通过希尔伯特空间嵌入的量子最大似然预测
Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings
AI总结：
研究量子最大似然预测任务，通过将经验概率分布嵌入量子态并最小化量子相对熵，提出统一框架，给出非渐近性能保证。
摘要：
最大似然预测是现代大型语言模型的核心任务。这里，我们作为第一步，针对由独立同分布样本组成的简化数据模型研究该任务的量子版本。量子最大似然预测器通过将经验概率分布嵌入量子态，并在给定状态类上最小化量子相对熵得到。当量子模型类具有足够表达能力时，我们从量子反向信息投影和量子勾股定理的角度给出了该预测器的解释。我们进一步推导了在迹范数和量子相对熵下的非渐近性能保证，包括收敛速度和集中不等式。我们的方法为处理经典和量子LLM中的MLP提供了统一框架。
英文摘要：
Maximum likelihood prediction (MLP) is a core task at the heart of modern large language models. Here, we study a quantum version of this task for a simplified data model consisting of independent and identically distributed samples, as a first step. The quantum maximum likelihood predictor (QMLP) is obtained by embedding of empirical probability distributions into quantum states and performing a minimization of quantum relative entropy over a given class of states. We derive non-asymptotic performance guarantees for QMLP in terms of convergence rates and concentration inequalities, both in trace norm and quantum relative entropy. Our approach provides a unified framework to handle MLP within both classical and quantum LLMs. We also consider the related problem of quantum information projection and generalize the quantum Pythagorean theorem to mixture families specified by possibly non-self-adjoint linear constraints. We further show that the Pythagorean inequality continues to hold in the infinite-dimensional setting whenever the convex information-projection problem attains a finite minimum.
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