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Suppose f and g are normalized distributions and consider the function f(x)·g(y). If we "collapse" (integrate) in the y-dimension we recover f(x). If we collaps
by scythe 6y ago
Suppose f and g are normalized distributions and consider the function f(x)·g(y). If we "collapse" (integrate) in the y-dimension we recover f(x). If we collapse in the x-dimension we recover g(y).
But if we collapse along the lines x+y = v, we obtain the convolution f⋆g(v).
This picture is a little more advanced, but it makes clear two key properties of convolution: symmetry (commutativity) and the fact that the total integral of the convolution is the total integral of f(x)·g(y).
- bitdizzy 6y agoThis generalizes to convolution in an arbitrary group (and even groupoid, and even category), not necessarily one that is commutative like the reals or integers are. For group convolution over a group G, the functions take values in the complex numbers and have domain the elements of G. The value of the convolution of two such functions f and g at a point c (a group element) is computed by taking the sums f(a)*g(b) where ab = c. This has applications in quantum mechanics where non-commutativity plays a prominent role.
- dsagal 6y agoThis is really cool. The original post is great in putting it in terms of a concrete example. It would be cool (challenging but possible) to have a 3D visualization of collapsing f(x)·g(y) like you describe in the context of that same example. It's certainly a useful mental model.
- vlasev 6y agoWoah this is really cool and an awesome point of view. Collapsing along the x-axis is just a projection along the vector [1, 0, 0] onto the the y-z plane defined by it and the origin. Then if you project along the vector [1, 1, 0] onto the plane defined by [1, -1, 0] and [0, 0, 1], you get the convolution.