math matrix machine-learning regression dimensions

Matrix dimension do not mach in regression formula

I'm trying to calculate this $argmax_j|X_j^{T}(y-X\beta^{(k-1)})|$ regression formula, but I have problem with the dimension calculation, they are not correct: Where:

X-a matrix with dimensions 200x20, n=200 samples, p=20 predictors,

y-a matrix with dimensions 200x1,

$\beta^{(k)}$ - a sequence of coefficients, dimensions 20x1, and k=1,2,3...

$X^{T}$ - dimensions 20x200

j- and value from 1...p so from 1...20,

The problem is when I calculate $y-X\beta^{(k-1)}$

For example for k=20, k-1=19 i have $y-X\beta^{(19)}$ and the dimensions do not match to do a substraction 200x1 - 200x20 x 1x1 =200x1 - 200x20 will not work.

If I take all the beta vector then it is correct. does this: $\beta^{(19)}$ mean to take the 19th value of Beta and to multiply it with the matrix X?

Source of the formula:

Solution

You should be using the entire beta vector at each stage of the calculation.

(Tibshirani has been a bit permissive with his use of notation, perhaps...)

The k is just a counter for which step of the algorithm we are on. Right at the start (k = 0 or "step 0") we initialise the entire beta vector to have all elements equal to zero:

At each step of the algorithm (steps k = 1, 2, 3... and so on) we use our previous estimate of the vector beta ( calculated in step k - 1) to calculate a new improved estimate for the vector beta (). The superscript number is not an index into the vector, rather it is a label telling us at which stage of the algorithm that beta vector was produced.

I hope this makes sense. The important point is that each of the values is a different 20x1 vector.