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Equation 5 · Part 10 · The Hardest Unsolved Problems in Mechanistic Interpretability

subscript

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,
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

Open the illustrated subscripts: which member of a family? guide →

Sources cited in the surrounding passage

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