The Local Landscape of Phase Retrieval Under Limited Samples

In this paper, we provide a fine-grained analysis of the local landscape of phase retrieval under the regime with limited samples. Our aim is to ascertain the minimal sample size necessary to guarantee a benign local landscape surrounding global minima in high dimensions. Let $n$ and $d$ denote the sample size and input dimension, respectively. We first explore the local convexity and establish that when $n=o(d\log d)$, for almost every fixed point in the local ball, the Hessian matrix must have negative eigenvalues as long as $d$ is sufficiently large. Consequently, the local landscape is highly non-convex. We next consider the one-point strong convexity and show that as long as $n=\omega(d)$, with high probability, the landscape is one-point strongly convex in the local annulus: $\{w\in\mathbb{R}^d: o_d(1)\leqslant \|w-w^*\|\leqslant c\}$, where $w^*$ is the ground truth and $c$ is an absolute constant. This implies that gradient descent initialized from any point in this domain can converge to an $o_d(1)$-loss solution exponentially fast. Furthermore, we show that when $n=o(d\log d)$, there is a radius of $\widetilde\Theta\left(\sqrt{1/d}\right)$ such that one-point convexity breaks in the corresponding smaller local ball. This indicates an impossibility to establish a convergence to exact $w^*$ for gradient descent under limited samples by relying solely on one-point convexity.Comment: 41 page

Distinct expression profile and histological distribution of NLRP3 inflammasome components in the tissues of Hainan black goat suggest a site-specific role in the inflammatory response

The NOD-like receptor protein 3 (NLRP3) inflammasome comprised of NLRP3, ASC and caspase-1 plays an important role in the inflammatory and innate immune response. However, little is known about the expression pattern and histological distribution of these genes in goat. Here, we first cloned the fulllength cDNAs of the NLRP3, ASC and caspase-1 genes of Hainan black goat and produced their polyclonal antibodies. Tissue-specific expression and histological distribution of these genes were analysed. Phylogenetic analysis revealed that these three goat genes had high homology with Bos taurus genes and low homology with avian or fish genes. After immunisations with these recombinant Histagged proteins, the titres of antiserum were higher than 1:1024 and purified IgG was obtained. These three genes were expressed in all examined tissues, the mRNA expression level of NLRP3 and caspase-1 was most abundant in the spleen and mesenteric lymph nodes (MLNs), while ASC was primary expressed in the liver, spleen and kidney. The histological distribution of NLRP3, ASC and caspase-1 was detected in myocardial cells, hepatocytes, focal lymphocytes, bronchiolar epithelial cells, renal tubular epithelial cells, cortical neurons and endothelial cells of the germinal centres in the MLNs. These results will be helpful in further investigations into the function of the NLRP3 inflammasome and in elucidating its role in caprine inflammatory diseases

miR-152-3p Represses the Proliferation of the Thymic Epithelial Cells by Targeting <i>Smad2</i>

MicroRNAs (miRNAs) control the proliferation of thymic epithelial cells (TECs) for thymic involution. Previous studies have shown that expression levels of miR-152-3p were significantly increased in the thymus and TECs during the involution of the mouse thymus. However, the possible function and potential molecular mechanism of miR-152-3p remains unclear. This study identified that the overexpression of miR-152-3p can inhibit, while the inhibition of miR-152-3p can promote, the proliferation of murine medullary thymic epithelial cell line 1 (MTEC1) cells. Moreover, miR-152-3p expression was quantitatively analyzed to negatively regulate Smad2, and the Smad2 gene was found to be a direct target of miR-152-3p, using the luciferase reporter assay. Importantly, silencing Smad2 was found to block the G1 phase of cells and inhibit the cell cycle, which was consistent with the overexpression of miR-152-3p. Furthermore, co-transfection studies of siRNA–Smad2 (siSmad2) and the miR-152-3p mimic further established that miR-152-3p inhibited the proliferation of MTEC1 cells by targeting Smad2 and reducing the expression of Smad2. Taken together, this study proved miR-152-3p to be an important molecule that regulates the proliferation of TECs and therefore provides a new reference for delaying thymus involution and thymus regeneration