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

What does this equation mean?

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)},

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start with1
Divide byB
This relates toL
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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L\mathcal{L}

Symbol L

In its softmax form the objective.

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BB

Symbol B

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

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ii

Symbol i

i appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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uiu_i

Symbol u_i

the image embedding.

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viv_i

Symbol v_i

the text embedding.

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τ\tau

Symbol τ

the learned temperature.

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jj

Symbol j

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

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vjv_j

Symbol v_j

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

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

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

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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11

Numerator: 1

The complete quantity above the fraction bar.

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i=1i=1

Starting index or lower bound: i=1

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.

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BB

Ending index or upper bound: B

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

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exp⁡(⟨ui,vi⟩/τ)\exp(\langle u_i, v_i \rangle / \tau)

Numerator: exp(langle u_i, v_i rangle / τ)

The complete quantity above the fraction bar.

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∑j=1Bexp⁡(⟨ui,vj⟩/τ)\sum_{j=1}^{B} \exp(\langle u_i, v_j \rangle / \tau)

Denominator: sum_j=1^B exp(langle u_i, v_j rangle / τ)

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

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j=1j=1

Starting index or lower bound: j=1

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.

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BB

Ending index or upper bound: B

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

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

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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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 training examples that network was trained on [ 1 ] . A later variant replaces the softmax with a pairwise sigmoid loss that does not require a global view of the batch’s pairwise similarities for normalisation, and the authors report training a model to 84.5% ImageNet zero-shot accuracy in two days on four TPUv4 chips, while also finding that the benefit of larger batches saturates around 32,000 examples [ 6 ] .

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