A semidefinite program for unbalanced multisection in the stochastic block model

We propose a semidefinite programming (SDP) algorithm for community detection in the stochastic block model, a popular model for networks with latent community structure. We prove that our algorithm achieves exact recovery of the latent communities, up to the information-theoretic limits determined by Abbe and Sandon (2015). Our result extends prior SDP approaches by allowing for many communities of different sizes. By virtue of a semidefinite approach, our algorithms succeed against a semirandom variant of the stochastic block model, guaranteeing a form of robustness and generalization. We further explore how semirandom models can lend insight into both the strengths and limitations of SDPs in this setting.Comment: 29 page

Computational Hardness of Certifying Bounds on Constrained PCA Problems

Given a random n×n symmetric matrix W drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form x⊤Wx over all vectors x in a constraint set S⊂Rn. For a certain class of normalized constraint sets S we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of W. A notable special case included in our results is the hypercube S={±1/n−−√}n, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics. Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over x∈S is much larger than that of a GOE matrix.ISSN:1868-896

Estimation under group actions: recovering orbits from invariants

Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe very noisy copies of an unknown signal, each acted upon by a random element of some group (such as Z/p or SO(3)). The goal is to recover the orbit of the signal under the group action in the high-noise regime. This generalizes problems of interest such as multi-reference alignment (MRA) and the reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain matching lower and upper bounds on the sample complexity of these problems in high generality, showing that the statistical difficulty is intricately determined by the invariant theory of the underlying symmetry group. In particular, we determine that for cryo-EM with noise variance $\sigma^2$ and uniform viewing directions, the number of samples required scales as $\sigma^6$. We match this bound with a novel algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from heterogeneous cryo-EM samples.Comment: 54 pages. This version contains a number of new result

Quantum repeaters with individual rare-earth ions at telecommunication wavelengths

We present a quantum repeater scheme that is based on individual erbium and europium ions. Erbium ions are attractive because they emit photons at telecommunication wavelength, while europium ions offer exceptional spin coherence for long-term storage. Entanglement between distant erbium ions is created by photon detection. The photon emission rate of each erbium ion is enhanced by a microcavity with high Purcell factor, as has recently been demonstrated. Entanglement is then transferred to nearby europium ions for storage. Gate operations between nearby ions are performed using dynamically controlled electric-dipole coupling. These gate operations allow entanglement swapping to be employed in order to extend the distance over which entanglement is distributed. The deterministic character of the gate operations allows improved entanglement distribution rates in comparison to atomic ensemble-based protocols. We also propose an approach that utilizes multiplexing in order to enhance the entanglement distribution rate.Comment: 13 pages, 4 figure

Reference concentrations for trace elements in urine for the Brazilian population based on q-ICP-MS with a simple dilute-and-shoot procedure

Biomonitoring of trace elements is of critical importance in human health assessment. However, trace element concentrations in biological fluids are affected by environmental and physiological parameters, and therefore considerable variations can occur between specific population subgroups. Brazil is a large country with large environmental diversity and with a limited knowledge of the reference values (baseline data) for trace elements in biological fluids. Atomic absorption spectrometry (AAS) and inductively coupled plasma emission spectrometry (ICP-OES) are still the dominant analytical techniques used for biomonitoring trace element analysis in clinical specimens. However, the use of ICP-MS is becoming more usual in clinical laboratory analysis. Then, we evaluated here a simple dilute-and-shoot method for sequential determination of Al, Ba, Be, Cd, Co, Cu, Cs, Mn, Ni, Pb, Pt, Sb, Se, Sn, Tl and U in urine by quadrupole inductively coupled plasma mass spectrometry (q-ICP-MS). Urine samples (500 µL) were accurately pipetted into conical tubes (15 mL) and diluted to 10 mL with a solution containing 0.5 % (v/v) HNO3 + 0.005% (v/v) Triton X-100. Diluted urine samples also contain rhodium, iridium and yttrium added as internal standards. After that, samples were directly analyzed by ICP-MS against matrix-matching calibration. Method detection limit (3s, n = 20) were in the ng L-1 range for all analytes. The method was applied to the analysis of 412 ordinary urine samples from Brazilian healthy and non-exposed subjects to establish reference values. Data validation was provided by the analysis of the standard reference material (SRM) 2670a toxic elements in urine (freeze-dried) (high and low levels) from the National Institute of Standards and Technology (NIST) and reference urine samples from the trace elements intercomparison program operated by the Institut National de Sante' Publique du Quebec, Canada.O biomonitoramento de elementos químicos é de extrema importância na avaliação da saúde humana. Entretanto, as concentrações dos elementos químicos nos fluidos biológicos são afetadas por parâmetros ambientais e fisiológicos e, consequentemente, consideráveis variações podem ocorrer entre subgrupos de populações específicas. O Brasil é um país com ampla diversidade ambiental e existe limitado conhecimento de valores de referência para elementos químicos em fluidos biológicos. A espectrometria de absorção atômica (AAS) e a espectrometria de emissão ótica com plasma acoplado indutivamente (ICP-OES) ainda são as técnicas analíticas mais comumente empregadas no biomonitoramento de elementos químicos em amostras clínicas. Entretanto, o uso da espectrometria de massas com plasma acoplado indutivamente (ICP-MS) está se tornando a cada dia mais comum nos laboratórios clínicos. Neste estudo, foi avaliado um método rápido envolvendo simples diluição da amostra para determinação de Al, Ba, Be, Cd, Co, Cu, Cs, Mn, Ni, Pb, Pt, Sb, Se, Sn, Tl e U em urina por ICP-MS. Amostras de urina (500 μL) foram pipetadas em frascos cônicos de 15 mL e diluídas para 10 mL com uma solução contendo 0,5 % (v/v) HNO3 + 0,005% (v/v) Triton X-100. Ródio, irídio e ítrio foram adicionados como padrões internos. Em seguida, as amostras foram diretamente analisadas por ICP-MS com calibração por ajuste de matriz. Os limites de detecção do método (3s, n = 20) foram da ordem de ng L-1 para todos os analitos em estudo. O método foi aplicado para a análise de 412 amostras de urina de brasileiros saudáveis e não expostos ambientalmente ou ocupacionalmente a metais para o estabelecimento de valores de referência na população brasileira. A validação dos resultados foi acompanhada pela análise de material de referência certificada de urina (SRM) 2670a proveniente do National Institute of Standards and Technology (NIST) e de materiais de referência provenientes do Institut National de Sante' Publique Du Quebec, no Canadá.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq