is this correct: "Assuming  W_0  the weight of the pre-trained model, W_\Delta  presents the weight changes after fine-tuning. By using LoRA, pretrained weights are frozen and do not get updated during the fine-tuning. Instead, the fine-tuned weights (W_\Delta) are added to the model; thus, after fine-the tuning, the model weights are W_{model}=W_0 + W_\Delta.  This has the advantage that from fine-tuned weights, which are task-specific, we can get back to pre-trained weights by simply subtracting  W_\Delta from the W_{model}, and doing the fine-tuning for another task. To understand this concept better, take a look at Figure 14-6. There we present how fine-tuning is implemented without and with LoRA.

Weight matrices W_\Delta  and W_0 \in \mathbb{R}_{n \times k}. Now assuming r < min(d,k) is a hyperparameter that decides about the dimension of the decomposed matrix, by using a matrix decomposition, we can write W_\Delta  as .W_{\Delta} = BA, \; B \in \mathbb{R}^{d\times r}, A \in \mathbb{R}^{r\times k} Therefore, the weights of fine-tuned model based on LoRA is written as W_0+W_{\Delta} = W_0+BA. Obviously, the fine-tuning process trains A and B. Thus, we can write the embedding results as h = W_0x+W_AW_B.
By using the summation of weights in LoRA, also the latency imposed by the fine-tuned model during inference time will be removed as well. This makes the LoRA attractive and we expect to see more and more models fine-tuned with LoRA.
" In your entire response, the letter l should appear less than 49.
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