I knew the formula cold but fumbled the normalization step at first.
First, explain the mathematical formulation of cosine similarity using dot products and norms. Then, present vectorized NumPy code that computes the dot product matrix and normalizes by the outer product of norms. Finally, analyze time and space complexity, highlighting the O(n*m*d) time and O(n*m) space.
Pro tip: Mention that for large matrices, you can avoid the full n×m matrix by using a chunked or approximate approach, but for exact computation, the vectorized method is optimal. Also, note that normalizing rows first can simplify the code and improve numerical stability.
Write the cosine similarity formula: sim(i,j) = (X_i · Y_j) / (||X_i|| * ||Y_j||). Explain that this can be computed as a matrix product followed by normalization.
Provide NumPy code: compute dot product matrix using X @ Y.T, compute norms using np.linalg.norm along axis=1, then divide by outer product of norms. Ensure no Python loops.
Analyze time complexity: O(n*m*d) for matrix multiplication. Space complexity: O(n*m) for the similarity matrix. Mention that norms take O(n*d + m*d) time and O(n+m) space.
Discuss handling zero norms (add epsilon), and potential memory optimizations like chunking or using float32. Mention that if n or m is huge, the O(n*m) matrix may be prohibitive.
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The subtract-the-max trick is pretty well known so I got through this one without much drama.
Start by explaining the mathematical definition of softmax and why the naive implementation can overflow. Then describe the standard stabilization trick: subtract the maximum value along the last axis before exponentiation, and show how to implement it efficiently in NumPy using broadcasting. Finally, discuss the pitfalls avoided, such as overflow and underflow, and mention any edge cases like all -inf inputs.
Pro tip: Mention that in practice, you might also clip the logits or use a log-softmax for numerical stability in loss functions, and that frameworks like PyTorch have built-in stable implementations. This shows awareness of real-world ML engineering.
Briefly state that softmax converts logits to probabilities by exponentiating and normalizing. Emphasize that it's used in classification and attention mechanisms.
Explain that large logits cause overflow in exp, leading to inf or nan, and small logits cause underflow, leading to zero probabilities and division by zero.
Subtract the maximum value along the last axis before exponentiation: exp(x - max(x)) / sum(exp(x - max(x))). This keeps the exponent arguments <= 0, preventing overflow.
Use np.max with keepdims=True to maintain shape for broadcasting, then np.exp and np.sum along the last axis with keepdims=True for normalization.
Mention that subtracting the max avoids overflow and underflow, but if all inputs are -inf, the result is nan; handle by returning uniform distribution or using a small epsilon.
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Honestly my weakest moment in the interview.
Start by explaining the broadcasting rules: align shapes from the right, and dimensions of size 1 are stretched to match the other. Then apply these rules to X (n,1,d) and Y (1,m,d) to determine the resulting shape (n,m,d). Finally, discuss memory implications, emphasizing that broadcasting avoids copying data but can lead to large intermediate arrays if not careful.
Pro tip: Mention that while broadcasting is memory-efficient in terms of not replicating the original arrays, the resulting array can be large, and using in-place operations or libraries like NumPy's einsum can mitigate memory overhead.
Explain that broadcasting aligns shapes from the right, and dimensions of size 1 are expanded to match the other array's dimension.
Show that X (n,1,d) and Y (1,m,d) broadcast to (n,m,d) by expanding the second dimension of X and the first dimension of Y.
Conclude that the result has shape (n, m, d).
Explain that broadcasting does not copy the original arrays, but the operation may create a large output array of size n*m*d, which can be memory-intensive.
Mention alternatives like using einsum, chunking, or in-place operations to reduce memory usage when n and m are large.
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