A Comparative Study of Matrix Factorization and Random Walk with Restart in Recommender Systems

Between matrix factorization or Random Walk with Restart (RWR), which method works better for recommender systems? Which method handles explicit or implicit feedback data better? Does additional information help recommendation? Recommender systems play an important role in many e-commerce services such as Amazon and Netflix to recommend new items to a user. Among various recommendation strategies, collaborative filtering has shown good performance by using rating patterns of users. Matrix factorization and random walk with restart are the most representative collaborative filtering methods. However, it is still unclear which method provides better recommendation performance despite their extensive utility. In this paper, we provide a comparative study of matrix factorization and RWR in recommender systems. We exactly formulate each correspondence of the two methods according to various tasks in recommendation. Especially, we newly devise an RWR method using global bias term which corresponds to a matrix factorization method using biases. We describe details of the two methods in various aspects of recommendation quality such as how those methods handle cold-start problem which typically happens in collaborative filtering. We extensively perform experiments over real-world datasets to evaluate the performance of each method in terms of various measures. We observe that matrix factorization performs better with explicit feedback ratings while RWR is better with implicit ones. We also observe that exploiting global popularities of items is advantageous in the performance and that side information produces positive synergy with explicit feedback but gives negative effects with implicit one.Comment: 10 pages, Appears in IEEE International Conference on Big Data 2017 (IEEE BigData 2017

Topological dephasing in the ν=2/3\nu=2/3 fractional Quantum Hall Regime

We study dephasing in electron transport through a large quantum dot (a Fabry-Perot interferometer) in the fractional quantum Hall regime with filling factor $2/3$. In the regime of sequential tunneling, dephasing occurs due to electron fractionalization into counterpropagating charge and neutral edge modes on the dot. In particular, when the charge mode moves much faster than the neutral mode, and at temperatures higher than the level spacing of the dot, electron fractionalization combined with tje fractional statistics of the charge mode leads to the dephasing selectively suppressing $h/e$ Aharonov-Bohm oscillations but not $h/(2e)$ oscillations, resulting in oscillation-period halving.Comment: 11pages, 3 figure

Incoherent transport on the ν=2/3\nu=2/3 quantum Hall edge

The nature of edge state transport in quantum Hall systems has been studied intensely ever since Halperin [1] noted its importance for the quantization of the Hall conductance. Since then, there have been many developments in the study of edge states in the quantum Hall effect, including the prediction of multiple counter-propagating modes in the fractional quantum Hall regime, the prediction of edge mode renormalization due to disorder, and studies of how the sample confining potential affects the edge state structure (edge reconstruction), among others. In this paper, we study edge transport for the $\nu_{\text{bulk}}=2/3$ edge in the disordered, fully incoherent transport regime. To do so, we use a hydrodynamic approximation for the calculation of voltage and temperature profiles along the edge of the sample. Within this formalism, we study two different bare mode structures with tunneling: the original edge structure predicted by Wen [2] and MacDonald [3], and the more complicated edge structure proposed by Meir [4], whose renormalization and transport characteristics were discussed by Wang, Meir and Gefen (WMG) [5]. We find that in the fully incoherent regime, the topological characteristics of transport (quantized electrical and heat conductance) are intact, with finite size corrections which are determined by the extent of equilibration. In particular, our calculation of conductance for the WMG model in a double QPC geometry reproduce conductance results of a recent experiment by R. Sabo, et al. [17], which are inconsistent with the model of MacDonald. Our results can be explained in the charge/neutral mode picture, with incoherent analogues of the renormalization fixed points of Ref. [5]. Additionally, we find diffusive $(\sim1/L)$ conductivity corrections to the heat conductance in the fully incoherent regime for both models of the edge.Comment: 40 pages, 10 figures. v2 added citations, now 41 page

Noise on complex quantum Hall edges: Chiral anomaly and heat diffusion

Electrical and thermal conductances of a quantum Hall bar reflect the topological structure of the incompressible bulk phase. Here we show that noise of electrical current carried through the edge evidences the interplay between these two topological observables. Transport through a structured edge is modeled by a voltage-biased line junction made up of two counter-propagating modes associated with respective filling factors. Specifically, we focus on the edge of a $\nu=2/3$ fractional quantum Hall state. Noise is generated at a point distinctly separated from the hot spot (where most of the Ohmic dissipation takes place) and reflects the competition between ballistically carried downstream current and diffusively carried heat (which can propagate also upstream). We propose specific setups where our predictions can be measured.Comment: 5pages, 5 figures, 5-page supplemental materia

