Quantum divide-and-conquer anchoring for separable non-negative matrix factorization

© 2018 International Joint Conferences on Artificial Intelligence. All right reserved. It is NP-complete to find non-negative factors W and H with fixed rank r from a non-negative matrix X by minimizing ||X − WHτ||2F. Although the separability assumption (all data points are in the conical hull of the extreme rows) enables polynomial-time algorithms, the computational cost is not affordable for big data. This paper investigates how the power of quantum computation can be capitalized to solve the non-negative matrix factorization with the separability assumption (SNMF) by devising a quantum algorithm based on the divide-and-conquer anchoring (DCA) scheme [Zhou et al., 2013]. The design of quantum DCA (QDCA) is challenging. In the divide step, the random projections in DCA is completed by a quantum algorithm for linear operations, which achieves the exponential speedup. We then devise a heuristic post-selection procedure which extracts the information of anchors stored in the quantum states efficiently. Under a plausible assumption, QDCA performs efficiently, achieves the quantum speedup, and is beneficial for high dimensional problems

Compressed learning

University of Technology, Sydney. Faculty of Engineering and Information Technology.There has been an explosion of data derived from the internet and other digital sources. These data are usually multi-dimensional, massive in volume, frequently incomplete, noisy, and complicated in structure. These "big data" bring new challenges to machine learning (ML), which has historically been designed for small volumes of clearly defined and structured data. In this thesis we propose new methods of "compressed learning", which explore the components and procedures in ML methods that are compressible, in order to improve their robustness, scalability, adaptivity, and performance for big data analysis. We will study novel methodologies that compress different components throughout the learning process, propose more interpretable general compressible structures for big data, and develop effective strategies to leverage these compressible structures to produce highly scalable learning algorithms. We present several new insights into popular learning problems in the context of compressed learning. The theoretical analyses are tested on real data in order to demonstrate the efficacy and efficiency of the methodologies in real-world scenarios. In particular, we propose "manifold elastic net (MEN)" and "double shrinking (DS)" as two fast frameworks extracting low-dimensional sparse features for dimension reduction and manifold learning. These methods compress the features on both their dimension and cardinality, and significantly improve their interpretation and performance in clustering and classification tasks. We study how to derive fewer "anchor points" for representing large datasets in their entirety by proposing "divide-and-conquer anchoring", in which the global solution is rapidly found for near-separable non-negative matrix factorization and completion in a distributed manner. This method represents a compression of the big data itself, rather than features, and the extracted anchors define the structure of the data. Two fast low-rank approximation methods, "bilateral random projections (BRP)" of fast computer closed-form and "greedy bilateral sketch (GreBske)", are proposed based on random projection and greedy augmenting update rules. They can be broadly applied to learning procedures that requires updates of a low-rank matrix variable and result in significant acceleration in performance. We study how to compress noisy data for learning by decomposing it into the sum mixture of low-rank part and sparse part. "GO decomposition (GoDec)" and the "greedy bilateral (GreB)" paradigm are proposed as two efficient approaches to this problem based on randomized and greedy strategies, respectively. Modifications of these two schemes result in novel models and extremely fast algorithms for matrix completion that aim to recover a low-rank matrix from a small number of its entries. In addition, we extend the GoDec problem in order to unmix more than two incoherent structures that are more complicated and expressive than low-rank or sparse matrices. The three proposed variants are not only novel and effective algorithms for motion segmentation in computer vision, multi-label learning, and scoring-function learning in recommendation systems, but also reveal new theoretical insights into these problems. Finally, a compressed learning method termed “compressed labelling (CL) on distilled label sets (DL)" is proposed for solving the three core problems in multi-label learning, namely high-dimensional labels, label correlation modeling, and sample imbalance for each label. By compressing the labels and the number of classifiers in multi-label learning, CL can generate an effective and efficient training algorithm from any single-label classifier

A quantum-inspired classical algorithm for separable Non-negative Matrix Factorization

