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In the image above, if you wantbto be more similar toathan toc, which measure should you pick?
Dot product
Correct! The dot product is proportional to both the cosine and the
lengths of vectors. So even though the cosine is higher for the angle
formed bybandcthan for the angle formed
bybanda, the greater vector length ofamakes the dot product ofaandbgreater than the dot product ofbandc.
Cosine
The cosine depends only on the angle between vectors, and the smaller
angle \(\theta_{bc}\) makes \(\cos(\theta_{bc})\) larger than
\(\cos(\theta_{ab})\).
Euclidean distance
The distance \(\vec{bc}\) is smaller than \(\vec{ab}\), leavingbcloser tocthan toa.
You are calculating similarity for music videos. The length of the
embedding vectors of music videos is proportional to their popularity. You
switch from cosine to dot product to calculate similarity. How does
similarity between music videos change?
Popular videos becomemore similarto all videos in general.
Since the dot product is affected by the lengths of both vectors, the
large vector length of popular videos will make them more similar to all
videos.
Popular videos only becomemore similarto other popular videos.
Recall that the dot product is calculated as \(|a||b|\cos(\theta)\).
Assumingais a popular music video, we know its
embedding length, \(|a|\), is larger than that of unpopular videos.
The larger length increases the
dot product irrespective of the value of \(|b|\). Hence, popular videos
become more similar to all other videos, not just other popular videos.
Popular videos becomeless similarto each other than to
less popular videos.
Since dot product increases with vector length, and popular videos have
high vector length, the similarity measure will increase, not decrease.
No change.
Dot product is affected by vector length. The high vector length of
popular videos will change the (previously cosine) similarity measure.
Given the same setup as the previous question, suppose you switch back
to cosine from dot product. How does similarity between music videos change?
Popular videos becomeless similarto all videos.
Because cosine is not affected by vector length, the large vector
length of popular video embeddings does not contribute to similarity.
Thus, switching to cosine from dot product reduces the similarity
between popular videos and all other videos.
Popular videos becomemore similarto all videos.
Cosine is not affected by the vector lengths, so switching from dot
product will cause the similarities for all popular videos to decrease.
Popular videos becomemore similaronly to each other.
Cosine is not affected by the vector lengths, so switching from dot
product will cause the similarities for all popular videos to decrease.
No change.
Since cosine is not affected by vector length, while dot product is,
switching from dot product to cosine will change similarities.
[[["Easy to understand","easyToUnderstand","thumb-up"],["Solved my problem","solvedMyProblem","thumb-up"],["Other","otherUp","thumb-up"]],[["Missing the information I need","missingTheInformationINeed","thumb-down"],["Too complicated / too many steps","tooComplicatedTooManySteps","thumb-down"],["Out of date","outOfDate","thumb-down"],["Samples / code issue","samplesCodeIssue","thumb-down"],["Other","otherDown","thumb-down"]],["Last updated 2025-08-25 UTC."],[[["\u003cp\u003eDot product considers both angle and magnitude of vectors, making vectors with larger magnitudes more similar despite the angle between them.\u003c/p\u003e\n"],["\u003cp\u003eCosine similarity only considers the angle between vectors, disregarding their magnitudes.\u003c/p\u003e\n"],["\u003cp\u003eSwitching from cosine to dot product for similarity calculation makes popular items (with larger vector magnitudes) more similar to all other items.\u003c/p\u003e\n"],["\u003cp\u003eSwitching back from dot product to cosine similarity normalizes the effect of magnitude, making popular items less biased towards similarity with all other items.\u003c/p\u003e\n"]]],[],null,["# Check your understanding\n\nIn the image above, if you want `b` to be more similar to `a` than to `c`, which measure should you pick? \nDot product \nCorrect! The dot product is proportional to both the cosine and the lengths of vectors. So even though the cosine is higher for the angle formed by `b` and `c` than for the angle formed by `b` and `a`, the greater vector length of `a` makes the dot product of `a` and `b` greater than the dot product of `b` and `c`. \nCosine \nThe cosine depends only on the angle between vectors, and the smaller angle \\\\(\\\\theta_{bc}\\\\) makes \\\\(\\\\cos(\\\\theta_{bc})\\\\) larger than \\\\(\\\\cos(\\\\theta_{ab})\\\\). \nEuclidean distance \nThe distance \\\\(\\\\vec{bc}\\\\) is smaller than \\\\(\\\\vec{ab}\\\\), leaving `b` closer to `c` than to `a`. \nYou are calculating similarity for music videos. The length of the embedding vectors of music videos is proportional to their popularity. You switch from cosine to dot product to calculate similarity. How does similarity between music videos change? \nPopular videos become **more similar** to all videos in general. \nSince the dot product is affected by the lengths of both vectors, the large vector length of popular videos will make them more similar to all videos. \nPopular videos only become **more similar** to other popular videos. \nRecall that the dot product is calculated as \\\\(\\|a\\|\\|b\\|\\\\cos(\\\\theta)\\\\). Assuming `a` is a popular music video, we know its embedding length, \\\\(\\|a\\|\\\\), is larger than that of unpopular videos. The larger length increases the dot product irrespective of the value of \\\\(\\|b\\|\\\\). Hence, popular videos become more similar to all other videos, not just other popular videos. \nPopular videos become **less similar** to each other than to less popular videos. \nSince dot product increases with vector length, and popular videos have high vector length, the similarity measure will increase, not decrease. \nNo change. \nDot product is affected by vector length. The high vector length of popular videos will change the (previously cosine) similarity measure. \nGiven the same setup as the previous question, suppose you switch back to cosine from dot product. How does similarity between music videos change? \nPopular videos become **less similar** to all videos. \nBecause cosine is not affected by vector length, the large vector length of popular video embeddings does not contribute to similarity. Thus, switching to cosine from dot product reduces the similarity between popular videos and all other videos. \nPopular videos become **more similar** to all videos. \nCosine is not affected by the vector lengths, so switching from dot product will cause the similarities for all popular videos to decrease. \nPopular videos become **more similar** only to each other. \nCosine is not affected by the vector lengths, so switching from dot product will cause the similarities for all popular videos to decrease. \nNo change. \nSince cosine is not affected by vector length, while dot product is, switching from dot product to cosine will change similarities."]]
