My understanding differs and I might be wrong. Here's what I inferred:
Let's say you finetune a Mistral-7B. Now, there are hundreds of other fine-tuned Mistral-7B's, which means it's easy to find the universal subspace U of the weights of all these models combined. You can then decompose the weights of your specific model using U and a coefficient matrix C specific to your model. Then you can convert any operation of the type `out=Wh` to `out=U(C*x)` Both U and C are much smaller dimension that W and so the number of matrix operations as well as the memory required is drastically lower.
Let's say you finetune a Mistral-7B. Now, there are hundreds of other fine-tuned Mistral-7B's, which means it's easy to find the universal subspace U of the weights of all these models combined. You can then decompose the weights of your specific model using U and a coefficient matrix C specific to your model. Then you can convert any operation of the type `out=Wh` to `out=U(C*x)` Both U and C are much smaller dimension that W and so the number of matrix operations as well as the memory required is drastically lower.