Gianna M. Del Corso

Alessandro Poggiali, Alessandro Berti, Anna Bernasconi et al.

Quantum computing is a promising paradigm based on quantum theory for performing fast computations. Quantum algorithms are expected to surpass their classical counterparts in terms of computational complexity for certain tasks, including machine learning. In this paper, we design, implement, and evaluate three hybrid quantum k-Means algorithms, exploiting different degree of parallelism. Indeed, each algorithm incrementally leverages quantum parallelism to reduce the complexity of the cluster assignment step up to a constant cost. In particular, we exploit quantum phenomena to speed up the computation of distances. The core idea is that the computation of distances between records and centroids can be executed simultaneously, thus saving time, especially for big datasets. We show that our hybrid quantum k-Means algorithms can be more efficient than the classical version, still obtaining comparable clustering results.

Giacomo Antonioli, Paola Boito, Gianna M. Del Corso et al.

Quantum block encoding (QBE) is a crucial step in the development of most quantum algorithms, as it provides an embedding of a given matrix into a suitable larger unitary matrix. Historically, the development of efficient techniques for QBE has mostly focused on sparse matrices; less effort has been devoted to data-sparse (e.g., rank-structured) matrices. In this work we examine a particular case of rank structure, namely, one-pair semiseparable matrices. We present a new block encoding approach that relies on a suitable factorization of the given matrix as the product of triangular and diagonal factors. To encode the matrix, the algorithm needs $2\log(N)+7$ ancillary qubits. This process takes polylogarithmic time and has an error of $\mathcal{O}(N^2)$, where $N$ is the matrix size.

Gianna M. Del Corso, Francesco Romani

The Nonnegative Matrix Factorization (NMF) of the rating matrix has shown to be an effective method to tackle the recommendation problem. In this paper we propose new methods based on the NMF of the rating matrix and we compare them with some classical algorithms such as the SVD and the regularized and unregularized non-negative matrix factorization approach. In particular a new algorithm is obtained changing adaptively the function to be minimized at each step, realizing a sort of dynamic prior strategy. Another algorithm is obtained modifying the function to be minimized in the NMF formulation by enforcing the reconstruction of the unknown ratings toward a prior term. We then combine different methods obtaining two mixed strategies which turn out to be very effective in the reconstruction of missing observations. We perform a thoughtful comparison of different methods on the basis of several evaluation measures. We consider in particular rating, classification and ranking measures showing that the algorithm obtaining the best score for a given measure is in general the best also when different measures are considered, lowering the interest in designing specific evaluation measures. The algorithms have been tested on different datasets, in particular the 1M, and 10M MovieLens datasets containing ratings on movies, the Jester dataset with ranting on jokes and Amazon Fine Foods dataset with ratings on foods. The comparison of the different algorithms, shows the good performance of methods employing both an explicit and an implicit regularization scheme. Moreover we can get a boost by mixed strategies combining a fast method with a more accurate one.

Gianna M. Del Corso, Francesco Romani

After the phenomenal success of the PageRank algorithm, many researchers have extended the PageRank approach to ranking graphs with richer structures beside the simple linkage structure. In some scenarios we have to deal with multi-parameters data where each node has additional features and there are relationships between such features. This paper stems from the need of a systematic approach when dealing with multi-parameter data. We propose models and ranking algorithms which can be used with little adjustments for a large variety of networks (bibliographic data, patent data, twitter and social data, healthcare data). In this paper we focus on several aspects which have not been addressed in the literature: (1) we propose different models for ranking multi-parameters data and a class of numerical algorithms for efficiently computing the ranking score of such models, (2) by analyzing the stability and convergence properties of the numerical schemes we tune a fast and stable technique for the ranking problem, (3) we consider the issue of the robustness of our models when data are incomplete. The comparison of the rank on the incomplete data with the rank on the full structure shows that our models compute consistent rankings whose correlation is up to 60% when just 10% of the links of the attributes are maintained suggesting the suitability of our model also when the data are incomplete.