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Equation 8 · Part 6 · Training on Your Own Output: Synthetic Data and What It Does to a Distribution

=

p^k+1=F(D0∪S1∪⋯∪Sk),\hat{p}_{k+1} = \mathcal{F}\left( D_0 \cup S_1 \cup \cdots \cup S_k \right),
=

What this part means

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Change one term and the analysis changes with it. Consider instead p^k+1=F(D0∪S1∪⋯∪Sk)\hat{p}_{k+1} = \mathcal{F}\left( D_0 \cup S_1 \cup \cdots \cup S_k \right). where D0D_0 is the original real corpus and SjS_j the synthetic output of generation j . Gerstgrasser and colleagues make exactly this substitution and report that while replacing real data with each generation’s synthetic data does tend toward collapse, accumulating successive generations alongside the original real data avoids it — across transformers, diffusion models and variational autoencoders — and they prove that under accumulation the test error has a finite upper bound independent of the number of iterations [ 2 ] . The accumulating case is also the more accurate description of the actual web, which…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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