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Let's say I feed a neural network with multiple string sentences that mean roughly the same thing but are formulated differently. Will the neural network be able to derive patterns of meaning in the same way that it is being done with images. Is this an approach currently used in Natural Language Processing?

With images of dogs the neural network will get the underlying patterns that define a dog. Could it be the same thing with sentences?

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No, it will not derive patterns of meaning, as the network has no understanding of language. What will happen, is that it picks up surface features (usually letter sequences) which are common between sentences with the same (or a similar) meaning.

This approach is often used in chatbots for intent recognition. Sometimes it picks up subtle patterns that humans would not notice, but they are also often not reliable: you get wrong classifications without knowing why.

Having said that, it works fine if you want to distinguish between a limited number of different intents (you don't even need a lot of training examples — 4 to 5 examples are often sufficient if they are well-selected). In this fairly limited scenario yes, otherwise no.

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