Evenly spaced vectors in N-dim

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In summary, the conversation is discussing the most convenient way to obtain a set of unit vectors in N dimensions with a fixed angle between neighboring vectors. Possible methods include using a N-dimensional cube or a cross-polytope. It is suggested that this problem is similar to a sphere-packing problem, where the normalized vectors are considered to be on a higher-dimensional sphere. The desired solution is an approximate one that covers all the surface area.
  • #1
Gerenuk
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What is the most convenient way to get a set of unit vectors in N dimensions, such that the angle between two neighboring vectors is about a fixed value?

To be more precise I'd like to find a small number of vectors so that all possible other unit vectors are at most at a certain angle from a vector of this set.

It doesn't have to be precise, which probably would work for platonic solids like cubes only. Some rough method is to take all points on a N-dimensional cube and normalize the vectors. A whacky idea would be to do the same with a cross-polytope, but I don't know how to parametrize that.
But maybe one can do better?
 
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  • #2
Sounds like a sphere-packing problem, where you're packing (N-2)-spheres into an (N-1)-sphere. What I mean is, in 3 dimensions, your normalized vectors live on S^2, and their angular neighborhoods are 1-spheres (circles).
 
  • #3
Yes, it is. However a full sphere packing problem is highly non-trivial, but I would be satisfied with a more approximate solution. The requirement is to find a reasonable solution which covers all the "surface area".
 

1. What are evenly spaced vectors in N-dim?

Evenly spaced vectors in N-dim are a set of vectors with equal spacing between each vector in an N-dimensional space. This means that the distance between any two adjacent vectors is the same.

2. How are evenly spaced vectors in N-dim different from evenly spaced vectors in 2-dim?

The main difference between evenly spaced vectors in N-dim and 2-dim is the number of dimensions. N-dim refers to an N-dimensional space, while 2-dim refers to a 2-dimensional space. In N-dim, there are N axes and N components for each vector, while in 2-dim, there are only 2 axes and 2 components for each vector.

3. What is the significance of evenly spaced vectors in N-dim?

Evenly spaced vectors in N-dim have many applications in mathematics, physics, and engineering. They can be used to represent physical quantities, such as position, velocity, and acceleration, in a multi-dimensional space. They also have important uses in data analysis and machine learning algorithms.

4. How are evenly spaced vectors in N-dim used in machine learning?

In machine learning, evenly spaced vectors in N-dim are often used to represent data points in a multi-dimensional feature space. This allows for more complex relationships between the data points to be captured and can improve the performance of machine learning models.

5. Can evenly spaced vectors in N-dim have negative components?

Yes, it is possible for evenly spaced vectors in N-dim to have negative components. In an N-dimensional space, each vector has N components, which can be positive, negative, or zero. The spacing between vectors is determined by the magnitude of the components, not their sign.

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