaining as much useful information as possible. However, there is no one-size-fits-all solution, and the best approach will depend on the specific data and research question at hand.

I recommend checking out some of the references below for more information on the curse of dimensionality and potential solutions:

1. "The Curse of Dimensionality: A Review" by David Donoho (2000)
2. "Pattern Recognition and Machine Learning" by Christopher M. Bishop (2006)
3. "Dimensionality Reduction: A Comparative Review" by Li, H., & Zhang, Z. (2017)
4. "Feature Selection for High-Dimensional Data: A Fast Correlation-Based Filter Solution" by Yu, L., & Liu, H. (2003)

I hope this helps and provides a good starting point for further research on this topic. Best of luck with your studies!



Scientist in [Your Field]
 

1. What is the curse of dimensionality?

The curse of dimensionality refers to the phenomenon where the distances between samples in high dimensions become less meaningful and reliable. This means that as the number of dimensions increases, the distance between any two points becomes less informative and it becomes more challenging to accurately measure the similarity between data points.

2. How does the curse of dimensionality affect data analysis?

The curse of dimensionality can greatly impact data analysis as it makes it difficult to find meaningful patterns and relationships between data points. It can also lead to overfitting in machine learning algorithms and make it challenging to visualize and interpret the data.

3. What causes the curse of dimensionality?

The curse of dimensionality is caused by the exponential increase in the volume of the sample space as the number of dimensions increases. As the number of dimensions grows, the amount of data required to fill that space also increases exponentially, making it difficult to effectively capture and represent the data.

4. How can the curse of dimensionality be mitigated?

One way to mitigate the curse of dimensionality is by using dimensionality reduction techniques, such as principal component analysis or t-distributed stochastic neighbor embedding, to reduce the number of dimensions and retain the most important features of the data. Another approach is to carefully select and engineer features to reduce the number of irrelevant dimensions.

5. Is the curse of dimensionality a major concern in all data analysis?

No, the curse of dimensionality is not a major concern in all data analysis. It primarily affects data with a high number of dimensions, such as images, text, or genetic data. Data with fewer dimensions, such as time series data, may not be as affected by the curse of dimensionality.

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