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Equation 6 · Part 10 · One Model, Many Modalities: What Multimodal Systems Actually Share

fraction

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)},
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

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 . Radford and colleagues showed that this simple pre-training task, applied to 400 million image–text pairs collected from the internet, matched the accuracy of the original ResNet-50 on ImageNet zero-shot without using any of the 1.28 million…

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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

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