Volume Concentration and the Edge Effect
When you explore shapes in high-dimensional spaces, you encounter a surprising phenomenon: most of the volume of objects like spheres and cubes is packed close to their surface rather than deep inside. This is very different from your everyday experience in two or three dimensions. For instance, in a 3D ball, you might expect a fair amount of space both near the center and near the surface. However, as the number of dimensions grows, the proportion of volume that lies close to the surface increases dramatically. This effect is not limited to spheres; it also appears in hypercubes and other familiar shapes.
The root of this phenomenon lies in the way distances and volumes behave as dimensions increase. In higher dimensions, the "middle" of the shape shrinks in relative importance, while the "edges" or "shells" dominate. This has profound consequences for data analysis and machine learning, because most randomly chosen points inside a high-dimensional object will be found very close to the boundary, not evenly scattered throughout the interior.
To build a deeper understanding, you can compare the proportion of volume near the surface for shapes in different dimensions.
In a circle, the area near the edge is a thin ring. If you take a ring of width 10% of the radius, it contains only about 19% of the total area. Most points are not close to the edge.
In a sphere, a shell near the surface (with thickness 10% of the radius) contains about 27% of the volume. More of the total volume is near the surface, but the interior is still significant.
In a 10-dimensional ball, a shell just 10% thick at the surface contains over 65% of the volume. Almost all points are close to the boundary.
In high dimensions, almost every point sampled uniformly from a ball or cube is close to the surface. This means that most of the action is at the edge, which affects how you interpret distances, densities, and the behavior of algorithms in high-dimensional spaces.
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Can you explain why the volume concentrates near the surface in higher dimensions?
How does this phenomenon affect machine learning algorithms?
Can you show a mathematical example comparing 3D and higher dimensions?
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When you explore shapes in high-dimensional spaces, you encounter a surprising phenomenon: most of the volume of objects like spheres and cubes is packed close to their surface rather than deep inside. This is very different from your everyday experience in two or three dimensions. For instance, in a 3D ball, you might expect a fair amount of space both near the center and near the surface. However, as the number of dimensions grows, the proportion of volume that lies close to the surface increases dramatically. This effect is not limited to spheres; it also appears in hypercubes and other familiar shapes.
The root of this phenomenon lies in the way distances and volumes behave as dimensions increase. In higher dimensions, the "middle" of the shape shrinks in relative importance, while the "edges" or "shells" dominate. This has profound consequences for data analysis and machine learning, because most randomly chosen points inside a high-dimensional object will be found very close to the boundary, not evenly scattered throughout the interior.
To build a deeper understanding, you can compare the proportion of volume near the surface for shapes in different dimensions.
In a circle, the area near the edge is a thin ring. If you take a ring of width 10% of the radius, it contains only about 19% of the total area. Most points are not close to the edge.
In a sphere, a shell near the surface (with thickness 10% of the radius) contains about 27% of the volume. More of the total volume is near the surface, but the interior is still significant.
In a 10-dimensional ball, a shell just 10% thick at the surface contains over 65% of the volume. Almost all points are close to the boundary.
In high dimensions, almost every point sampled uniformly from a ball or cube is close to the surface. This means that most of the action is at the edge, which affects how you interpret distances, densities, and the behavior of algorithms in high-dimensional spaces.
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