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Efficient Deep Approximation of GMMs
The universal approximation theorem states that any regular function can be approximated closely using a single hidden layer neural network.
Efficient Deep Learning of GMMs
We show that a collection of Gaussian mixture models (GMMs) in $R^{n}$ can be optimally classified using $O(n)$ neurons in a neural network with two hidden layers (deep neural network), whereas in contrast, a neural network with a single hidden layer (shallow neural network) would require at least $O(\exp(n))$ neurons or possibly exponentially large coefficients.
Linear Time Clustering for High Dimensional Mixtures of Gaussian Clouds
Consequently, the expected overall running time of the algorithm is linear in $n$ and quasi-linear in $p$ at $o(\ln{p})O(np)$, and the sample complexity is independent of $p$.
A New Family of Near-metrics for Universal Similarity
We show that for structured data including categorical and continuous data, the near-metrics corresponding to normalized forward k-step diffusion (k small) work as one of the best performing similarity measures; for vector representations of text and images including those extracted from deep learning, the near-metrics derived from normalized and reverse k-step graph diffusion (k very small) exhibit outstanding ability to distinguish data points from different classes.
