Math for Deep Learning by Ronald T. Kneusel
Author:Ronald T. Kneusel
Language: eng
Format: epub
Publisher: No Starch Press
Published: 2022-10-15T00:00:00+00:00
L-Norms and Distance Metrics
For an n-dimensional vector, x, we define the p-norm of the vector to be
where p is a real number. Although we use p in the definition, people generally refer to these as Lp norms. We saw one of these norms in Chapter 5 when we defined the magnitude of a vector. In that case, we were calculating the L2-norm,
which is the square root of the inner product of x with itself.
The norms we use most often in deep learning are the L2-norm and the L1-norm,
which is nothing more than the sum of the absolute values of the components of x. Another norm youâll encounter is the Lâ-norm,
Lâ = max |xi|
the maximum absolute value of the components of x.
If we replace x with the difference of two vectors, x â y, we can treat the norms as distance measures between the two vectors. Alternatively, we can picture the process as computing the vector norm on the vector that is the difference between x and y.
Switching from norm to distance makes a trivial change in Equation 6.6:
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