Geometry and Linear Algebraic Operations

The classifier used in the text seems rather unnatural. A more natural way is to flatten the images, normalize them, then take the dot product as discussed in the text as a measure of similarity. The predicted label is the label of the average image that is more similar to the test image, hence the argmax.

# normalize matrices using broadcasting
W = torch.stack([ave_0.flatten().t(), ave_1.flatten().t()], dim=1)
W = W / torch.norm(W, dim=0).reshape(1, -1)
X_test = X_test.reshape(-1, 784)
X_test = X_test / torch.norm(X_test, dim=1).reshape(-1, 1)

# predict and evaluate
y_pred = torch.argmax(X_test @ W, dim=1)
print((y_test == y_pred).type(torch.float).mean())

This obtains an accuracy of ~0.95. :grinning:

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What’s the interpretation of A^4 in exercise 7?

I’ve typed up solutions to the exercises in this chapter here (see bottom of the notebook). I’m still seeking guidance on exercise 7.

Any help will be greatly appreciated!

Hey, maybe a little late here; but as far as I understand it, A^4 means a matrix with 4 dimensions.

Based on this section, I think A^4 means A * A * A * A, that is, matrix A multiplied 4 times by itself. It’s the power operator (A to the power of 4) for matrices, if I’m not mistaken.

In Section 18.1.3. Hyperplanes, it says - “The set of all points where this is true is a line at right angles to the vector w”. Which condition is “this” referring to here? Is it referring to all vectors (or rather, points here) whose projection is equal to 1/||w||?