Planting trees in graphs, and finding them back

In this paper we study detection and reconstruction of planted structures in Erd\H{o}s-R\'enyi random graphs. Motivated by a problem of communication security, we focus on planted structures that consist in a tree graph. For planted line graphs, we establish the following phase diagram. In a low density region where the average degree $\lambda$ of the initial graph is below some critical value $\lambda_c=1$, detection and reconstruction go from impossible to easy as the line length $K$ crosses some critical value $f(\lambda)\ln(n)$, where $n$ is the number of nodes in the graph. In the high density region $\lambda>\lambda_c$, detection goes from impossible to easy as $K$ goes from $o(\sqrt{n})$ to $\omega(\sqrt{n})$, and reconstruction remains impossible so long as $K=o(n)$. For $D$-ary trees of varying depth $h$ and $2\le D\le O(1)$, we identify a low-density region $\lambda<\lambda_D$, such that the following holds. There is a threshold $h*=g(D)\ln(\ln(n))$ with the following properties. Detection goes from feasible to impossible as $h$ crosses $h*$. We also show that only partial reconstruction is feasible at best for $h\ge h*$. We conjecture a similar picture to hold for $D$-ary trees as for lines in the high-density region $\lambda>\lambda_D$, but confirm only the following part of this picture: Detection is easy for $D$-ary trees of size $\omega(\sqrt{n})$, while at best only partial reconstruction is feasible for $D$-ary trees of any size $o(n)$. These results are in contrast with the corresponding picture for detection and reconstruction of {\em low rank} planted structures, such as dense subgraphs and block communities: We observe a discrepancy between detection and reconstruction, the latter being impossible for a wide range of parameters where detection is easy. This property does not hold for previously studied low rank planted structures

A non-backtracking method for long matrix and tensor completion

We consider the problem of low-rank rectangular matrix completion in the regime where the matrix $M$ of size $n\times m$ is ``long", i.e., the aspect ratio $m/n$ diverges to infinity. Such matrices are of particular interest in the study of tensor completion, where they arise from the unfolding of a low-rank tensor. In the case where the sampling probability is $\frac{d}{\sqrt{mn}}$, we propose a new spectral algorithm for recovering the singular values and left singular vectors of the original matrix $M$ based on a variant of the standard non-backtracking operator of a suitably defined bipartite weighted random graph, which we call a \textit{non-backtracking wedge operator}. When $d$ is above a Kesten-Stigum-type sampling threshold, our algorithm recovers a correlated version of the singular value decomposition of $M$ with quantifiable error bounds. This is the first result in the regime of bounded $d$ for weak recovery and the first for weak consistency when $d\to\infty$ arbitrarily slowly without any polylog factors. As an application, for low-rank orthogonal $k$-tensor completion, we efficiently achieve weak recovery with sample size $O(n^{k/2})$, and weak consistency with sample size $\omega(n^{k/2})$

Non-backtracking spectra of weighted inhomogeneous random graphs

We study a model of random graphs where each edge is drawn independently (but not necessarily identically distributed) from the others, and then assigned a random weight. When the mean degree of such a graph is low, it is known that the spectrum of the adjacency matrix $A$ deviates significantly from that of its expected value $\mathbb E A$. In contrast, we show that over a wide range of parameters the top eigenvalues of the non-backtracking matrix $B$ -- a matrix whose powers count the non-backtracking walks between two edges -- are close to those of $\mathbb E A$, and all other eigenvalues are confined in a bulk with known radius. We also obtain a precise characterization of the scalar product between the eigenvectors of $B$ and their deterministic counterparts derived from the model parameters. This result has many applications, in domains ranging from (noisy) matrix completion to community detection, as well as matrix perturbation theory. In particular, we establish as a corollary that a result known as the Baik-Ben Arous-P\'ech\'e phase transition, previously established only for rotationally invariant random matrices, holds more generally for matrices $A$ as above under a mild concentration hypothesis.Comment: 60 page

Escaping mediocrity: how two-layer networks learn hard single-index models with SGD

This study explores the sample complexity for two-layer neural networks to learn a single-index target function under Stochastic Gradient Descent (SGD), focusing on the challenging regime where many flat directions are present at initialization. It is well-established that in this scenario $n=O(d\log{d})$ samples are typically needed. However, we provide precise results concerning the pre-factors in high-dimensional contexts and for varying widths. Notably, our findings suggest that overparameterization can only enhance convergence by a constant factor within this problem class. These insights are grounded in the reduction of SGD dynamics to a stochastic process in lower dimensions, where escaping mediocrity equates to calculating an exit time. Yet, we demonstrate that a deterministic approximation of this process adequately represents the escape time, implying that the role of stochasticity may be minimal in this scenario

