On the existence of 0/1 polytopes with high semidefinite extension complexity

In Rothvoß (Math Program 142(1–2):255–268, 2013) it was shown that there exists a 0/1 polytope (a polytope whose vertices are in {0, 1}n) such that any higher-dimensional polytope projecting to it must have 2Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high positive semidefinite extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations

Reverse Chv\'atal-Gomory rank

We introduce the reverse Chv\'atal-Gomory rank r*(P) of an integral polyhedron P, defined as the supremum of the Chv\'atal-Gomory ranks of all rational polyhedra whose integer hull is P. A well-known example in dimension two shows that there exist integral polytopes P with r*(P) equal to infinity. We provide a geometric characterization of polyhedra with this property in general dimension, and investigate upper bounds on r*(P) when this value is finite.Comment: 21 pages, 4 figure

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs

Toward sharing brain images: Differentially private TOF-MRA images with segmentation labels using generative adversarial networks

Sharing labeled data is crucial to acquire large datasets for various Deep Learning applications. In medical imaging, this is often not feasible due to privacy regulations. Whereas anonymization would be a solution, standard techniques have been shown to be partially reversible. Here, synthetic data using a Generative Adversarial Network (GAN) with differential privacy guarantees could be a solution to ensure the patient's privacy while maintaining the predictive properties of the data. In this study, we implemented a Wasserstein GAN (WGAN) with and without differential privacy guarantees to generate privacy-preserving labeled Time-of-Flight Magnetic Resonance Angiography (TOF-MRA) image patches for brain vessel segmentation. The synthesized image-label pairs were used to train a U-net which was evaluated in terms of the segmentation performance on real patient images from two different datasets. Additionally, the Fréchet Inception Distance (FID) was calculated between the generated images and the real images to assess their similarity. During the evaluation using the U-Net and the FID, we explored the effect of different levels of privacy which was represented by the parameter ϵ. With stricter privacy guarantees, the segmentation performance and the similarity to the real patient images in terms of FID decreased. Our best segmentation model, trained on synthetic and private data, achieved a Dice Similarity Coefficient (DSC) of 0.75 for ϵ = 7.4 compared to 0.84 for ϵ = ∞ in a brain vessel segmentation paradigm (DSC of 0.69 and 0.88 on the second test set, respectively). We identified a threshold of ϵ <5 for which the performance (DSC <0.61) became unstable and not usable. Our synthesized labeled TOF-MRA images with strict privacy guarantees retained predictive properties necessary for segmenting the brain vessels. Although further research is warranted regarding generalizability to other imaging modalities and performance improvement, our results mark an encouraging first step for privacy-preserving data sharing in medical imaging

Calcium-dependent dynamics of cadherin interactions at cell–cell junctions

Cadherins play a key role in the dynamics of cell–cell contact formation and remodeling of junctions and tissues. Cadherin–cadherin interactions are gated by extracellular Ca^(2+), which serves to rigidify the cadherin extracellular domains and promote trans junctional interactions. Here we describe the direct visualization and quantification of spatiotemporal dynamics of N-cadherin interactions across intercellular junctions in living cells using a genetically encodable FRET reporter system. Direct measurements of transjunctional cadherin interactions revealed a sudden, but partial, loss of homophilic interactions (τ = 1.17 ± 0.06 s^(−1)) upon chelation of extracellular Ca^(2+). A cadherin mutant with reduced adhesive activity (W2A) exhibited a faster, more substantial loss of homophilic interactions (τ = 0.86 ± 0.02 s^(−1)), suggesting two types of native cadherin interactions—one that is rapidly modulated by changes in extracellular Ca^(2+) and another with relatively stable adhesive activity that is Ca^(2+) independent. The Ca^(2+)-sensitive dynamics of cadherin interactions were transmitted to the cell interior where β-catenin translocated to N-cadherin at the junction in both cells. These data indicate that cadherins can rapidly convey dynamic information about the extracellular environment to both cells that comprise a junction

The Potential Involvement of E-cadherin and β-catenins in Meningioma

To investigate the potential involvements of E-cadherin and β-catenin in meningioma.Immunohistochemistry staining was performed on samples from patients with meningioma. The results were graded according to the positive ratio and intensity of tissue immunoreactivity. The expression of E-cadherin and β-catenin in meningioma was analyzed by its relationship with WHO2007 grading, invasion, peritumoral edema and postoperative recurrence.The positive rates of E-cadherin in meningioma WHO I, II, III were 92.69%, 33.33% and 0, respectively, (P<0.05); while the positive rates of β-catenin in meningioma WHO I, II, III were 82.93%, 33.33% and 20.00%, respectively, (P<0.05). The positive rate of E-cadherin in meningioma without invasion (94.12%) was higher than that with invasion (46.67%) (P<0.05). The difference in the positive rate of β-catenin between meningioma without invasion (88.24%) and meningioma with invasion (33.33%, P<0.05) was also statically significant. The positive rates of E-cadherin in meningioma with peritumoral edema 0, 1, 2, 3 were 93.75%, 85.71%, 60.00% and 0 respectively, (P<0.05); the positive rates of β-catenin in meningioma with peritumoral edema 0, 1, 2, 3 were 87.50%, 85.71%, 30.00% and 0 respectively, (P<0.01). The positive rates of E- cadherin in meningioma with postoperative recurrence were 33.33%, and the positive rate with postoperative non-recurrence was 90.00% (P<0.01). The positive rates of β-catenin in meningioma with postoperative recurrence and non-recurrence were 11.11%, 85.00%, respectively (P<0.01).The expression levels of E- cadherin and β-catenin correlated closely to the WHO 2007 grading criteria for meningioma. In atypical or malignant meningioma, the expression levels of E-cadherin and β-catenin were significantly lower. The expression levels of E- cadherin and β-catenin were also closely correlated with the invasion status of meningioma, the size of the peritumoral edema and the recurrent probabilities of the meningioma, all in an inverse correlationship. Taken together, the present study provided novel molecular targets in clinical treatments to meningioma