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Indexed notation is commonly used to represent a subsequence of elements, such as vectors. A sequence starting from index $$m$$ and ending at index $$len$$ is written as $$x_m, x_{m+1}, \dots, x_{len-1}, x_{len}$$. In the context of sequence models, this notation is frequently used for different components. For instance, a sequence of hidden states is often denoted as $$h_m, h_{m+1}, \dots, h_{len}$$, while a sequence of embeddings might be represented as $$\mathbf{e}_m, \mathbf{e}_{m+1}, \dots, \mathbf{e}_{len}$$.

Indexed Sequence Notation

A sequence of embedding vectors, indexed from a starting point $m$ to an ending point $len$, is represented as: $\mathbf{e}_m, \mathbf{e}_{m+1}, \mathbf{e}_{m+2}, \dots, \mathbf{e}_{len-1}, \mathbf{e}_{len}$. This notation serves as a specific application of indexed sequence representation for embedding vectors within sequence models.

Example of Indexed Embedding Sequence Notation

A sequence of hidden states, indexed from an initial point $$m$$ to a final point $$len$$, can be represented as a series of vectors: $$\mathbf{h}_m, \mathbf{h}_{m+1}, \mathbf{h}_{m+2}, \dots, \mathbf{h}_{len-1}, \mathbf{h}_{len}$$. This notation is commonly used in sequence models to denote a specific subsequence of hidden state vectors.

Example of Indexed Hidden State Sequence Notation

A model generates a sequence of 12 output vectors, denoted as $\mathbf{o}_1, \mathbf{o}_2, \dots, \mathbf{o}_{12}$. Using the standard indexed notation, write out the subsequence that starts with the fifth vector and ends with the ninth vector.

Applying Indexed Sequence Notation

A sequence of vectors is represented by the notation $\mathbf{e}_5, \mathbf{e}_6, \dots, \mathbf{e}_{11}$. This notation represents a subsequence containing exactly 6 vectors.

A model processes a sequence of seven input vectors, represented as $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4, \mathbf{v}_5, \mathbf{v}_6, \mathbf{v}_7$. Which of the following correctly uses indexed notation to represent the subsequence containing only the third, fourth, and fifth vectors?

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