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Equation 5 · Part 22 · A History of Multimodal AI

Ending index or upper bound: B

L=−12B∑i=1B[log⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨ui,vj⟩/τ)+log⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨uj,vi⟩/τ)],\mathcal{L} = -\frac{1}{2B}\sum_{i=1}^{B}\left[\log\frac{\exp(\langle u_i, v_i\rangle/\tau)}{\sum_{j=1}^{B}\exp(\langle u_i, v_j\rangle/\tau)} + \log\frac{\exp(\langle u_i, v_i\rangle/\tau)}{\sum_{j=1}^{B}\exp(\langle u_j, v_i\rangle/\tau)}\right],
BB

What this part means

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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

…The method itself is not exotic: encode an image, encode its paired caption, and train both encoders so that the true pairing scores higher than every mismatched pairing drawn from the same batch. For a batch of B image-text pairs with normalised image embedding uiu_i , text embedding viv_i , and a learned temperature τ\tau , the symmetric form of the objective is L=−12B∑i=1B[log⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨ui,vj⟩/τ)+log⁡exp⁡(⟨ui,vi⟩/τ)∑j=1Bexp⁡(⟨uj,vi⟩/τ)]\mathcal{L} = -\frac{1}{2B}\sum_{i=1}^{B}\left[\log\frac{\exp(\langle u_i, v_i\rangle/\tau)}{\sum_{j=1}^{B}\exp(\langle u_i, v_j\rangle/\tau)} + \log\frac{\exp(\langle u_i, v_i\rangle/\tau)}{\sum_{j=1}^{B}\exp(\langle u_j, v_i\rangle/\tau)}\right]. an image-to-text and a text-to-image cross-entropy averaged together, each treating every other pairing in the batch as a…

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Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the surrounding passage

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