34. Distance Matrices, Procrustes Problem | Summary and Q&A

TL;DR

The video discusses the failure of the triangle inequality and introduces the Procrustes' problem as a method to find the best orthogonal transformation between two sets of vectors.

Key Insights

• 🔺 The triangle inequality is violated when distances in a given matrix violate the condition, leading to the failure of finding satisfactory points.
• 😫 The Procrustes' problem addresses the need for transforming one set of vectors to match another set, seeking the best orthogonal transformation based on the Frobenius norm.
• 🍹 The Frobenius norm can be computed using different expressions, including the sum of squares, A transpose A, AA transpose, or the sum of squares of singular values.
• 🖐️ Traces and eigenvalues play a significant role in understanding properties of matrices and the Frobenius norm.

Transcript

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Questions & Answers

Q: What happens when the triangle inequality fails?

When the distances violate the triangle inequality, the matrix that connects the distance matrix to the matrix of dot products fails to be positive definite, making it impossible to find the desired points.

Q: How does the Procrustes' problem aim to solve the discrepancy between two sets of vectors?

The Procrustes' problem seeks to find the best orthogonal matrix that can transform one set of vectors to closely match another set of vectors. By minimizing the Frobenius norm, the closest transformation can be achieved.

Q: What are the different formulas for computing the Frobenius norm?

The Frobenius norm can be computed as the sum of squares of all entries in the matrix, as the trace of the matrix squared, or as the sum of squares of singular values.

Q: How are orthogonal matrices and singular values related?

Orthogonal matrices do not change the singular values of a matrix when multiplied, thus preserving the Frobenius norm. Similarly, changing the order of matrices in the trace operation does not affect the Frobenius norm.

Summary & Key Takeaways

• The triangle inequality fails when given distances violate the inequality, which leads to the failure of finding a matrix that satisfies the given distances.

• The Procrustes' problem involves finding an orthogonal matrix that can best transform one set of vectors to match another set of vectors, minimizing the Frobenius norm.