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Equation 6 · Part 2 · One Model, Many Modalities: What Multimodal Systems Actually Share

Symbol B

L=−1B∑i=1Blog⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨ui,vj⟩/τ),\mathcal{L} = -\frac{1}{B} \sum_{i=1}^{B} \log \frac{\exp(\langle u_i, v_i \rangle / \tau)}{\sum_{j=1}^{B} \exp(\langle u_i, v_j \rangle / \tau)},
BB

What this part means

B occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Its job in the formula

B occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

Alignment is the step that made cross-modal retrieval work, and it has an unusually clean formulation. Contrastive language–image pretraining takes a batch of B image–text pairs, encodes each side separately into a normalised vector, and trains both encoders so that matched pairs score higher than mismatched ones. In its softmax form the objective is L=−1B∑i=1Blog⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨ui,vj⟩/τ)\mathcal{L} = -\frac{1}{B} \sum_{i=1}^{B} \log \frac{\exp(\langle u_i, v_i \rangle / \tau)}{\sum_{j=1}^{B} \exp(\langle u_i, v_j \rangle / \tau)}. with image embedding uiu_i , text embedding viv_i , and a learned temperature τ\tau .…

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A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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