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Equation 5 · Part 24 · A History of Multimodal AI

Denominator: sum_j=1^Bexp(langle u_j, v_irangle/τ)

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=1Bexp⁡(⟨uj,vi⟩/τ)\sum_{j=1}^{B}\exp(\langle u_j, v_i\rangle/\tau)

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

sumjm_j=1^Bexp(langle uju_j, viv_irangle/τ) 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

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.