Non-backtracking spectra of weighted inhomogeneous random graphs

80 pagesWe study a model of random graphs where each edge is drawn independently (but not necessarily identically distributed) from the others, and then assigned a random weight. When the mean degree of such a graph is low, it is known that the spectrum of the adjacency matrix $A$ deviates significantly from that of its expected value $\mathbb E A$. In contrast, we show that over a wide range of parameters the top eigenvalues of the non-backtracking matrix $B$ -- a matrix whose powers count the non-backtracking walks between two edges -- are close to those of $\mathbb E A$, and all other eigenvalues are confined in a bulk with known radius. We also obtain a precise characterization of the scalar product between the eigenvectors of $B$ and their deterministic counterparts derived from the model parameters. This result has many applications, in domains ranging from (noisy) matrix completion to community detection, as well as matrix perturbation theory. In particular, we establish as a corollary that a result known as the Baik-Ben Arous-P\'ech\'e phase transition, previously established only for rotationally invariant random matrices, holds more generally for matrices $A$ as above under a mild concentration hypothesis

Learning Two-Layer Neural Networks, One (Giant) Step at a Time

We study the training dynamics of shallow neural networks, investigating the conditions under which a limited number of large batch gradient descent steps can facilitate feature learning beyond the kernel regime. We compare the influence of batch size and that of multiple (but finitely many) steps. Our analysis of a single-step process reveals that while a batch size of $n = O(d)$ enables feature learning, it is only adequate for learning a single direction, or a single-index model. In contrast, $n = O(d^2)$ is essential for learning multiple directions and specialization. Moreover, we demonstrate that ``hard'' directions, which lack the first $\ell$ Hermite coefficients, remain unobserved and require a batch size of $n = O(d^\ell)$ for being captured by gradient descent. Upon iterating a few steps, the scenario changes: a batch-size of $n = O(d)$ is enough to learn new target directions spanning the subspace linearly connected in the Hermite basis to the previously learned directions, thereby a staircase property. Our analysis utilizes a blend of techniques related to concentration, projection-based conditioning, and Gaussian equivalence that are of independent interest. By determining the conditions necessary for learning and specialization, our results highlight the interaction between batch size and number of iterations, and lead to a hierarchical depiction where learning performance exhibits a stairway to accuracy over time and batch size, shedding new light on feature learning in neural networks

Clinicians' Attitudes Toward the Use of Long-Acting Injectable Antipsychotics

Depot formulations are not widely used in everyday practice. This study aimed to assess psychiatrists' attitudes toward the use of long-acting injectable (LAI) antipsychotics in schizophrenia. We interviewed 113 French psychiatrists about the factors that influenced their prescription of LAI antipsychotics. Multidimensional and cluster analyses were used to detect correlations. The most important factor against the use of LAI antipsychotics is a sufficient estimated compliance with the oral formulation. For first-generation LAI, the main factor is the risk for extrapyramidal symptoms; and for second-generation LAI, it is the unavailability of the equivalent oral formulation. Four factors incite the psychiatrists to prescribe LAI. Two different clusters of patients can also be identified. Most factors influencing the clinicians' attitudes toward the use of LAI antipsychotics are shared in many countries. Conversely, some attitudes related to organizational aspects, particularly the relevance of health care costs, may vary from one country to another

Clinicians' Attitudes Toward the Use of Long-Acting Injectable Antipsychotics

Depot formulations are not widely used in everyday practice. This study aimed to assess psychiatrists' attitudes toward the use of long-acting injectable (LAI) antipsychotics in schizophrenia. We interviewed 113 French psychiatrists about the factors that influenced their prescription of LAI antipsychotics. Multidimensional and cluster analyses were used to detect correlations. The most important factor against the use of LAI antipsychotics is a sufficient estimated compliance with the oral formulation. For first-generation LAI, the main factor is the risk for extrapyramidal symptoms; and for second-generation LAI, it is the unavailability of the equivalent oral formulation. Four factors incite the psychiatrists to prescribe LAI. Two different clusters of patients can also be identified. Most factors influencing the clinicians' attitudes toward the use of LAI antipsychotics are shared in many countries. Conversely, some attitudes related to organizational aspects, particularly the relevance of health care costs, may vary from one country to another

Non-CG DNA methylation is a biomarker for assessing endodermal differentiation capacity in pluripotent stem cells.

