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The Gram Matrix Is Positive Semi-Definite (property)

Mon, 21 Sep 2026 08:52:00 +0900

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The matrix of pairwise inner products of a set of vectors has a property that constrains what similarity patterns are possible: it is symmetric and positive semi-definite, so its eigenvalues are non-negative and its diagonal entries are the squared lengths of the vectors.

Cross-Entropy Is Convex in the Logits (property)

Mon, 21 Sep 2026 08:02:00 +0900

Convexity of a loss in the parameters is rare in deep learning, but convexity in the scores is not, and the cross-entropy of a softmax has it. The property is what guarantees that the inner optimisation — fitting the output layer, probing a frozen representation — has no local minima to fall into.