Is your feature request related to a problem? Please describe.

I think there might be an issue with the maxvol function. https://github.com/tensorly/tensorly/blob/main/tensorly/contrib/decomposition/_tt_cross.py
You referred to "The Greedy approximation algorithm for MAX-VOL" in the paper "Ali Çivril, Malik Magdon-Ismail: On selecting a maximum volume sub-matrix of a matrix and related problems."

In that paper, they mention "Remove the projection of v from every element of A." When I read this part, I interpreted it as calculating an orthogonal vector to the vector 𝑣 with the largest norm.
However, in your code, it doesn't seem to calculate the orthogonal vector. Instead, it uses something that looks strange to me.

Maybe I am misunderstanding your code or the paper.
Could you clarify whether I am wrong or if there is indeed an issue with the code?

Describe alternatives you've considered

This is your code

#Find the row of max norm"
max_row_idx = np.argmax(rows_norms, axis=0)
max_row = A[rest_row_idx[max_row_idx], :]

projection = np.dot(A_new, max_row.T)
projection = projection / np.sum(max_row**2)
        
#calculate orthogonal vector"
A_new = A_new - np.outer(projection, max_row)

This is my code

Additional context

When I compared the performance between my code and yours, here are the results:

• The blue line represents my code, while the orange line represents your code.
• You can see that the volume achieved by my code is almost always larger than yours.
• Occasionally, your code outperforms mine, but this happens in only 11% of the iterations.

The values on the graph were calculated using a randomly generated matrix.

Could you clarify the cause of this discrepancy? Is there an issue with the logic in your code, or is it something I might be missing?
Let me know if this works!