For some reason, people who delve into data science tend to focus more on certain aspects of the craft at the expense of others. One of these things that often doesn’t get nearly enough attention is the concept of distance. If you ask a data scientist (especially one who is fairly new to the craft or overspecialized in one aspect of it), they’ll tell you about the distance metrics they are familiar with and how distance is a kind of similarity metric. Although all of this is true, it only portrays just one part of the picture. I’ve delved into the topic for several years now and since my Ph.D. is based on transductive systems (i.e. data science systems that are based on distances), I’ve come to have a particular perspective on the matter, one that helps me see the incompleteness of it all. After all, no matter how many distance heuristics we develop, the way distance is perceived will remain limited until we look at it through a more holistic angle. So, let’s look at the different kinds of distances out there and how they are useful in data science. Distances of the first kind are those most commonly used and are expressed through the various distance heuristics people have devised over the centuries. The most common ones are Euclidean distance and Manhattan distance. Mathematically, it is defined as the norm of a vector connecting two points. Another kind of distances is the normalized ones. Every distance metric out there that is not in this category is crude and limited to the particular set of dimensions it was calculated in. This makes comparisons of distances between two datasets of different dimensionality impossible (if the meaning is to be maintained), even if mathematically it’s straightforward. Normalizing the matrix of distances of the various data points in the dataset requires finding the largest distance, something feasible when the number of data points is small but quite challenging otherwise. What if we need the normalized distances of a sample of data points only because the whole dataset is too large? That’s a fundamental question that needs to be answered efficiently (i.e. at a fairly low big O complexity) if normalized distances are to be practical. The last and most interesting kind of distances is the weighted distance. Although this kind of distance is already welldocumented, the way it has been tackled is fairly rudimentary, considering the plethora of possibilities it offers. For example, by warping the feature space based on the discernibility scores of the various features, you can improve the feature set’s predictive potential in various transductive systems. Also, using a specialized weighted distance, you can better pinpoint the signal of a dataset and refine the similarity between two data points in a large dimensionality space, effectively rendering the curse of dimensionality a nonissue. However, all this is possible only through a different kind of data analytics paradigm, one that is not limited by the unnecessary assumptions of the current one. Naturally, you can have a combination of the latter two kinds of distances for an even more robust distance measure. Whatever the case, understanding the limitations of the first kind of distances is crucial for gaining a deeper understanding of the concept and apply it more effectively. Note that all this is my personal take on the matter. You are advised to approach this whole matter with skepticism and arrive at your own conclusions. After all, the intention of this post is to make you think more (and hopefully more deeply) about this topic, instead of spoonfeeding you canned answers. So, experiment with distances instead of limiting your thinking to the stuff that’s already been documented already. Otherwise, the distance between what you can do and what you are capable of doing, in data science, will remain depressingly large...
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Zacharias Voulgaris, PhD
Passionate data scientist with a foxy approach to technology, particularly related to A.I. Archives
January 2019
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