Tail estimates for norms of sums of log-concave random vectors

We establish new tail estimates for order statistics and for the Euclidean norms of projections of an isotropic log-concave random vector. More generally, we prove tail estimates for the norms of projections of sums of independent log-concave random vectors, and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub-matrices in the setting of a log-concave ensemble. This is used to study a quantity $A_{k,m}$ that controls uniformly the operator norm of the sub-matrices with $k$ rows and $m$ columns of a matrix $A$ with independent isotropic log-concave random rows. We apply our tail estimates of $A_{k,m}$ to the study of Restricted Isometry Property that plays a major role in the Compressive Sensing theory

Chevet type inequality and norms of submatrices

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of expectation of the supremum of "symmetric exponential" processes compared to the Gaussian ones in the Chevet inequality. This is used to give sharp upper estimate for a quantity $\Gamma_{k,m}$ that controls uniformly the Euclidean operator norm of the sub-matrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the Restricted Isometry Constant of a random matrix with independent log-concave unconditional rows. We show also that our Chevet type inequality does not extend to general isotropic log-concave random matrices

Combining X-ray CT and 3D printing technology to produce microcosms with replicable, complex pore geometries

Measurements in soils have been traditionally used to demonstrate that soil architecture is one of the key drivers of soil processes. Major advances in the use of X-ray Computed Tomography (CT) afford significant insight into the pore geometry of soils, but until recently no experimental techniques were available to reproduce this complexity in microcosms. This article describes a 3D additive manufacturing technology that can print physical structures with pore geometries reflecting those of soils. The process enables printing of replicated structures, and the printing materials are suitable to study fungal growth. This technology is argued to open up a wealth of opportunities for soil biological studies

Regulation of Contact with Offspring by Domestic Sows: Temporal Patterns and Individual Variation

We used a sow-controlled housing system to examine temporal and individual variation in the tendency of sows to associate with young. During a 5-week lactation, 22 sows and litters were housed in a pen where the sow could freely leave and re-enter the piglets\u27 area by stepping over a barrier that the piglets could not cross. Despite this option, the sows remained with the piglets almost constantly during the 1st day after birth. Nineteen sows (\u27leavers\u27) changed to spending most of their time away from the litter at some point in the lactation. The change was rapid, often within a single week, and occurred in week 2, 3, 4 or 5, depending on the individual. The time of rapid increase in time away was not related to characteristics of the sow or litter, including parity, litter size and sex ratio. Three sows (\u27stayers\u27) did not increase their time away as lactation advanced, and rarely spent more than 15% of their day in the piglet-free area. Nearly all sows showed a clear preference to defecate in the piglet-free area. This study shows 1. that sows voluntarily reduce their contact with the young; 2. that the timing of this reduction varies greatly amongst sows for reasons that may relate to differences in maternal motivation, and 3. that sows do not abandon the litter if the young cannot follow. The clear preference that most sows developed for the piglet-free area reinforces physiological evidence that constant confinement with older litters is aversive for many sows

The Relationship between Creep Feeding Behavior of Piglets and Adaptation to Weaning: Effect of Diet Quality

Individual variation in creep feed intake and its relation to adaptation to weaning were studied in piglets weaned at 4 wk of age. The animals received either a low-complexity creep-starter diet based on corn, barley and soybean meal (12 litters), or a high-complexity , medicated, commercial diet without soybean meal (12 litters). Diets were fed as creep feed during the 2 wk before weaning, and as the sole diet during the 2 wk after weaning. Creep feeding behavior of piglets was monitored by video recording. Pigs fed the high-complexity diet consumed more creep feed (P \u3c 0.05), tended to gain more during the week before weaning (P \u3c 0.10), and converted feed more efficiently and gained more weight in the 2 wk after weaning (P \u3c 0.01). Use of creep feed varied greatly among individual littermates. Multiple regression analysis showed that on the high-complexity diet, pigs that used creep feed more than their littermates tended to be those with low gains in weeks 1-3 after birth (P \u3c 0.001), and tended to gain more weight during the week before and during the 2 wk after weaning (P \u3c 0.01). The trends were consistent but weaker with the low-complexity diet. However, predictive power was low, with creep feeding accounting for only 4% of individual variation in post-weaning gain on the high-complexity diet and 1% on the low. Regardless of diet quality, therefore, creep feeding remained highly variable and only weakly related to weight gains during the 2 wk after weaning

Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles

Let $K$ be an isotropic convex body in $\R^n$. Given \eps>0, how many independent points $X_i$ uniformly distributed on $K$ are needed for the empirical covariance matrix to approximate the identity up to \eps with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits. More precisely, let $X\in\R^n$ be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector $X$ is a random point in an isotropic convex body. We show that for any \eps>0, there exists C(\eps)>0, such that if N\sim C(\eps) n and $(X_i)_{i\le N}$ are i.i.d. copies of $X$, then \Big\|\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \Id\Big\| \le \epsilon, with probability larger than $1-\exp(-c\sqrt n)$.Comment: Exposition changed, several explanatory remarks added, some proofs simplifie

Sharp bounds on the rate of convergence of the empirical covariance matrix

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has \sup_{x\in S^{n-1}} \Big|\frac{1/N}\sum_{i=1}^N (||^2 - \E||^2\r)\Big| \leq C \sqrt{\frac{n/N}}, where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $()_{i\leq N, x\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\sqrt n)_{i\leq N}$ exhibit uniformly a sub-exponential decay. As a consequence, if $A$ denotes the random matrix with columns $(X_i)$, then with overwhelming probability, the extremal singular values $\lambda_{\rm min}$ and $\lambda_{\rm max}$ of $AA^\top$ satisfy the inequalities $1 - C\sqrt{{n/N}} \le {\lambda_{\rm min}/N} \le \frac{\lambda_{\rm max}/N} \le 1 + C\sqrt{{n/N}}$ which is a quantitative version of Bai-Yin theorem \cite{BY} known for random matrices with i.i.d. entries