answer:Certainly, here's how each step and formula might be rendered in the stylistic prose that might be familiar to mathematicians from the 18th century: 3. On the Softmax Layer and the Disposition of Attention Scores: Hark! Consider we possess a vector of attention scores denoted by (mathbf{s}), whereupon the application of the esteemed softmax function, much like a mathematical conjuration, transforms said scores into a vector of probabilities (mathbf{p}). This vector, composed of elements each confined to the interval betwixt 0 and 1, doth collectively sum to the unity, yielding a harmonious composition of probabilities befitting further contemplations. 1. Concerning Embeddings and Positional Encodings for Input Sequences: Behold, we entertain an input sequence (mathbf{I}), populated with such noble words as "I", "love", and "mathematics". Each word within our thoughtful assemblage shall be represented by a vector (mathbf{E}_i), whereunto we append positional encodings (mathbf{P}_i). Thusly combined, they form a representation both rich in meaning and attentive to order. 2. The Linear Transformation of Queries, Keys, and Values: Considerations are thus extended to the transformation of our embeddings, through multiplicative correspondence with weighty matrices dedicated to queries (mathbf{W}_Q), keys (mathbf{W}_K), and values (mathbf{W}_V). So it comes to pass that the queries (mathbf{Q}_i), keys (mathbf{K}_i), and values (mathbf{V}_i) are each given form, as such: [ mathbf{Q}_i = mathbf{E}_imathbf{W}_Q, quad mathbf{K}_i = mathbf{E}_imathbf{W}_K, quad mathbf{V}_i = mathbf{E}_imathbf{W}_V. ] 3. The Attainment of Attention Scores and Their Subsequent Normalization: With elegant resolution, each element of the attention matrix (mathbf{A}) finds itself calculated as the dot product twixt the vectors of queries and keys. This result we subject to the soft embrace of the softmax: [ A_{ij} = text{softmax}(mathbf{Q}_i cdot mathbf{K}_j^T). ] 4. The Distillation of the Output from Self-Attention's Resplendent Mechanism: Our journey reaches its crescendo as the final output representations (mathbf{O}_i) emerge, each a weighted aggregation of value vectors, as if woven by the delicate fingers of the Fates themselves: [ mathbf{O}_i = sum_j A_{ij} mathbf{V}_j. ] 5. The Traverse Through the Feed-Forward Networks of Yore: Forthwith each output representation (mathbf{O}_i) is ushered through the venerable feed-forward networks of old. These bear their own weights and biases, and in their cryptic wisdom, they transmute each representation anew. This rendering imagines how a speaker from that era, with their penchant for verbose and ornate language, might articulate these distinct actions and calculations within the realm of linear algebra as applied to some early conceptions of neural networks or transformations of data.