Non-negative Matrix Factorization (NMF) asks to decompose a (entry-wise) non-negative matrix into the product of two smaller-sized nonnegative matrices, which has been shown intractable in general. In order to overcome this issue, separability assumption is introduced which assumes all data points are in a conical hull. This assumption makes NMF tractable and is widely used in text analysis and image processing, but still impractical for huge-scale datasets. In this paper, inspired by recent development on dequantizing techniques, we propose a new classical algorithm for separable NMF problem. Our new algorithm runs in polynomial time in the rank and logarithmic in the size of input matrices, which achieves an exponential speedup in the low-rank setting

Quantum differentially private sparse regression learning

Differentially private (DP) learning, which aims to accurately extract patterns from the given dataset without exposing individual information, is an important subfield in machine learning and has been extensively explored. However, quantum algorithms that could preserve privacy, while outperform their classical counterparts, are still lacking. The difficulty arises from the distinct priorities in DP and quantum machine learning, i.e., the former concerns a low utility bound while the latter pursues a low runtime cost. These varied goals request that the proposed quantum DP algorithm should achieve the runtime speedup over the best known classical results while preserving the optimal utility bound. The Lasso estimator is broadly employed to tackle the high dimensional sparse linear regression tasks. The main contribution of this paper is devising a quantum DP Lasso estimator to earn the runtime speedup with the privacy preservation, i.e., the runtime complexity is $\tilde{O}(N^{3/2}\sqrt{d})$ with a nearly optimal utility bound $\tilde{O}(1/N^{2/3})$, where $N$ is the sample size and $d$ is the data dimension with $N\ll d$. Since the optimal classical (private) Lasso takes $\Omega(N+d)$ runtime, our proposal achieves quantum speedups when $N<O(d^{1/3})$. There are two key components in our algorithm. First, we extend the Frank-Wolfe algorithm from the classical Lasso to the quantum scenario, {where the proposed quantum non-private Lasso achieves a quadratic runtime speedup over the optimal classical Lasso.} Second, we develop an adaptive privacy mechanism to ensure the privacy guarantee of the non-private Lasso. Our proposal opens an avenue to design various learning tasks with both the proven runtime speedups and the privacy preservation

Symbolic geography in John Ruskin's modern painters, Volumes III, IV, V

Modern Painters vol. III, IV et V est une œuvre tripartite développée après 1850, en synchronie avec la guerre de Crimée (1854-56), la répression de la révolte indienne (1857-59) et la deuxième guerre d’indépendance d’Italie (1859). Marqué par ces événements politiques, Ruskin met en œuvre une stratégie complexe pour configurer dans un langage symbolique les frontières et les taxonomies impériales de l’espace européen. Il développe des stratégies de présentation qui combinent le texte et les illustrations pour créer des allégories mentales et visuelles, construites à partir des stéréotypes littéraires et culturels véhiculés dans l’espace britannique. L’auteur met ses derniers volumes de Modern Painters sous le signe de « la crise de la civilisation » représentée par les conflagrations de Crimée, d’Inde et d’Italie, en exprimant son soutien pour la nouvelle alliance entre l’Angleterre et la France. Un autre motif est son obsession avec la réforme sociale via un retour aux valeurs chrétiennes traditionnelles.John Ruskin writes Modern Painters Volumes III, IV, and V as events such as The Crimean War (1854-56), the Indian Mutiny (1857-59), and the Second Italian War of Independence (1859) unfold. As such, Ruskin’s work tends to reflect and respond to the political context of his time. In these works, Ruskin tries to symbolically interpret and represent geopolitical and taxonomical characteristics of the European continent, generally in an imperial narrative, paying particular attention to British identity and national stereotypes. Ruskin articulates his ideas using a unique style that combines visual and written elements to create powerful allegories. In these volumes, Ruskin is especially concerned with what he sees as an impending “crisis of civilization” of which the aforementioned conflicts are symptoms. As a response, Ruskin strongly advocates societal reform in the form of a return to old Christian values. He also supports a military alliance between Britain and France