What does (n,) mean in the context of numpy and vectors?

Question

I've tried searching StackOverflow, googling, and even using symbolhound to do character searches, but was unable to find an answer. Specifically, I'm confused about Ch. 1 of Nielsen's Neural Networks and Deep Learning, where he says "It is assumed that the input a is an (n, 1) Numpy ndarray, not a (n,) vector."

At first I thought (n,) referred to the orientation of the array - so it might refer to a one-column vector as opposed to a vector with only one row. But then I don't see why we need (n,) and (n, 1) both - they seem to say the same thing. I know I'm misunderstanding something but am unsure.

For reference a refers to a vector of activations that will be input to a given layer of a neural network, before being transformed by the weights and biases to produce the output vector of activations for the next layer.

EDIT: This question equivocates between a "one-column vector" (there's no such thing) and a "one-column matrix" (does actually exist). Same for "one-row vector" and "one-row matrix".

A vector is only a list of numbers, or (equivalently) a list of scalar transformations on the basis vectors of a vector space. A vector might look like a matrix when we write it out, if it only has one row (or one column). Confusingly, we will sometimes refer to a "vector of activations" but actually mean "a single-row matrix of activation values transposed so that it is a single-column."

Be aware that in neither case are we discussing a one-dimensional vector, which would be a vector defined by only one number (unless, trivially, n==1, in which case the concept of a "column" or "row" distinction would be meaningless).

Answer 1

In numpy an array can have a number of different dimensions, 0, 1, 2 etc.

The typical 2d array has dimension (n,m) (this is a Python tuple). We tend to describe this as having n rows, m columns. So a (n,1) array has just 1 column, and a (1,m) has 1 row.

But because an array may have just 1 dimension, it is possible to have a shape (n,) (Python notation for a 1 element tuple: see here for more).

For many purposes (n,), (1,n), (n,1) arrays are equivalent (also (1,n,1,1) (4d)). They all have n terms, and can be reshaped to each other.

But sometimes that extra 1 dimension matters. A (1,m) array can multiply a (n,1) array to produce a (n,m) array. A (n,1) array can be indexed like a (n,m), with 2 indices, x[:,0] where as a (n,) only accepts x[0].

MATLAB matrices are always 2d (or higher). So people transfering ideas from MATLAB tend to expect 2 dimensions. There is a np.matrix subclass that supposed to imitate that.

For numpy programmers the distinctions between vector, row vector, column vector, matrix are loose and relatively unimportant. Or the use is derived from the application rather than from numpy itself. I think that's what's happening with this network book - the notation and expectations come from outside of numpy.

See as well this answer for how to interpret the shapes with respect to the data stored in ndarrays. It also provides insight on how to use .reshape: https://stackoverflow.com/a/22074424/3277902

Answer 2

(n,) is a tuple of length 1, whose only element is n. (The syntax isn't (n) because that's just n instead of making a tuple.)

If an array has shape (n,), that means it's a 1-dimensional array with a length of n along its only dimension. It's not a row vector or a column vector; it doesn't have rows or columns. It's just a vector.