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 you’ll encounter is the L∞-norm,

L∞ = max |xi|

the maximum absolute value of the components of x.

If we replace x with the difference of two vectors, x − 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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