Renormalization group flows of Hamiltonians using tensor networks

A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization ([G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)], [S. Yang, Z.-C. Gu, and X.-G Wen, Phys. Rev. Lett. 118, 110504 (2017)]) we obtain approximate fixed point tensor networks at criticality. Our formalism however preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.Comment: accepted version for Phys. Rev. Lett, main text: 5 pages, 3 figures, appendices: 9 pages, 1 figur

A multilevel approach for nonnegative matrix factorization

Nonnegative Matrix Factorization (NMF) is the problem of approximating a nonnegative matrix with the product of two low-rank nonnegative matrices and has been shown to be particularly useful in many applications, e.g., in text mining, image processing, computational biology, etc. In this paper, we explain how algorithms for NMF can be embedded into the framework of multi- level methods in order to accelerate their convergence. This technique can be applied in situations where data admit a good approximate representation in a lower dimensional space through linear transformations preserving nonnegativity. A simple multilevel strategy is described and is experi- mentally shown to speed up significantly three popular NMF algorithms (alternating nonnegative least squares, multiplicative updates and hierarchical alternating least squares) on several standard image datasets.nonnegative matrix factorization, algorithms, multigrid and multilevel methods, image processing

Multichannel high resolution NMF for modelling convolutive mixtures of non-stationary signals in the time-frequency domain

Several probabilistic models involving latent components have been proposed for modeling time-frequency (TF) representations of audio signals such as spectrograms, notably in the nonnegative matrix factorization (NMF) literature. Among them, the recent high-resolution NMF (HR-NMF) model is able to take both phases and local correlations in each frequency band into account, and its potential has been illustrated in applications such as source separation and audio inpainting. In this paper, HR-NMF is extended to multichannel signals and to convolutive mixtures. The new model can represent a variety of stationary and non-stationary signals, including autoregressive moving average (ARMA) processes and mixtures of damped sinusoids. A fast variational expectation-maximization (EM) algorithm is proposed to estimate the enhanced model. This algorithm is applied to piano signals, and proves capable of accurately modeling reverberation, restoring missing observations, and separating pure tones with close frequencies

Transfer Learning across Networks for Collective Classification

This paper addresses the problem of transferring useful knowledge from a source network to predict node labels in a newly formed target network. While existing transfer learning research has primarily focused on vector-based data, in which the instances are assumed to be independent and identically distributed, how to effectively transfer knowledge across different information networks has not been well studied, mainly because networks may have their distinct node features and link relationships between nodes. In this paper, we propose a new transfer learning algorithm that attempts to transfer common latent structure features across the source and target networks. The proposed algorithm discovers these latent features by constructing label propagation matrices in the source and target networks, and mapping them into a shared latent feature space. The latent features capture common structure patterns shared by two networks, and serve as domain-independent features to be transferred between networks. Together with domain-dependent node features, we thereafter propose an iterative classification algorithm that leverages label correlations to predict node labels in the target network. Experiments on real-world networks demonstrate that our proposed algorithm can successfully achieve knowledge transfer between networks to help improve the accuracy of classifying nodes in the target network.Comment: Published in the proceedings of IEEE ICDM 201

Adaptive Graph via Multiple Kernel Learning for Nonnegative Matrix Factorization

Nonnegative Matrix Factorization (NMF) has been continuously evolving in several areas like pattern recognition and information retrieval methods. It factorizes a matrix into a product of 2 low-rank non-negative matrices that will define parts-based, and linear representation of nonnegative data. Recently, Graph regularized NMF (GrNMF) is proposed to find a compact representation,which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure. In GNMF, an affinity graph is constructed from the original data space to encode the geometrical information. In this paper, we propose a novel idea which engages a Multiple Kernel Learning approach into refining the graph structure that reflects the factorization of the matrix and the new data space. The GrNMF is improved by utilizing the graph refined by the kernel learning, and then a novel kernel learning method is introduced under the GrNMF framework. Our approach shows encouraging results of the proposed algorithm in comparison to the state-of-the-art clustering algorithms like NMF, GrNMF, SVD etc.Comment: This paper has been withdrawn by the author due to the terrible writin

Nehari problems and equalizing vectors for infinite-dimensional systems

For a class of infinite-dimensional systems we obtain a simple frequency domain solution for the suboptimal Nehari extension problem. The approach is via $J$-spectral factorization, and it uses the concept of equalizing vectors. Moreover, the connection between the equalizing vectors and the Nehari extension problem is given. \u