Consider the formula `(i - j) / 10000^(2k/d)`, which is a core component for generating positional signals. Explain the mechanism by which this formula causes the resulting signals to have a longer wavelength (lower frequency) for higher-indexed dimensions (larger `k`) compared to lower-indexed dimensions (smaller `k`).

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This formula calculates a value based on the relative distance `(i - j)` between two sequence positions. This distance is then scaled by a denominator, $10000^{2k/d}$, which serves as a wavelength term and is a core component of the sinusoidal positional encoding scheme from the original Transformer model. The full formula is: $$(i-j)/10000^{2k/d}$$ In this expression, `k` typically represents the dimension index within the embedding, and `d` is the total dimensionality of the model's embeddings. The resulting value is commonly used as an input to sine and cosine functions to generate a final positional encoding vector or bias.

Formula for Relative Position Scaled by Sinusoidal Wavelength

In the formula `(i - j) / 10000^(2k/d)`, used for calculating a scaled relative position, `k` represents a specific dimension index within a `d`-dimensional embedding. Analyze the relationship between the dimension index `k` and the denominator `10000^(2k/d)`. What is the effect of this relationship on the resulting positional signal?

Based on the provided scenario, which part of the positional signal calculation is most likely misconfigured, and what is the nature of the error? Explain your reasoning by connecting the formula's components to the model's symptoms.

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