I've posted before about "compactification", and especially about its simplest example called the Alexandroff 1-point compactification, which "closes" a real interval (turning it into a circle).
en.wikipedia.org
I will now show a different and more powerful version of this mathematical tool, in terms of defining a natural relationship between intrinsic and extrinsic geometry. It is a version of "projection", which is a powerful tool in and of itself.
To define intrinsic and extrinsic geometry, imagine you have a hollow sphere (a "curved surface"). Mathematicians call this S2 (a "two dimensional" sphere), even though most of the ways we describe it are 3-dimensional. For example, how do we usually describe this sphere in terms of Cartesian coordinates? Well, the first thing we do is, we center the sphere at the origin. Then we compute the vectors r = sqrt (x^2 + y^2 + z^2) to assign coordinates to each point on the surface of the sphere.
But note, that the origin, is not part of the sphere. To assign the origin, we are "embedding" our 2-sphere into 3 dimensions. The origin is "extrinsic" to the sphere.
Whereas, if we weren't allowed to do this, or if we wanted to remain "on" the sphere for our math, we would have to be the bug walking on the surface of the sphere, living in Flatland as it were (even though Flatland in this case is curved - and the only way the bug could tell is if it walked in a straight line in one direction, and discovered itself returning to the starting point from the other direction).
Similarly, defining a tangent to any point on the surface of the sphere is "extrinsic", because the tangent plane takes us off the surface of the sphere. If we wish to remain "intrinsic" to the surface, we have to find a different way of describing our tangent slopes. For example, Einstein's general relativity is "intrinsic" because there is nothing outside of our universe, there is no way to magically "create an extra dimension" for our universe to live in.
Topologists have created methods for "lifting" a surface into higher dimensions, if such a lifting is permissible. In its simplest form, you can just draw another coordinate axis and assign the new coordinate to be a constant value. This retains the original curve and one can always "project back down" by simply eliminating (omitting or annihilating) the extra coordinate.
So now, consider a real interval (0, 2Ï€), which is an open line segment of length 2Ï€. Now bend the edges (upward or downward) till they meet. The line segment has now become a circle. (A circle is what mathematicians call S1 - a 1 dimensional sphere). Formally, since we started with an open interval (a proper topological subset), we must now add a point, at the location where the ends of the interval meet, if we want the result to be a properly closed circle. We can arbitrarily call this point "the North Pole", or equivalently, the "point at Infinity". This is the Alexandroff 1-point compactification, it makes our circle "compact" in the topological sense, so if we're the bug walking on the surface in one direction, we really CAN return to the starting point from the other direction, without encountering a discontinuity. (Without "falling off the edge of the earth", so to speak).
But look what this has done to our geometry. By employing this compactification, we have INDUCED an extrinsic geometry. If I ask you "where is the origin" you will most likely put a dot in the middle of the circle. Which is not a part of the circle itself, therefore, extrinsic. And, if I ask you to draw the tangent at some point on the circle, you will most likely draw a straight line that touches the circle at exactly one point - all other points being "off the circle", therefore, extrinsic.
The surface of the circle is 1 dimensional, just like the line segment it came from. But we have now induced a 2-dimensional embedding, which we can naturally represent in terms of a Cartesian X-Y coordinate system.
NOTE that all of our trigonometry is EXTERNAL, to define or measure rise and run for sines and cosines requires x and y. Similarly, the concept of "radius" is EXTERNAL, in other words polar coordinates are external too, just like Cartesian coordinates. If we didn't have the second dimension, the best we could do is describe our circle in terms of symmetry groups, by defining key points on the surface and transformations (rotations, permutations) between them. (And note I didn't say reflections, which are external since they require the definition of an axis of reflection containing points "not on the circle").
As indicated in the link, this type of compactification is an example of "stereographic projection", and topologically we can speak of "gluing" the edges of the line interval using the extra point, or "cutting" the circle at that point, to transition between 1-dimensional and 2-dimensional representations.
It turns out, there is also a different way of creating a second dimension for our line segment. It's called a fiber bundle. What you do is, at each point along the line interval, draw a vertical axis and make sure all such vertical lines use the same measure. You can now draw an arbitrary curve "above" the original interval, and map it straight down to an "origin" on the original interval. Done this way, you have induced a coordinate system and mapped the fibers into it. So you now have a 2-dimensional plane. But note this plane is NOT COMPACT. To make it compact you'd have to roll it up into a cylinder, and add a "line at Infinity", which in turn induces a further embedding into 3 dimensions. (Only one of which ends up being compact).
So you have these two powerful mathematical tools to extend dimensionality, and you can use them either alone or in combination.
To show that you understand, and as a bit of a brain teaser, I ask you the following question:
When a neural network increases the dimensionality of its feature space by identifying a new feature, which of the above methods does it use, and in what combination?
