How Many Features Can a Language Model Store Under the Linear Representation Hypothesis?

We introduce a mathematical framework for the linear representation hypothesis (LRH), which asserts that intermediate layers of language models store features linearly. We separate the hypothesis into two claims: linear representation (features are linearly embedded in neuron activations) and linear accessibility (features can be linearly decoded). We then ask: How many neurons $d$ suffice to both linearly represent and linearly access $m$ features? Classical results in compressed sensing imply that for $k$-sparse inputs, $d = O(k\log (m/k))$ suffices if we allow non-linear decoding algorithms (Candes and Tao, 2006; Candes et al., 2006; Donoho, 2006). However, the additional requirement of linear decoding takes the problem out of the classical compressed sensing, into linear compressed sensing. Our main theoretical result establishes nearly-matching upper and lower bounds for linear compressed sensing. We prove that $d = \Omega_\epsilon(\frac{k^2}{\log k}\log (m/k))$ is required while $d = O_\epsilon(k^2\log m)$ suffices. The lower bound establishes a quantitative gap between classical and linear compressed setting, illustrating how linear accessibility is a meaningfully stronger hypothesis than linear representation alone. The upper bound confirms that neurons can store an exponential number of features under the LRH, giving theoretical evidence for the ``superposition hypothesis'' (Elhage, 2022).