#<GRADED> def l2distance(X,Z=None): """ function D=l2distance(X,Z) Computes the Euclidean distance matrix. Syntax: D=l2distance(X,Z) Input: X: nxd data matrix with n vectors (rows) of dimensionality d Z: mxd data matrix with m vectors (rows) of dimensionality d Output: Matrix D of size nxm D(i,j) is the Euclidean distance of X(i,:) and Z(j,:) call with only one input: l2distance(X)=l2distance(X,X) """ if Z is None: Z=X; n,d1=X.shape m,d2=Z.shape assert (d1==d2), "Dimensions of input vectors must match!" # Your code goes here .. raise NotImplementedError('Your code goes here!') return D # ... until here #</GRADED> without using for-loop

Sure, here's how you can implement the l2distance function without using a for loop:

import numpy as np

def l2distance(X, Z=None):
    if Z is None:
        Z = X

    n, d1 = X.shape
    m, d2 = Z.shape
    assert (d1 == d2), "Dimensions of input vectors must match!"

    # Compute the sum of squares for rows in X and Z
    X_sq = np.sum(X**2, axis=1)
    Z_sq = np.sum(Z**2, axis=1)

    # Compute the dot product between rows in X and Z
    XZ = np.dot(X, Z.T)

    # Compute the Euclidean distance matrix
    D = np.sqrt(X_sq[:, np.newaxis] + Z_sq - 2*XZ)

    return D

In this implementation, we first check if Z is None and if so, we set Z = X. Then, we get the shapes of X and Z and check if their dimensions match.

Next, we compute the sum of squares for each row in X and Z. We also compute the dot product between rows in X and Z.

Finally, we compute the Euclidean distance matrix D using the formula sqrt(sum(X^2) + sum(Z^2) - 2*dot(X, Z)). This formula comes from the definition of the Euclidean distance and allows us to compute the distance matrix without using a for loop.

This implementation takes advantage of numpy's broadcasting and vectorized operations to compute the distance matrix efficiently.