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Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. The matrix of pairwise inner products of a set of vectors has a property that constrains what similarity patterns are possible: it is symmetric and positive semi-definite, so its eigenvalues are non-negative and its diagonal entries are the squared lengths of the vectors. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. The largest singular value of a weight matrix is one of the numbers used to judge how a layer transforms its input, and it is usually estimated rather than computed: repeated multiplication by the matrix makes any starting vector align with the leading eigenvector, after which the Rayleigh quotient gives the value. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. Layer normalisation rescales each token’s activations using statistics computed across the features of that token alone, which is what separates it from batch normalisation: nothing in the computation depends on the other examples in the batch. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. A claim is a statement put forward as true and argued for, not yet established as a theorem. The one here concerns temperature scaling: raising the temperature flattens a distribution, and it cannot make the most probable token more probable than it already was. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. Convexity of a loss in the parameters is rare in deep learning, but convexity in the scores is not, and the cross-entropy of a softmax has it. The property is what guarantees that the inner optimisation — fitting the output layer, probing a frozen representation — has no local minima to fall into. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. Gradient accumulation is the standard way to train with a batch larger than memory allows: several small batches are processed in turn and their gradients added before any parameter moves. The gradient of a sum is the sum of the gradients, so the result equals one step on the concatenated batch as long as the per-batch losses are averaged. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. A language model ends in a softmax: the raw scores of the last layer become a probability distribution over the vocabulary, and everything downstream — the training loss, sampling, beam search — reads that distribution rather than the scores themselves. ...

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct. Long-context attention is usually chosen on cost rather than on quality. What decides whether a variant is usable at a given sequence length is how the score cost and the per-token cache grow with \(T\), and both are easiest to compare side by side in one table rather than across several paragraphs. ...