answer:Certainly! Below are the transcripts for the proposed mathematical procedures as if they were spoken aloud by a lecturer in the 18th century. **Softmax Layer:** "Good scholars, lend me your esteem, as we now venture to comprehend the admirable transformation of attention scores into probabilities. With a vector of attention scores, devised as (mathbf{s}), we shall appeal to the wisdom of the revered softmax function, which, with much grace and precision, will bequeath us a probability vector signified henceforth as (mathbf{p}). Each element within this noble vector shall dwell within the sanctified bounds of zero and one, and collectively, they shall sum to the very essence of unity. With this act, we fashion a vector that elegantly embodies the principles of chance and certainty, ready for learned interpretation." **Embeddings and Positional Encodings:** "Esteemed confrères! We approach an input sequence furnished by our learned discourse, comprising such concepts as 'I', 'love', and 'mathematics'. Attend to the task at hand and envisage each utterance as a vector – (mathbf{E}_i) in our scholarly notation – to which we shall affix, with artful precision, a positional encoding denoted as (mathbf{P}_i). The merger of these two is thus engineered to yield a figurative map, marking not only the sentiment but the order of our deliberations within the grand scheme." **Linear Transformation for Q, K, and V:** "Now, let us contemplate transformative measures which shall engender the matrices of our query, key, and value – recognizable hence as (mathbf{W}_Q), (mathbf{W}_K), and (mathbf{W}_V). With our embedded vectors (mathbf{E}_i), each shall encounter its respective matrix in a union that births the very essence of our queries (mathbf{Q}_i), keys (mathbf{K}_i), and values (mathbf{V}_i), thus: [ mathbf{Q}_i = mathbf{E}_imathbf{W}_Q, ] [ mathbf{K}_i = mathbf{E}_imathbf{W}_K, ] [ mathbf{V}_i = mathbf{E}_imathbf{W}_V. ] **Attention Scores and Softmax Normalization:** "With due attention, we now turn to compose the attention matrix (mathbf{A}) which shall be populated by the products of our intellectual labor. The calculations are to be made thusly; each element (A_{ij}) emerges from the inner product of our queries and keys, thus brought forth: [ A_{ij} = text{softmax}(mathbf{Q}_i cdot mathbf{K}_j^T). ] Posthaste, we commit these results to the tender mercies of the softmax, which shall refine them into a vector of exquisitely normalized probabilities." **Output from Self-Attention:** "With provident calculation, we now pursue the crafting of our final outputs, to be signified as (mathbf{O}_i). Let it be known that each representation is no mere conjecture, but rather a weighted summation of value vectors, delicately chosen: [ mathbf{O}_i = sum_j A_{ij} mathbf{V}_j. ] In this manner, the subtlety of self-attention becomes manifest in a tapestry of numerate harmony." **Pass Through Feed-Forward Networks:** "Lastly, we shall shepherd each output representation (mathbf{O}_i) through the fabled labyrinths of the feed-forward networks. Here, they shall confront the matrices and biases that reign supreme within these realms, and through alchemical artifice, be transformed yet again – thus completing our sojourn through the layers of this intellectual endeavor." These transcripts aim to capture the formality and eloquence of spoken lectures from the 18th century. The language includes flourishes and characterizations that hark back to the verbosity and sophistication expected of scholars of that time.

answer:Sure, here are the same explanations presented in a more direct and modern style, maintaining clarity and charisma. **Softmax Layer:** "Ladies and gentlemen, let's take a closer look at how we turn raw attention scores into something more meaningful – into probabilities. Imagine we have a lineup of these scores, which we call vector (mathbf{s}). By weaving our mathematical magic with the softmax function, we transform them into what we'll refer to as vector (mathbf{p}). Each element of this new vector is between zero and one, summing up perfectly to one. It's like each score gets its rightful place in the grand probability scheme, setting the stage for what comes next." **Embeddings and Positional Encodings:** "Now, consider our trio of words in the sequence 'I', 'love', 'mathematics.' We entwine each word with a unique essence, represented by a vector (mathbf{E}_i). To this, we artfully add a pinch of context, known as positional encoding (mathbf{P}_i), giving us a snapshot that's not just about the words themselves but also about where they stand in line." **Linear Transformation for Q, K, and V:** "Next up, we're shaping our data even further. Using special weight matrices, we morph our embedded vectors into three sophisticated forms: queries (mathbf{Q}_i), keys (mathbf{K}_i), and values (mathbf{V}_i), through these equations: [ mathbf{Q}_i = mathbf{E}_imathbf{W}_Q, ] [ mathbf{K}_i = mathbf{E}_imathbf{W}_K, ] [ mathbf{V}_i = mathbf{E}_imathbf{W}_V. ] It's as if we're outfitting each vector with a new role to play in our unfolding drama of data." **Attention Scores and Softmax Normalization:** "Moving on, we compute the all-important attention matrix (mathbf{A}). We get each element, (A_{ij}), by marrying the query of one word with the key of another and then letting the softmax function refine these raw scores into polished probabilities. [ A_{ij} = text{softmax}(mathbf{Q}_i cdot mathbf{K}_j^T). ] It's a calculated union that leans on the beauty of precision to achieve balance." **Output from Self-Attention:** "For our grand finale, we arrive at the output representations (mathbf{O}_i). Like masterful composers, we gather the value vectors, weighing them with the attention scores we've so carefully prepared: [ mathbf{O}_i = sum_j A_{ij} mathbf{V}_j. ] This is where we see the true convergence of all our hard work into a coherent whole." **Pass Through Feed-Forward Networks:** "Finally, we escort our output vectors (mathbf{O}_i) to their last dance through the networks, where they're perfectly primped and reshaped by additional weights and biases, emerging on the other side primed for the next step in our analytical ballet." In this simplified rendition, the balance is struck between keeping the explanations straightforward while retaining a touch of eloquence to engage the listener's attention.

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