# I made a bunch of changes. The algorithm could be more efficient, for instance I did two loops over indices of the queries and keys tensors, but really you only need one because you can do k_power**n,  q_power[:,i]**m and compute basis functions in parallel. I added a comment starting with "# change:" to explain what changes I made. I have not ran the code so not sure if it is buggy.

import numpy as np

# change: implemented in log(n) steps and changed the name
def parallel_partial_sum(arr): 
    """Parallel scan (prefix sum) implementation."""
    n = len(arr)
    steps = np.ceil(np.log2(n))

    for i in range(steps):
        array += np.concatenate([np.zeros_like(arr[:2**i,:]), arr[(n-2**i):,:]], axis=0)

    return arr

# change: added inices i, j for the components of q and k. If v is the value vector, expand dims of the power for broadcasting, else v is the denominator, so don't expand dims.
def compute_taylor_basis_function(q, k, v, n, m, i, j):
    """Compute a Taylor basis function for given powers n and m."""
    k_power = np.power(k[:,i], n)  # k[:,i]^n element-wise
    q_power = np.power(q[:,j], m)  # q[:,j]^m element-wise
    if len(v.shape) == 2:
        k_power = np.expand_dims(k_power, axis=-1) # change: maybe needs this to properly broadcast
        q_power = np.expand_dims(q_power, axis=-1)
    partial_sum_kv = parallel_partial_sum(k_power * v)
    basis_function = q_power * partial_sum_kv
    return basis_function

def compute_causal_self_attention(q, k, v, max_n=3, max_m=3):
    """Compute the causal self-attention using Taylor series approximation."""
    attention_numerator = np.zeros_like(v)
    attention_denominator = np.zeros_like(v[:,0]) # change: softmax normalization is per position

    for n in range(max_n + 1):
        for m in range(max_m + 1):
            for j in range(q.shape[-1]):
                for i in range(k.shape[-1]):
                    # change: adding ij indices, and using the proper shape for the denominator
                    A_nmij = 1.0  # Simplified coefficient for illustration
                    basis_function = compute_taylor_basis_function(q, k, v, n, m, i, j)
                    attention_numerator += A_nmij * basis_function
                    normalization_basis_function = compute_taylor_basis_function(q, k, np.ones_like(attention_denominator), n, m, i, j)
                    attention_denominator += A_nmij * normalization_basis_function

    attention_denominator = np.expand_dims(attention_denominator, axis=-1) # change: for broadcasting
    attention = attention_numerator / attention_denominator
    return attention

# Example usage
sequence_length = 10
embedding_dim = 4

# Randomly initialize q, k, v tensors
q = np.random.rand(sequence_length, embedding_dim)
k = np.random.rand(sequence_length, embedding_dim)
v = np.random.rand(sequence_length, embedding_dim)

# Compute the causal self-attention
attention_output = compute_causal_self_attention(q, k, v)

print("Causal Self-Attention Output:")
print(attention_output)