Abstract: Isomap algorithm is a representative manifold learning algorithm. The algorithm simplifies the data analysis process and is widely used in neuroimaging, spectral analysis and other fields. However, the classic Isomap algorithm becomes unwieldy when dealing with large data sets. Our object is to accelerate the classical algorithm with quantum computing, and propose the quantum Isomap algorithm. The algorithm consists of two sub-algorithms.

Abstract: The weighted Euclidean distance between two vectors is a Euclidean distance where the contribution of each dimension is scaled by a given non-negative weight. The Johnson-Lindenstrauss (JL) lemma can be easily adapted to the weighted Euclidean distance if weights are known at construction time. Given a set of n vectors with dimension d, it suffices to scale each dimension of the input vectors according to the weights, and then apply any standard JL reduction: the weighted Euclidean distance between pairs of vectors is preserved within a multiplicative factor ε with high probability. However, this is not the case when weights are provided after the dimensionality reduction. In this paper, we show that by applying a linear map from real vectors to a complex vector space, it is possible to update the compressed vectors so that the weighted Euclidean distances between pairs of points can be computed within a multiplicative factor ε, even when weights are provided after the dimensionality reduction.

The Isomap is a well-known nonlinear dimensionality reduction method that highly suffers from computational complexity. Its computational complexity mainly arises from two stages; a) embedding a full graph on the data in the ambient space, and b) a complete eigenvalue decomposition. Although the reduction of the computational complexity of the graphing stage has been investigated, yet the eigenvalue decomposition stage remains a bottleneck in the problem. In this paper, we propose the Low-Rank Isomap algorithm by introducing a projection operator on the embedded graph from the ambient space to a low-rank latent space to facilitate applying the partial eigenvalue decomposition. This approach leads to reducing the complexity of Isomap to a linear order while preserving the structural information during the dimensionality reduction process. The superiority of the Low-Rank Isomap algorithm compared to some state-of-art algorithms is experimentally verified on facial image clustering in terms of speed and accuracy.

Suppose the data consist of a set $S$ of points $x_j, 1 \leq j \leq J$, distributed in a bounded domain $D \subset R^N$, where $N$ and $J$ are large numbers. In this paper an algorithm is proposed for checking whether there exists a manifold $\mathbb{M}$ of low dimension near which many of the points of $S$ lie and finding such $\mathbb{M}$ if it exists. There are many dimension reduction algorithms, both linear and non-linear. Our algorithm is simple to implement and has some advantages compared with the known algorithms. If there is a manifold of low dimension near which most of the data points lie, the proposed algorithm will find it. Some numerical results are presented illustrating the algorithm and analyzing its performance compared to the classical PCA (principal component analysis) and Isomap.