Recurrent Neural Network Hidden Vector

How is it possible to add two vectors of different sizes for the hidden vector formula? Simple vector/matrix addition is defined only for vectors/matrices of the same size. If the hidden vector and input vector are different sizes (ie. number of rows for the column vectors), then how is addition defined between them?

I am having difficulty understanding the following problem in the Artificial Neural Networks course (Recurrent Neural Networks Quiz 1 Problem 10).

The problem is the following:

"If our input vector has size 5, our hidden vector has size 10, and our output vector has size 7, how many parameters constitute our recurrent neural network? Remember that the recurrence is

Note by Matt Bryk
1 week, 4 days ago

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Hi Matt;

Thank you for asking this question.

You are correct to note that vectors cannot be added unless their dimensions match. This is the reason why we need to look at the transformation \(W_{hx}\). In other words, this matrix should be such that it can modify the input vector from size 5 to size 10 with some suitable transform.

Agnishom Chattopadhyay Staff - 1 week, 2 days ago

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Thank you for the response Agnishom. In other words, we choose the number of rows and columns, for Whx and Whh, such that the products of matrices (Whx)(Xt) and (Whh)(Ht-1) have the same dimensions? So that way we can now sum the products (Whx)(Xt) and (Whh)(ht-1).

Are the dimensions of the input vector and hidden vector allowed to change through time? For example, can ht have different dimensions than ht-1? Alternatively, can we have xt with different dimensions than xt-1? If yes, how do we construct Whx and Whh with changing input and hidden vector dimensions?

Matt Bryk - 6 days, 10 hours ago

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Hi Matt, Your first statement looks correct, the recurrence relation works because we pick out our matrices to make sure all products match.

Changing our dimensions with time is a very interesting thought, but in our case, because of how we've defined recurrent networks, it is not possible. The dimensions of our vectors can't change with time because the transformation matrices in our recurrence relation are constant throughout all time steps. They can only ever take in the same number of inputs and outputs, so there's no way to increase or decrease the dimensions of our hidden or input vectors.

Theoretically, you could define many transformation matrices, or come up with some novel way of handling the problem, but this isn't an approach that's been studied, and the architecture would no longer be a normal recurrent network. We can't say much about how you'd go about constructing this, or what its properties would be.

Andrew Dickson Staff - 2 days, 16 hours ago

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