Semi-supervised Eigenvectors for Large-scale Locally-biased Learning

In many applications, one has side information, e.g., labels that are provided in a semi-supervised manner, about a specific target region of a large data set, and one wants to perform machine learning and data analysis tasks "nearby" that prespecified target region. For example, one might be interested in the clustering structure of a data graph near a prespecified "seed set" of nodes, or one might be interested in finding partitions in an image that are near a prespecified "ground truth" set of pixels. Locally-biased problems of this sort are particularly challenging for popular eigenvector-based machine learning and data analysis tools. At root, the reason is that eigenvectors are inherently global quantities, thus limiting the applicability of eigenvector-based methods in situations where one is interested in very local properties of the data. In this paper, we address this issue by providing a methodology to construct semi-supervised eigenvectors of a graph Laplacian, and we illustrate how these locally-biased eigenvectors can be used to perform locally-biased machine learning. These semi-supervised eigenvectors capture successively-orthogonalized directions of maximum variance, conditioned on being well-correlated with an input seed set of nodes that is assumed to be provided in a semi-supervised manner. We show that these semi-supervised eigenvectors can be computed quickly as the solution to a system of linear equations; and we also describe several variants of our basic method that have improved scaling properties. We provide several empirical examples demonstrating how these semi-supervised eigenvectors can be used to perform locally-biased learning; and we discuss the relationship between our results and recent machine learning algorithms that use global eigenvectors of the graph Laplacian

Observation of the Fractional Quantum Hall Effect in Graphene

When electrons are confined in two dimensions and subjected to strong magnetic fields, the Coulomb interactions between them become dominant and can lead to novel states of matter such as fractional quantum Hall liquids. In these liquids electrons linked to magnetic flux quanta form complex composite quasipartices, which are manifested in the quantization of the Hall conductivity as rational fractions of the conductance quantum. The recent experimental discovery of an anomalous integer quantum Hall effect in graphene has opened up a new avenue in the study of correlated 2D electronic systems, in which the interacting electron wavefunctions are those of massless chiral fermions. However, due to the prevailing disorder, graphene has thus far exhibited only weak signatures of correlated electron phenomena, despite concerted experimental efforts and intense theoretical interest. Here, we report the observation of the fractional quantum Hall effect in ultraclean suspended graphene, supporting the existence of strongly correlated electron states in the presence of a magnetic field. In addition, at low carrier density graphene becomes an insulator with an energy gap tunable by magnetic field. These newly discovered quantum states offer the opportunity to study a new state of matter of strongly correlated Dirac fermions in the presence of large magnetic fields

Visualizing and Quantifying Intracellular Behavior and Abundance of the Core Circadian Clock Protein PERIOD2

SummaryTranscriptional-translational feedback loops (TTFLs) are a conserved molecular motif of circadian clocks. The principal clock in mammals is the suprachiasmatic nucleus (SCN) of the hypothalamus. In SCN neurons, auto-regulatory feedback on core clock genes Period (Per) and Cryptochrome (Cry) following nuclear entry of their protein products is the basis of circadian oscillation [1, 2]. In Drosophila clock neurons, the movement of dPer into the nucleus is subject to a circadian gate that generates a delay in the TTFL, and this delay is thought to be critical for oscillation [3, 4]. Analysis of the Drosophila clock has strongly influenced models of the mammalian clock, and such models typically infer complex spatiotemporal, intracellular behaviors of mammalian clock proteins. There are, however, no direct measures of the intracellular behavior of endogenous circadian proteins to support this: dynamic analyses have been limited and often have no circadian dimension [5–7]. We therefore generated a knockin mouse expressing a fluorescent fusion of native PER2 protein (PER2::VENUS) for live imaging. PER2::VENUS recapitulates the circadian functions of wild-type PER2 and, importantly, the behavior of PER2::VENUS runs counter to the Drosophila model: it does not exhibit circadian gating of nuclear entry. Using fluorescent imaging of PER2::VENUS, we acquired the first measures of mobility, molecular concentration, and localization of an endogenous circadian protein in individual mammalian cells, and we showed how the mobility and nuclear translocation of PER2 are regulated by casein kinase. These results provide new qualitative and quantitative insights into the cellular mechanism of the mammalian circadian clock