Volume, Sampling, and the Vanishing Middle
As you move to higher-dimensional spaces, the proportion of the total volume that lies near the center of a region shrinks dramatically. This phenomenon builds on the idea of volume concentration discussed earlier: in high dimensions, the bulk of the volume of shapes like hypercubes and hyperspheres is not in the middle, but instead is concentrated near the boundaries. Imagine a simple 2D square or 3D cube — most of the area or volume feels evenly distributed. However, as dimensions increase, the fraction of the total space that can be considered central becomes vanishingly small. The vast majority of the space is close to the outer shell, making the middle almost irrelevant for many practical purposes.
The vanishing middle effect describes how, in high-dimensional spaces, the central region occupies an increasingly negligible portion of the total volume, while most of the space is located near the boundaries.
This vanishing middle has important consequences for random sampling. When you randomly sample points in a high-dimensional region — say, a hypercube or hypersphere — almost all those points will end up close to the edge rather than near the center. This means that in high dimensions, the idea of a typical or average sample being centrally located no longer holds. As a result, algorithms and analyses that assume data will be clustered around the center can become misleading or ineffective. Understanding that most data points are near the boundary is essential for interpreting results and designing robust methods for high-dimensional data.
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Can you explain why the volume concentrates near the boundaries in high dimensions?
What are some practical implications of this phenomenon in data analysis or machine learning?
Can you provide an example or visualization to help illustrate this concept?
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As you move to higher-dimensional spaces, the proportion of the total volume that lies near the center of a region shrinks dramatically. This phenomenon builds on the idea of volume concentration discussed earlier: in high dimensions, the bulk of the volume of shapes like hypercubes and hyperspheres is not in the middle, but instead is concentrated near the boundaries. Imagine a simple 2D square or 3D cube — most of the area or volume feels evenly distributed. However, as dimensions increase, the fraction of the total space that can be considered central becomes vanishingly small. The vast majority of the space is close to the outer shell, making the middle almost irrelevant for many practical purposes.
The vanishing middle effect describes how, in high-dimensional spaces, the central region occupies an increasingly negligible portion of the total volume, while most of the space is located near the boundaries.
This vanishing middle has important consequences for random sampling. When you randomly sample points in a high-dimensional region — say, a hypercube or hypersphere — almost all those points will end up close to the edge rather than near the center. This means that in high dimensions, the idea of a typical or average sample being centrally located no longer holds. As a result, algorithms and analyses that assume data will be clustered around the center can become misleading or ineffective. Understanding that most data points are near the boundary is essential for interpreting results and designing robust methods for high-dimensional data.
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