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Equation 11 · Adapters, Native Pretraining, and Unified Tokens: The Main Multimodal Architectures, Compared

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ztz_t

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allowed to denote. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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ztz_t

Symbol z_t

allowed to denote.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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The same cross-entropy objective a text-only language model uses is the entire training objective here — the only thing that changed is what ztz_t is allowed to denote. Whether index ztz_t names a subword, a 16-by-16 image patch code, or a discretised robot joint angle is invisible to the loss function; the difficulty this section documents is not in the objective but in keeping training numerically stable once the vocabulary spans quantities of such different statistical character.

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