Thanks for the follow up. I've been following your circuits thread for several years now. I find the linear representation hypothesis very compelling, and I have a draft of a review for Toy Models of Superposition sitting in my notes. Circuits I find less compelling, since the analysis there feels very tied to the transformer architecture in specific, but what do I know.
Re linear representation hypothesis, surely it depends on the architecture? GANs, VAEs, CLIP, etc. seem to explicitly model manifolds. And even simple models will, due to optimization pressure, collapse similar-enough features into the same linear direction. I suppose it's hard to reconcile the manifold hypothesis with the empirical evidence that simple models will place similar-ish features in orthogonal directions, but surely that has more to do with the loss that is being optimized? In Toy Models of Superposition, you're using a MSE which effectively makes the model learn an autoencoder regression / compression task. Makes sense then that the interference patterns between co-occurring features would matter. But in a different setting, say a contrastive loss objective, I suspect you wouldn't see that same interference minimization behavior.
Re linear representation hypothesis, surely it depends on the architecture? GANs, VAEs, CLIP, etc. seem to explicitly model manifolds. And even simple models will, due to optimization pressure, collapse similar-enough features into the same linear direction. I suppose it's hard to reconcile the manifold hypothesis with the empirical evidence that simple models will place similar-ish features in orthogonal directions, but surely that has more to do with the loss that is being optimized? In Toy Models of Superposition, you're using a MSE which effectively makes the model learn an autoencoder regression / compression task. Makes sense then that the interference patterns between co-occurring features would matter. But in a different setting, say a contrastive loss objective, I suspect you wouldn't see that same interference minimization behavior.