Spectral density of the non-backtracking operator

The non-backtracking operator was recently shown to provide a significant improvement when used for spectral clustering of sparse networks. In this paper we analyze its spectral density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero spectral density, that is stable outside a circle of radius $\sqrt{\rho}$, where $\rho$ is the leading eigenvalue of the non-backtracking operator. We observe a second-order phase transition at the edge of this circle, between a zero and a non-zero spectral density. That fact that this phase transition is absent in the spectral density of other matrices commonly used for spectral clustering provides a physical justification of the performances of the non-backtracking operator in spectral clustering.Comment: 6 pages, 6 figures, submitted to EP

Matrix Completion from Fewer Entries: Spectral Detectability and Rank Estimation

The completion of low rank matrices from few entries is a task with many practical applications. We consider here two aspects of this problem: detectability, i.e. the ability to estimate the rank $r$ reliably from the fewest possible random entries, and performance in achieving small reconstruction error. We propose a spectral algorithm for these two tasks called MaCBetH (for Matrix Completion with the Bethe Hessian). The rank is estimated as the number of negative eigenvalues of the Bethe Hessian matrix, and the corresponding eigenvectors are used as initial condition for the minimization of the discrepancy between the estimated matrix and the revealed entries. We analyze the performance in a random matrix setting using results from the statistical mechanics of the Hopfield neural network, and show in particular that MaCBetH efficiently detects the rank $r$ of a large $n\times m$ matrix from $C(r)r\sqrt{nm}$ entries, where $C(r)$ is a constant close to $1$. We also evaluate the corresponding root-mean-square error empirically and show that MaCBetH compares favorably to other existing approaches.Comment: NIPS Conference 201

Clustering from Sparse Pairwise Measurements

We consider the problem of grouping items into clusters based on few random pairwise comparisons between the items. We introduce three closely related algorithms for this task: a belief propagation algorithm approximating the Bayes optimal solution, and two spectral algorithms based on the non-backtracking and Bethe Hessian operators. For the case of two symmetric clusters, we conjecture that these algorithms are asymptotically optimal in that they detect the clusters as soon as it is information theoretically possible to do so. We substantiate this claim for one of the spectral approaches we introduce

Epigenetics: What it is about?

Epigenetics has captured the attention of scientists in the past decades, yet its scope has been continuously changing. In this paper, we give an overview on how and why its definition has evolved and suggest several clarification on the concepts used in this field, in particular, on the notions of epigenetic information, epigenetic stability and epigenetic templating. Another issue that we address is the role of epigenetic information. Not only it is important in allowing alternative interpretations of genetic information, but it appears to be important in protecting the genetic information, moreover, we suggest that this function appeared first in evolution and only later on the epigenetic mechanisms were recruited to play a role in cell differentiation.Comment: 7 pages, 3 figure

Spectral Clustering of Graphs with the Bethe Hessian

Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the non-backtracking walk on the graph, leads to improved performance in detecting clusters, and even to optimal performance for the stochastic block model. Here, we propose to use instead a simpler object, a symmetric real matrix known as the Bethe Hessian operator, or deformed Laplacian. We show that this approach combines the performances of the non-backtracking operator, thus detecting clusters all the way down to the theoretical limit in the stochastic block model, with the computational, theoretical and memory advantages of real symmetric matrices.Comment: 8 pages, 2 figure

Exploring the latitude of attitude: Intentions to breastfeed among adolescents in Lebanese schools.

School-based breastfeeding education (SBBE) may help improve breastfeeding rates in the long-term by targeting children and adolescents' knowledge, attitudes, skills, and intentions. Breastfeeding rates in Lebanon are suboptimal. Psychosocial drivers of breastfeeding intention among the youth are unknown. We administered a survey to 658 high school students (448 females; 210 males) at two large Lebanese schools to understand intentions, intention drivers, and views on SBBE as means to guide SBBE programme design on the basis of the theory of planned behaviour. We collected information on demographics, intention to breastfeed/support wife to breastfeed future. Intention was predicted by attitude related to breastfeeding health outcomes and family normative beliefs-χ2 (25) = 115, P < .001 for males, and χ2 (39.3) = 186, P < .001 for females. Among females, intention was also positively associated with being breastfed, higher socio-economic status, and being more accepting of public breastfeeding. Seventy-eight per cent of students felt they were not learning enough about breastfeeding in school but were interested in SBBE through didactic teaching methods and interactive experiences. Findings indicate that breastfeeding intention among adolescent students is not merely influenced by the extent of knowledge but by more complicated psychosocial drivers that may differ by gender. Our findings also suggest a misalignment exists between what schools are providing with what students feel they need, thereby opening up a potential space for intervention