Macroscopic Quantum Entanglement of a Kondo Cloud at Finite Temperature

We propose a variational approach for computing the macroscopic entanglement in a many-body mixed state, based on entanglement witness operators, and compute the entanglement of formation (EoF), a mixed-state generalization of the entanglement entropy, in single- and two-channel Kondo systems at finite temperature. The thermal suppression of the EoF obeys power-law scaling at low temperature. The scaling exponent is halved from the single- to the two-channel system, which is attributed, using a bosonization method, to the non-Fermi liquid behavior of a Majorana fermion, a "half" of a complex fermion, emerging in the two-channel system. Moreover, the EoF characterizes the size and power-law tail of the Kondo screening cloud of the single-channel system.Comment: Supplementary Material include

Topological vacuum bubble by anyon braiding

According to a basic rule of fermionic and bosonic many-body physics, known as the linked cluster theorem, physical observables are not affected by vacuum bubbles, which represent virtual particles created from vacuum and self-annihilating without interacting with real particles. Here, we show that this conventional knowledge must be revised for anyons, quasiparticles that obey fractional exchange statistics intermediate between fermions and bosons. We find that a certain class of vacuum bubbles of Abelian anyons does affect physical observables. They represent virtually excited anyons which wind around real anyonic excitations. These topological bubbles result in a temperature-dependent phase shift of Fabry-Perot interference patterns in the fractional quantum Hall regime accessible in current experiments, thus providing a tool for direct and unambiguous observation of elusive fractional statistics.Comment: 7 pages, 4 figures, and 9 pages of Supplementary Informatio

Negative-UU Anisotropic Charge Kondo Effect in a Triple Quantum Dot

We predict a new type of the negative-$U$ Anderson impurity formed in a triple quantum dot. The two dots of the system behave as a negative-$U$ impurity preferring zero or double electron occupancy rather than single occupancy, and the third dot stabilizes the attractive interaction of $U < 0$ via Coulomb repulsion. Using a bosonization method, we find that the system has the two different phases of massive or vanishing charge fluctuations between the two occupancies at low temperature, which are equivalent with the antiferromagnetic and ferromagnetic phases of the anisotropic Kondo model, respectively. The phase transition is experimentally accessible and identifiable by electron conductance, offering the possibility of experimentally exploring the anisotropic Kondo model

How to directly measure a Kondo cloud's length

We propose a method to directly measure, by electrical means, the Kondo screening cloud formed by an Anderson impurity coupled to semi-infinite quantum wires, on which an electrostatic gate voltage is applied at distance L from the impurity. We show that the Kondo cloud, and hence the Kondo temperature and the electron conductance through the impurity, are affected by the gate voltage, as L decreases below the Kondo cloud length. Based on this behavior, the cloud length can be experimentally identified by changing L with a keyboard type of gate voltage or tuning the coupling strength between the impurity and the wires.Comment: 5 pages, 4 figure

Anisotropic charge Kondo effect in a triple quantum dot

We predict that an anisotropic charge Kondo effect appears in a triple quantum dot, when the system has two-fold degenerate ground states of (1,1,0) and (0,0,1) charge configurations. Using bosonization and refermionization methods, we find that at low temperature, the system has the two different phases of massive charge fluctuations between the two charge configurations and vanishing fluctuations, which are equivalent with the Kondo-screened and ferromagnetic phases of the anisotropic Kondo model, respectively. The phase transition is identifiable by electron conductance measurement, offering the possibility of experimentally exploring the anisotropic Kondo model. Our charge Kondo effect has similar origin to that in a negative-U Anderson impurity.Comment: 5 pages, 2 figure

Symmetry-related transport on a fractional quantum Hall edge

Low-energy transport in quantum Hall states is carried through edge modes, and is dictated by bulk topological invariants and possibly microscopic Boltzmann kinetics at the edge. Here we show how the presence or breaking of symmetries of the edge Hamiltonian underlie transport properties, specifically d.c. conductance and noise. We demonstrate this through the analysis of hole-conjugate states of the quantum Hall effect, specifically the $\nu=2/3$ case in a quantum point-contact (QPC) geometry. We identify two symmetries, a continuous $SU(3)$ and a discrete $Z_3$, whose presence or absence (different symmetry scenarios) dictate qualitatively different types of behavior of conductance and shot noise. While recent measurements are consistent with one of these symmetry scenarios, others can be realized in future experiments.Comment: 14 pages, 6 figure