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The paper controls the convergence rate of the NPMLE by sharply characterizing the bracketing entropy of finite Gaussian mixture classes under local Hellinger distance.
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10/22/2026
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UnverifiedBracketing entropy is defined as the logarithm of the minimum number of brackets needed to cover a function class, and is commonly used to bound the supremum of empirical processes via the Dudley entropy integral, thereby controlling the convergence rate of the MLE.75% similarUnverifiedThe paper proves that when the true mixture measure satisfies the sub-Gaussian assumption, the unconstrained NPMLE achieves the minimax optimal rate log(n)/n under squared Hellinger distance.72% similarUnverifiedCross-entropy loss often converges faster than MSE in classification tasks because, when paired with softmax output, gradients do not vanish in the saturation region.71% similarUnverifiedFor a stationary ergodic process, the entropy rate limit exists and equals the conditional entropy limit lim H(X_n|X_{n-1},...,X_1).66% similarUnverifiedComputing Graph Entropy typically involves constructing a k-nearest-neighbor graph and computing eigenvalues or random-walk entropy; its complexity is higher than simple distance averaging, but it is feasible on medium-sized datasets.66% similar
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