  Informally, the 'linear representation hypothesis' is the idea that
high-level concepts are represented linearly as directions in some
representation space. In this paper, we address two closely related questions:
What does "linear representation" actually mean? And, how do we make sense of
geometric notions (e.g., cosine similarity or projection) in the representation
space? To answer these, we use the language of counterfactuals to give two
formalizations of "linear representation", one in the output (word)
representation space, and one in the input (sentence) space. We then prove
these connect to linear probing and model steering, respectively. To make sense
of geometric notions, we use the formalization to identify a particular
(non-Euclidean) inner product that respects language structure in a sense we
make precise. Using this causal inner product, we show how to unify all notions
of linear representation. In particular, this allows the construction of probes
and steering vectors using counterfactual pairs. Experiments with LLaMA-2
demonstrate the existence of linear representations of concepts, the connection
to interpretation and control, and the fundamental role of the choice of inner
product.
