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Equation 5 · Part 25 · A History of Multimodal AI

Starting index or lower bound: j=1

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],
j=1j=1

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

Its job in the formula

j=1 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

Radford and colleagues’ 2021 paper, “Learning Transferable Visual Models From Natural Language Supervision,” is usually remembered simply as CLIP, and the compression loses the specific thing that made it a turning point rather than an incremental improvement. 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…

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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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