Robust Correlation Clustering

In this paper, we introduce and study the Robust-Correlation-Clustering problem: given a graph G = (V,E) where every edge is either labeled + or - (denoting similar or dissimilar pairs of vertices), and a parameter m, the goal is to delete a set D of m vertices, and partition the remaining vertices V D into clusters to minimize the cost of the clustering, which is the sum of the number of + edges with end-points in different clusters and the number of - edges with end-points in the same cluster. This generalizes the classical Correlation-Clustering problem which is the special case when m = 0. Correlation clustering is useful when we have (only) qualitative information about the similarity or dissimilarity of pairs of points, and Robust-Correlation-Clustering equips this model with the capability to handle noise in datasets. In this work, we present a constant-factor bi-criteria algorithm for Robust-Correlation-Clustering on complete graphs (where our solution is O(1)-approximate w.r.t the cost while however discarding O(1) m points as outliers), and also complement this by showing that no finite approximation is possible if we do not violate the outlier budget. Our algorithm is very simple in that it first does a simple LP-based pre-processing to delete O(m) vertices, and subsequently runs a particular Correlation-Clustering algorithm ACNAlg [Ailon et al., 2005] on the residual instance. We then consider general graphs, and show (O(log n), O(log^2 n)) bi-criteria algorithms while also showing a hardness of alpha_MC on both the cost and the outlier violation, where alpha_MC is the lower bound for the Minimum-Multicut problem

Pneumococcal vaccination in inflammatory rheumatic disease and in splenectomy patients. From antibody response to memory cells.

Objectives:The overall aim of the dissertation is to examine antibody response to immunization with pneumococcal vaccine in patients with inflammatory rheumatic disease (IRD), in relation to disease-modifying antirheumatic drug (DMARD) treatments, and in postsplenectomy patients.Methods:(I) Splenectomized patients without previous pneumococcal conjugate vaccine (PCV) immunization were invited to receive one dose 13-valent PCV (PCV13). Blood was drawn before and 4-6 weeks after PCV13. Serotype-specific antibody responses were determined using a multiplex fluorescent microsphere immunoassay (MFMI). (II and III) Consecutive patients with systemic vasulitis, rheumatoid arthritis (RA), and primary Sjögren’s syndrome (pSS), and healthy controls (HC) were invited to receive immunization with one dose PCV13. Serotype 6B and 23F IgG were determined before and 4-6 weeks after PCV13 using enzyme-linked immunosorbent assay (ELISA) and functionality of antibodies (23F) with an opsonophagocytic activity (OPA) assay. Positive antibody response (AR) was defined as ≥2-fold rise in pre- to postvaccination IgG. (IV) Patients with RA or systemic vasculitis and HC were invited to receive PCV and a booster dose with 23-valent pneumococcal polysaccharride vaccine (PCV23) after at least 8 weeks. IgG was determined before PCV and PPV23 and 4-6 weeks after using MFMI and OPA assay. (V) RA patients planned to start methotrexate (MTX) treatment, patients without DMARD and HC were included. Blood was obtained at inclusion, at immunization with PCV13 (after at least 6 weeks on MTX) and 7 days after for flow cytometric phenotyping of lymphocytes, and 4-6 weeks after for MFMI.Results:Splenectomy patients (n=24) with previous PPV23, received a dose of PCV13, and geometric mean concentration (GMC) increased for 9/12 serotypes. Patients with systemic vasculitis (n=49) and ongoing standard of care therapy received one dose of PCV13, IgG GMC for serotypes 6B and 23F increased, and there was no significant difference in antibody response (≥2-fold rise in IgG) compared to HC. Although OPA increased after PCV13, it was lower in patients compared to HC (p=0.001). In patients with RA (n=50) and pSS (n=15) without ongoing DMARD treatment IgG GMC for 6B and 23F and OPA increased, and the proportions with positive antibody responses for RA (52%) were similar to HC (55%, n=49). Patients with IRD treated with rituximab (RTX, n=30), abatacept (n=23), conventional DMARD (cDMARD, n=27) and HC (n=28) received immunization with PCV+PPV23. Antibody response improved after PPV23 in cDMARD (both 2-fold AR and OPA), and ABT (2-fold AR but not OPA), but no improvement was seen in RTX treated patients. Start of MTX treatment in RA patients resulted in decreased Th17 cells, and impaired memory B cell and plasmablast responses after PCV13.Conclusions:PCV is immunogenic as a booster dose in splenectomized patients with previous PPV23 immunization. PCV is immunogenic in systemic vaculitis patients with ongoing standard of care treatment, although functionality is lower compared to HC. Antibody response is not impaired in RA and pSS patients without DMARD treatment compared to HC. A PPV23 booster could be recommended in IRD patients with cDMARD, and ABT, but vaccination needs to be completed before starting RTX. MTX treatment can have negative effects on memory B cells following PCV13

