我们引入了一个数学框架，用于线性表示假设（LRH），该假设认为语言模型的中间层以线性方式存储特征。我们将该假设分为两个主张：线性表示（特征以线性嵌入在神经元激活中）和线性可访问性（特征可以线性解码）。然后我们提出问题：多少个神经元d足以同时线性表示和线性访问m个特征？压缩感知的经典结果表明，对于k-稀疏输入，如果允许非线性解码算法，则d=O(k log(m/k))足够（Candes和Tao，2006；Candes等，2006；Donoho，2006）。然而，线性解码的额外要求将问题带出经典压缩感知，进入线性压缩感知领域。我们的主要理论结果建立了线性压缩感知的几乎匹配的上界和下界。我们证明需要d=Ω_ε(k²/log k log(m/k))，而d=O_ε(k² log m)则足够。下界确立了经典与线性压缩设置之间的定量差距，说明线性可访问性比线性表示本身是一个更有意义的假设。上界证实了在LRH下神经元可以存储指数级数量的特征，为“叠加假设”（Elhage等，2022）提供了理论证据。上界证明使用标准的随机构造具有近正交列的矩阵。下界证明使用近单位矩阵的秩界（Alon，2003）以及图论中的Turán定理（限制无圈图的边数）。我们还展示了我们的结果如何约束特征表示的几何结构，并将结果扩展到允许具有激活函数和偏置的解码器。
英文摘要：
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 $d = Ω_ε(\frac{k^2}{\log k}\log (m/k))$ is required while $d = O_ε(k^2\log m)$ 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 et al., 2022).
The upper bound proof uses standard random constructions of matrices with approximately orthogonal columns. The lower bound proof uses rank bounds for near-identity matrices (Alon, 2003) together with Turán's theorem (bounding the number of edges in clique-free graphs). We also show how our results do and do not constrain the geometry of feature representations and extend our results to allow decoders with an activation function and bias.
Scientific Hypothesis Generation by a Large Language Model: Laboratory Validation in Breast Cancer Treatment
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）；The National Institute of Agricultural Botany（国家农业植物学研究所）；Arctoris Ltd（Arctoris有限公司）；Goldsmiths, University of London（伦敦大学金史密斯学院）；Chalmers University（查尔姆斯理工大学）
