论文标题
结构化先验的张量估计
Tensor estimation with structured priors
论文作者
论文摘要
当张量被高斯噪声损坏时，我们考虑排名衡量的对称张量估计，而形成张量的尖峰是来自广义线性模型的结构化信号。后者是信号中非平凡隐藏的低维度潜在结构的数学可牵引模型。我们在较大的维度方向上工作，具有信噪比空间维度的固定比率。值得注意的是，在这个渐近方案中，尖峰和观测值之间的相互信息可以表示为有限维的变分问题，并且可以从其溶液中推断出最小均值的error。我们在示例中讨论了相变的特性，这是信噪比的函数。通常，临界信噪比随着信噪比的增加而降低。我们讨论了信噪比空间尺寸消失比率的极限，并确定限制张量估计问题。我们还指出了与矩阵情况的相似性和差异。
We consider rank-one symmetric tensor estimation when the tensor is corrupted by Gaussian noise and the spike forming the tensor is a structured signal coming from a generalized linear model. The latter is a mathematically tractable model of a non-trivial hidden lower-dimensional latent structure in a signal. We work in a large dimensional regime with fixed ratio of signal-to-latent space dimensions. Remarkably, in this asymptotic regime, the mutual information between the spike and the observations can be expressed as a finite-dimensional variational problem, and it is possible to deduce the minimum-mean-square-error from its solution. We discuss, on examples, properties of the phase transitions as a function of the signal-to-noise ratio. Typically, the critical signal-to-noise ratio decreases with increasing signal-to-latent space dimensions. We discuss the limit of vanishing ratio of signal-to-latent space dimensions and determine the limiting tensor estimation problem. We also point out similarities and differences with the case of matrices.