Non-CG methylation is an unexplored epigenetic hallmark of pluripotent stem cells. Here we report that a reduction in non-CG methylation is associated with impaired differentiation capacity into endodermal lineages. Genome-wide analysis of 2,670 non-CG sites in a discovery cohort of 25 phenotyped human induced pluripotent stem cell (hiPSC) lines revealed unidirectional loss (Δβ=13%, P<7.4 × 10(-4)) of non-CG methylation that correctly identifies endodermal differentiation capacity in 23 out of 25 (92%) hiPSC lines. Translation into a simplified assay of only nine non-CG sites maintains predictive power in the discovery cohort (Δβ=23%, P<9.1 × 10(-6)) and correctly identifies endodermal differentiation capacity in nine out of ten pluripotent stem cell lines in an independent replication cohort consisting of hiPSCs reprogrammed from different cell types and different delivery systems, as well as human embryonic stem cell (hESC) lines. This finding infers non-CG methylation at these sites as a biomarker when assessing endodermal differentiation capacity as a readout.We thank Kerra Pearce (UCL Genomics) for array processing, and Tim Fell and Jonathan Best (CellCentric), Jason Wray (UCL) and Rosemary Drake (TAP Biosystems) for discussions. We also thank Minal Patel, Chris Kirton, Anja Kolb-Kokocinski, Willem H. Ouwehand, Richard Durbin and Fiona M. Watt on behalf of the Human Induced Pluripotent Stem Cell Initiative (HipSci) funded by grant WT098503 from the Wellcome Trust and the Medical Research Council, for sharing data and materials. This work was supported in part by a TSB/EPSRC grant (TS/H000933/1). The Vallier lab is supported by the Cambridge Hospitals National Institute for Health Research Biomedical Research Center and an ERC Starting Grant (Relieve IMDS). F.A.C.S. is funded by a PhD studentship from Fundação para a Ciência e a Tecnologia (SFRH/BD/69033/2010). The Ferguson-Smith lab is supported by grants from the MRC and Wellcome Trust, and EU-FP7 projects EPIGENESYS (257082) and BLUEPRINT (282510). The Beck lab is supported by the Wellcome Trust (084071), a Royal Society Wolfson Research Merit Award (WM100023), and EU-FP7 projects EPIGENESYS (257082) and BLUEPRINT (282510).This is the final version of the article. It first appeared from Nature Publishing Group via http://dx.doi.org/10.1038/ncomms1045

Signs and Symptoms of Mercury-Exposed Gold Miners

Objectives: Gold miners use mercury to extract gold from ore adding liquid mercury to the milled gold-containing ore. This results in a mercury-gold compound, called amalgam. Miners smelt this amalgam to obtain gold, vaporizing it and finally inhaling the toxic mercury fumes. The objective was to merge and analyze data from different projects, to identify typical signs and symptoms of chronic inorganic mercury exposure. Material and Methods: Miners and community members from various artisanal small-scale gold mining areas had been examined (Philippines, Mongolia, Tanzania, Zimbabwe, Indonesia). Data of several health assessments were pooled. Urine, blood and hair samples were analyzed for mercury (N = 1252). Questionnaires, standardized medical examinations and neuropsychological tests were used. Participants were grouped into: Controls (N = 209), living in an exposed area (N = 408), working with mercury as panners (N = 181), working with mercury as amalgam burners (N = 454). Ch(i)2 test, linear trend test, Mann-Whitney test, Kruskal-Wallis test, correlation coefficient, Spearman's rho, and analysis of variance tests were used. An algorithm was used to define participants with chronic mercury intoxication. Results: Mean mercury concentrations in all exposed subgroups were elevated and above threshold limits, with amalgam burners showing highest levels. Typical symptoms of chronic metallic mercury intoxication were tremor, ataxia, coordination problems, excessive salivation and metallic taste. Participants from the exposed groups showed poorer results in different neuropsychological tests in comparison to the control group. Fifty-four percent of the high-exposed group (amalgam burners) were diagnosed as being mercury-intoxicated, compared to 0% within the control group (Chi(2) p < 0.001). Conclusions: Chronic mercury intoxication, with tremor, ataxia and other neurological symptoms together with a raised body burden of mercury was clinically diagnosed in exposed people in artisanal small-scale mining areas. The mercury exposure needs to be urgently reduced. Health care systems need to be prepared for this emerging problem of chronic mercury intoxication among exposed people