基于线性判别分析的量子降维
Quantum Dimensionality Reduction by Linear Discriminant Analysis
AI总结：
提出一种量子LDA降维算法及电路，通过改进类间散布矩阵不可逆问题获取低维目标态，在向量数和维度上实现指数和二次加速，并可作为其他量子机器学习任务的子模块。
摘要：
数据的降维（DR）是许多机器学习任务（如模式识别和数据分类）的关键问题。本文提出了一种量子算法和量子电路，以高效执行线性判别分析（LDA）进行降维。首先，提出的算法改进了现有的量子LDA算法，避免了原始算法中类间散布矩阵 $S_B$ 不可逆所引起的误差。其次，提出了一种量子算法和量子电路，以获取对应于低维数据的目标状态。与最著名的经典算法相比，当原始数据集被投影到多重对数低维空间时，量子线性判别分析降维（QLDADR）算法在向量数量 $M$ 上具有指数级加速，在原始数据空间的维度 $D$ 上具有二次加速。此外，我们的算法获得的目标状态可用作其他量子机器学习任务的子模块。它具有摆脱维度灾难的实际应用价值。
英文摘要：
Dimensionality reduction (DR) of data is a crucial issue for many machine learning tasks, such as pattern recognition and data classification. In this paper, we present a quantum algorithm and a quantum circuit to efficiently perform linear discriminant analysis (LDA) for dimensionality reduction. Firstly, the presented algorithm improves the existing quantum LDA algorithm to avoid the error caused by the irreversibility of the between-class scatter matrix $S_B$ in the original algorithm. Secondly, a quantum algorithm and quantum circuits are proposed to obtain the target state corresponding to the low-dimensional data. Compared with the best-known classical algorithm, the quantum linear discriminant analysis dimensionality reduction (QLDADR) algorithm has exponential acceleration on the number $M$ of vectors and a quadratic speedup on the dimensionality $D$ of the original data space, when the original dataset is projected onto a polylogarithmic low-dimensional space. Moreover, the target state obtained by our algorithm can be used as a submodule of other quantum machine learning tasks. It has practical application value of make that free from the disaster of dimensionality.
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