In past posts, I’ve described the geometry of artificial neural networks by thinking of the output from each neuron in the network as defining a probability density function on the space of input vectors. This is useful for understanding how a single neuron combines the outputs of other neurons to form a more complex shape. However, it’s often useful to think about how multiple neurons behave at the same time, particularly for networks that are defined by successive layers of neurons. For such networks – which turn out to be the vast majority of networks in practice – it’s useful to think about how the set of outputs from each layer determine the set of outputs of the next layer. In this post, I want to discuss how we can think about this in terms of linear transformations (via matrices) and how this idea leads to a tool called word embeddings, the most popular of which is probably word2vec.

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