Given that η refers to learning rate and xi​ refers to the ith input to the…

Key idea: the SOM weight update moves each neuron's weight vector toward the input vector by a fraction determined by the learning rate and the neighborhood function.

• General vector form: w_j(new) = w_j(old) + η * h_j * (x − w_j(old)), where h_j is the neighborhood function (depends on distance from the winner).

• Component-wise (i-th component): w_ij(new) = w_ij(old) + η * h_j * (x_i − w_ij(old)).

• For the winning neuron the neighborhood function equals 1, so w_ij(new) = w_ij(old) + η * (x_i − w_ij(old)) = (1−η) * w_ij(old) + η * x_i.

Therefore the expression (1−η) * w_ij(old) + η * x_i correctly represents the updated weight for the winning neuron.

Common mistakes to avoid:

• Using swapped coefficients like η * w_ij(old) + (1−η) * x_i is incorrect because it does not equal the standard η * (x_i − w_ij(old)) update.

• Omitting η on the input term (giving (1−η) * w_ij(old) + x_i) removes control over step size and causes overly large updates.

• Adding η * x_i directly to the old weight without subtracting η * w_ij(old) (i.e., w_ij(old) + η * x_i) is not equivalent to moving toward x_i by a controlled fraction of the difference.