Closest point to a set with l1 norm

In summary, the speaker attempted to find the element of best approximation ||t_0||≤||t||, ∀ y ∈ π. They then stated that |x_0|+|y_0|+|z_0| ≤|x|+|y|+|z| and provided two equations x_0+2y_0+z=1 and x+2y+z=1. However, they were unsure of how to continue and mentioned considering different combinations of signs for the coordinates. They also added that there is a symmetry that can reduce the number of cases to consider. Finally, they mentioned that based on their visualization, it is likely that all coordinates will be positive.
  • #1
CCMarie
11
1
Homework Statement
Given the plane (π):x+2y+z-1=0, what is the element closest to the origin when the distance is measured with l1 norm.
Relevant Equations
The l1 norm for a vector t=(x_0,y_0,z_0) is
||t|| = |x_0|+|y_0|+|z_0|
I tried to find the element of best approximation
||t_0||≤||t||, ∀ y ∈ π

Then |x_0|+|y_0|+|z_0| ≤|x|+|y|+|z| and we have x_0+2y_0+z=1 and x+2y+z=1.

But I don't know hoe to continue...
 
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  • #2
The only rigorous way I know to solve this would be to consider the various combinations of signs the coordinates could have. For each, the metric can be treated as a linear function. In principle there would be eight cases, but there is a symmetry which allows you to whittle that down.
 
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  • #3
Being a little less rigorous, from visualising the plane, it looks very likely all coordinates will be positive.
 
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  • #4
Thank you!
 

1. What is the definition of the "closest point to a set with l1 norm?"

The closest point to a set with l1 norm is the point that has the minimum sum of absolute differences between its coordinates and the coordinates of any point in the set. In other words, it is the point that minimizes the sum of the absolute values of the differences between its coordinates and the coordinates of any other point in the set.

2. How is the closest point to a set with l1 norm calculated?

The closest point to a set with l1 norm is calculated by finding the point that minimizes the sum of the absolute differences between its coordinates and the coordinates of any other point in the set. This can be done using various mathematical algorithms, such as linear programming or gradient descent.

3. What is the difference between l1 norm and l2 norm?

L1 norm, also known as Manhattan norm, calculates the distance between two points by summing the absolute differences between their coordinates. L2 norm, also known as Euclidean norm, calculates the distance between two points by taking the square root of the sum of the squared differences between their coordinates. L1 norm is often used in cases where the data may have outliers, as it is less sensitive to extreme values compared to l2 norm.

4. Can the closest point to a set with l1 norm be calculated for non-numeric data?

No, the closest point to a set with l1 norm can only be calculated for numeric data. This is because the sum of absolute differences can only be calculated for numbers, and the concept of distance does not apply to non-numeric data.

5. In what fields or applications is the concept of closest point to a set with l1 norm commonly used?

The concept of closest point to a set with l1 norm is commonly used in fields such as statistics, machine learning, and signal processing. It can be applied in various applications, including data clustering, image and signal denoising, and outlier detection.

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