You are building an anomaly-detection service for a wearable device that streams 3-D acceleration vectors x ∈ ℝ³. Because the sensor can be mounted in any orientation, the raw data may later be multiplied by an unknown orthonormal rotation matrix R before they reach your model. You need a distance function d(x, y) whose numerical value stays exactly the same when evaluated on the rotated vectors (Rx, Ry). Which of the following commonly used distance metrics fails to meet this rotation-invariance requirement and therefore should be avoided in this situation?
Rotation (an orthonormal transform) preserves inner-product-based quantities such as vector length and the angle between vectors. As a result, measures derived from the Euclidean norm (including the squared Euclidean form that appears in Gaussian radial functions) and the cosine distance remain unchanged after any rigid rotation. In contrast, the Manhattan (L1) distance adds absolute coordinate differences along fixed axes; rotating the coordinate system changes those coordinate-wise differences and therefore changes the L1 distance between the same two physical vectors. Because the Manhattan metric depends on axis orientation, it violates the stated requirement, whereas the other three options do not.
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What is the difference between Euclidean (L2) and Manhattan (L1) distance?
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What does it mean for a distance metric to be rotation invariant?
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What is an orthonormal rotation matrix, and why is it significant in this problem?