The Universality of Power Law Slopes in the Solar Photosphere and Transition Region Observed with HMI and IRIS

We compare the size distributions of self-organized criticality (SOC) systems in the solar photosphere and the transition region, using magnetogram data from Helioseismic and Magnetic Imager (HMI) and Interface Region Imaging Spectrograph (IRIS)} data. For each dataset we fit a combination of a Gaussian and a power law size distribution function, which yields information on four different physical processes: (i) Gaussian random noise in IRIS data; (ii) spicular events in the plages of the transition region (described by power law size distribution in IRIS data); (iii) salt-and-pepper small-scale magnetic structures (described by the random noise in HMI magnetograms); and (iv) magnetic reconnection processes in flares and nanoflares (described by power law size distributions in HMI data). We find a high correlation (CCC=0.90) between IRIS and HMI data. Datasets with magnetic flux balance are generally found to match the SOC-predicted power law slope a_F=1.80 (for mean fluxes F), but exceptions occur due to arbitrary choices of the HMI field-of-view. The matching cases confirm the universality of SOC-inferred flux size distributions, and agree with the results of Parnell et al.~(2009), a_F=1.85 +/- 0.14.Comment: text 17 pages, 3 Tables, 8 Figure

Greedy Pruning with Group Lasso Provably Generalizes for Matrix Sensing

Pruning schemes have been widely used in practice to reduce the complexity of trained models with a massive number of parameters. In fact, several practical studies have shown that if a pruned model is fine-tuned with some gradient-based updates it generalizes well to new samples. Although the above pipeline, which we refer to as pruning + fine-tuning, has been extremely successful in lowering the complexity of trained models, there is very little known about the theory behind this success. In this paper, we address this issue by investigating the pruning + fine-tuning framework on the overparameterized matrix sensing problem with the ground truth $U_\star \in \mathbb{R}^{d \times r}$ and the overparameterized model $U \in \mathbb{R}^{d \times k}$ with $k \gg r$. We study the approximate local minima of the mean square error, augmented with a smooth version of a group Lasso regularizer, $\sum_{i=1}^k \| U e_i \|_2$. In particular, we provably show that pruning all the columns below a certain explicit $\ell_2$-norm threshold results in a solution $U_{\text{prune}}$ which has the minimum number of columns $r$, yet close to the ground truth in training loss. Moreover, in the subsequent fine-tuning phase, gradient descent initialized at $U_{\text{prune}}$ converges at a linear rate to its limit. While our analysis provides insights into the role of regularization in pruning, we also show that running gradient descent in the absence of regularization results in models which {are not suitable for greedy pruning}, i.e., many columns could have their $\ell_2$ norm comparable to that of the maximum. To the best of our knowledge, our results provide the first rigorous insights on why greedy pruning + fine-tuning leads to smaller models which also generalize well.Comment: 49 pages, 2 figure

Missing Mass of Rank-2 Markov Chains

Estimation of missing mass with the popular Good-Turing (GT) estimator is well-understood in the case where samples are independent and identically distributed (iid). In this article, we consider the same problem when the samples come from a stationary Markov chain with a rank-2 transition matrix, which is one of the simplest extensions of the iid case. We develop an upper bound on the absolute bias of the GT estimator in terms of the spectral gap of the chain and a tail bound on the occupancy of states. Borrowing tail bounds from known concentration results for Markov chains, we evaluate the bound using other parameters of the chain. The analysis, supported by simulations, suggests that, for rank-2 irreducible chains, the GT estimator has bias and mean-squared error falling with number of samples at a rate that depends loosely on the connectivity of the states in the chain