Learning and Inference in Hilbert Space with Quantum Graphical Models
Learning and Inference in Hilbert Space with Quantum Graphical Models
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发表时间：
2018-10
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通讯作者：
Siddarth Srinivasan;Carlton Downey;Byron Boots
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作者：
Siddarth Srinivasan;Carlton Downey;Byron Boots
量子图模型（QGM）通过采用形式主义来推理量子力学的不确定性，从而概括了经典图模型。与经典图模型不同，QGM 表示复杂希尔伯特空间中密度矩阵的不确定性。希尔伯特空间嵌入（HSE）也推广了希尔伯特空间中的贝叶斯推理。我们研究了 QGM 和 HSE 之间的联系，并表明 QGM 的求和规则和贝叶斯规则分别相当于 HSE 中的核求和规则和 Nadaraya-Watson 核回归的特殊情况。我们证明这些操作可以被核化，并利用这些见解提出隐藏量子马尔可夫模型的希尔伯特空间嵌入（HSE-HQMM）来建模动力学。我们提供的实验结果表明，HSE-HQMM 在多个数据集上与 LSTM 和 PSRNN 等最先进的模型具有竞争力，同时还提供了一种非参数方法来维持连续值特征的概率分布。
Quantum Graphical Models (QGMs) generalize classical graphical models by adopting the formalism for reasoning about uncertainty from quantum mechanics. Unlike classical graphical models, QGMs represent uncertainty with density matrices in complex Hilbert spaces. Hilbert space embeddings (HSEs) also generalize Bayesian inference in Hilbert spaces. We investigate the link between QGMs and HSEs and show that the sum rule and Bayes rule for QGMs are equivalent to the kernel sum rule in HSEs and a special case of Nadaraya-Watson kernel regression, respectively. We show that these operations can be kernelized, and use these insights to propose a Hilbert Space Embedding of Hidden Quantum Markov Models (HSE-HQMM) to model dynamics. We present experimental results showing that HSE-HQMMs are competitive with state-of-the-art models like LSTMs and PSRNNs on several datasets, while also providing a nonparametric method for maintaining a probability distribution over continuous-valued features.