加速经典和量子张量PCA
Accelerating Classical and Quantum Tensor PCA
AI总结：
本文提出了一种加速经典和量子张量PCA的方法，实现经典算法二次加速和量子算法八次加速，并探讨了其在检测和恢复中的应用。
摘要：
谱方法是处理具有“突起”高斯张量的张量PCA的主要方法。这些方法利用向量空间中指数级高维线性算子的谱。在参考文献1中已证明，量子算法可以带来指数级的空间节省以及相对于经典方法的四次方加速。在这里，我们展示了如何通过二次加速经典和量子算法，同时保持两者之间的四次方分离。也就是说，本文中的经典算法比原始经典算法快二次方，而量子算法比原始经典算法快八次方。我们然后对量子算法进行了进一步的修改，使其相对于修改后的经典算法的加速速度提高到六次方。我们仅证明了这些加速效果适用于检测，而非恢复，但给出了一个强有力的可信论据，表明我们的算法也实现了恢复。注：在本文准备完成后，A. Schmidhuber指出参考文献3。这改进了某种随机算子的谱范数的最佳现有界限。由于该算子的范数进入运行时间，有了这个范数的改进，我们不再有可证明的多项式加速。我们的结果是基于该算子某些谱性质（不仅仅是最大特征值，还包括态密度）。因此，如果这些性质仍然成立，加速效果仍然成立。与其修改论文，我选择保持原样，但添加了一个讨论所需态密度性质的结尾部分，并考虑了哪些问题（存在多个问题，其中这种四次方量子加速已被使用，而本文中的技术可能适用于其中的多个问题）该性质可能成立。
英文摘要：
Spectral methods are a leading approach for tensor PCA with a ``spiked" Gaussian tensor. The methods use the spectrum of a linear operator in a vector space with exponentially high dimension and in Ref. 1 it was shown that quantum algorithms could then lead to an exponential space saving as well as a quartic speedup over classical. Here we show how to accelerate both classical and quantum algorithms quadratically, while maintaining the same quartic separation between them. That is, our classical algorithm here is quadratically faster than the original classical algorithm, while the quantum algorithm is eigth-power faster than the original classical algorithm. We then give a further modification of the quantum algorithm, increasing its speedup over the modified classical algorithm to the sixth power. We only prove these speedups for detection, rather than recovery, but we give a strong plausibility argument that our algorithm achieves recovery also. Note added: After this paper was prepared, A. Schmidhuber pointed out to me Ref. 3. This improves the best existing bounds on the spectral norm of a certain random operator. Because the norm of this operator enters into the runtime, with this improvement on the norm, we no longer have a provable polynomial speedup. Our results are phrased in terms of certain properties of the spectrum of this operator (not merely the largest eigenvalue but also the density of states). So, if these properties still hold, the speedup still holds. Rather than modify the paper, I have left it unchanged but added a section at the end discussing the needed property of density of states and considering for which problems (there are several problems for which this kind of quartic quantum speedup has been used and the techniques here will likely be applicable to several of them) the property is likely to hold.
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