Alexandroff extension - Wikipedia
I will now show a different and more powerful version of this mathematical tool, in terms of defining a natural relationship between intrinsic and extrinsic geometry. It is a version of "projection", which is a powerful tool in and of itself.
To define intrinsic and extrinsic geometry, imagine you have a hollow sphere (a "curved surface"). Mathematicians call this S2 (a "two dimensional" sphere), even though most of the ways we describe it are 3-dimensional. For example, how do we usually describe this sphere in terms of Cartesian coordinates? Well, the first thing we do is, we center the sphere at the origin. Then we compute the vectors r = sqrt (x^2 + y^2 + z^2) to assign coordinates to each point on the surface of the sphere.
But note, that the origin, is not part of the sphere. To assign the origin, we are "embedding" our 2-sphere into 3 dimensions. The origin is "extrinsic" to the sphere.
Whereas, if we weren't allowed to do this, or if we wanted to remain "on" the sphere for our math, we would have to be the bug walking on the surface of the sphere, living in Flatland as it were (even though Flatland in this case is curved - and the only way the bug could tell is if it walked in a straight line in one direction, and discovered itself returning to the starting point from the other direction).
Similarly, defining a tangent to any point on the surface of the sphere is "extrinsic", because the tangent plane takes us off the surface of the sphere. If we wish to remain "intrinsic" to the surface, we have to find a different way of describing our tangent slopes. For example, Einstein's general relativity is "intrinsic" because there is nothing outside of our universe, there is no way to magically "create an extra dimension" for our universe to live in.
Topologists have created methods for "lifting" a surface into higher dimensions, if such a lifting is permissible. In its simplest form, you can just draw another coordinate axis and assign the new coordinate to be a constant value. This retains the original curve and one can always "project back down" by simply eliminating (omitting or annihilating) the extra coordinate.
So now, consider a real interval (0, 2Ï€), which is an open line segment of length 2Ï€. Now bend the edges (upward or downward) till they meet. The line segment has now become a circle. (A circle is what mathematicians call S1 - a 1 dimensional sphere). Formally, since we started with an open interval (a proper topological subset), we must now add a point, at the location where the ends of the interval meet, if we want the result to be a properly closed circle. We can arbitrarily call this point "the North Pole", or equivalently, the "point at Infinity". This is the Alexandroff 1-point compactification, it makes our circle "compact" in the topological sense, so if we're the bug walking on the surface in one direction, we really CAN return to the starting point from the other direction, without encountering a discontinuity. (Without "falling off the edge of the earth", so to speak).
But look what this has done to our geometry. By employing this compactification, we have INDUCED an extrinsic geometry. If I ask you "where is the origin" you will most likely put a dot in the middle of the circle. Which is not a part of the circle itself, therefore, extrinsic. And, if I ask you to draw the tangent at some point on the circle, you will most likely draw a straight line that touches the circle at exactly one point - all other points being "off the circle", therefore, extrinsic.
The surface of the circle is 1 dimensional, just like the line segment it came from. But we have now induced a 2-dimensional embedding, which we can naturally represent in terms of a Cartesian X-Y coordinate system.
NOTE that all of our trigonometry is EXTERNAL, to define or measure rise and run for sines and cosines requires x and y. Similarly, the concept of "radius" is EXTERNAL, in other words polar coordinates are external too, just like Cartesian coordinates. If we didn't have the second dimension, the best we could do is describe our circle in terms of symmetry groups, by defining key points on the surface and transformations (rotations, permutations) between them. (And note I didn't say reflections, which are external since they require the definition of an axis of reflection containing points "not on the circle").
As indicated in the link, this type of compactification is an example of "stereographic projection", and topologically we can speak of "gluing" the edges of the line interval using the extra point, or "cutting" the circle at that point, to transition between 1-dimensional and 2-dimensional representations.
It turns out, there is also a different way of creating a second dimension for our line segment. It's called a fiber bundle. What you do is, at each point along the line interval, draw a vertical axis and make sure all such vertical lines use the same measure. You can now draw an arbitrary curve "above" the original interval, and map it straight down to an "origin" on the original interval. Done this way, you have induced a coordinate system and mapped the fibers into it. So you now have a 2-dimensional plane. But note this plane is NOT COMPACT. To make it compact you'd have to roll it up into a cylinder, and add a "line at Infinity", which in turn induces a further embedding into 3 dimensions. (Only one of which ends up being compact).
So you have these two powerful mathematical tools to extend dimensionality, and you can use them either alone or in combination.
To show that you understand, and as a bit of a brain teaser, I ask you the following question:
When a neural network increases the dimensionality of its feature space by identifying a new feature, which of the above methods does it use, and in what combination?
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