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Equation 5 · Part 2 · The Hardest Unsolved Problems in Mechanistic Interpretability

Symbol j

di(m2)≈∑j=1m1gj dj(m1),∥g∥0≪m1,m1<m2,d_i^{(m_2)} \approx \sum_{j=1}^{m_1} g_j\, d_j^{(m_1)}, \qquad \lVert g \rVert_0 \ll m_1, \quad m_1 < m_2,
jj

What this part means

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

Its job in the formula

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

The passage around this formula

Leask, Nanda and colleagues then tested the assumption directly with two new techniques, and the result undercuts the idea that there is a “right” dictionary size waiting to be found by scaling further. Stitching the dictionaries of differently sized sparse autoencoders together, they show that larger dictionaries recover latents genuinely missing from smaller ones — the smaller dictionary is incomplete. Training a second, “meta” sparse autoencoder on the decoder directions of a first one, they show that what looks like a single, atomic feature in a large dictionary is itself well approximated by a sparse combination of directions from a smaller one — a reported example decomposes an…

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