Large Dimensional Analysis and Optimization of Robust Shrinkage Covariance Matrix Estimators

This article studies two regularized robust estimators of scatter matrices proposed (and proved to be well defined) in parallel in (Chen et al., 2011) and (Pascal et al., 2013), based on Tyler's robust M-estimator (Tyler, 1987) and on Ledoit and Wolf's shrinkage covariance matrix estimator (Ledoit and Wolf, 2004). These hybrid estimators have the advantage of conveying (i) robustness to outliers or impulsive samples and (ii) small sample size adequacy to the classical sample covariance matrix estimator. We consider here the case of i.i.d. elliptical zero mean samples in the regime where both sample and population sizes are large. We demonstrate that, under this setting, the estimators under study asymptotically behave similar to well-understood random matrix models. This characterization allows us to derive optimal shrinkage strategies to estimate the population scatter matrix, improving significantly upon the empirical shrinkage method proposed in (Chen et al., 2011).Comment: Journal of Multivariate Analysi

Fracture toughness of SiC/Al metal matrix composite

An experimental study was conducted to evaluate fracture toughness of SiC/Al metal matrix composite (MMC). The material was a 12.7 mm thick extrusion of 6061-T6 aluminum alloy with 40 v/o SiC particulates. Specimen configuration and test procedure conformed to ASTM E399 Standard for compact specimens. It was found that special procedures were necessary to obtain fatigue cracks of controlled lengths in the preparation of precracked specimens for the MMC material. Fatigue loading with both minimum and maximum loads in compression was used to start the precrack. The initial precracking would stop by self-arrest. Afterwards, the precrack could be safely extended to the desired length by additional cyclic tensile loading. Test results met practically all the E399 criteria for the calculation of plane strain fracture toughness of the material. A valid K sub IC value of the SiC/Al composite was established as K sub IC = 8.9 MPa square root of m. The threshold stress intensity under which crack would cease to grow in the material was estimated as delta K sub th = 2MPa square root of m for R = 0.09 using the fatigue precracking data. Fractographic examinations show that failure occurred by the micromechanism involved with plastic deformation although the specimens broke by brittle fracture. The effect of precracking by cyclic loading in compression on fracture toughness is included in the discussion

Weighted sampling of outer products

This note gives a simple analysis of the randomized approximation scheme for matrix multiplication of Drineas et al (2006) with a particular sampling distribution over outer products. The result follows from a matrix version of Bernstein's inequality. To approximate the matrix product $AB^\top$ to spectral norm error $\varepsilon\|A\|\|B\|$, it suffices to sample on the order of $(\mathrm{sr}(A) \vee \mathrm{sr}(B)) \log(\mathrm{sr}(A) \wedge \mathrm{sr}(B)) / \varepsilon^2$ outer products, where $\mathrm{sr}(M)$ is the stable rank of a matrix $M$

Decompositions of ideals of minors meeting a submatrix

We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices.Comment: 10 pages. v2: minor corrections. v3: minor changes, final version to appear in Comm. Al

On the Burer-Monteiro method for general semidefinite programs

Consider a semidefinite program (SDP) involving an $n\times n$ positive semidefinite matrix $X$. The Burer-Monteiro method uses the substitution $X=Y Y^T$ to obtain a nonconvex optimization problem in terms of an $n\times p$ matrix $Y$. Boumal et al. showed that this nonconvex method provably solves equality-constrained SDPs with a generic cost matrix when $p \gtrsim \sqrt{2m}$, where $m$ is the number of constraints. In this note we extend their result to arbitrary SDPs, possibly involving inequalities or multiple semidefinite constraints. We derive similar guarantees for a fixed cost matrix and generic constraints. We illustrate applications to matrix sensing and integer quadratic minimization.Comment: 10 page

Manufacturing of Composites by Pressure Infiltration, Structure and Mechanical Properties

This paper presents the possibility of composite block production by using pressure infiltration technology. This method uses the pressure of an inert gas (usually argon or nitrogen) to force the melted matrix material to infiltrate the reinforcing elements. Two types of materials were considered: metal matrix syntactic foam and carbon fibre reinforced metal matrix composite. Physical and mechanical investigations – such as optical microscopy, scanning electron microscopy (SEM), X-ray diffractography (XRD), tensile and upsetting tests (considering aspect ratio) – were performed. The results of measurements are summarized briefly here. Microscopic investigations showed almost perfect infiltration. XRD measurements and tensile tests revealed negative effect of an intermetallic phase (Al(4)C(3)) on ultimate tensile strength (UTS). Syntactic foams showed plateau region in their upsetting diagrams. The effect of aspect ratio was also investigated. Specimens with higher aspect ratios showed higher peak stress and higher modulus of elasticity. In the case of carbon fibre reinforced metal matrix composites Al(4)C(3) ensured high compressive fracture strength

Remarks on the multi-species exclusion process with reflective boundaries

We investigate one of the simplest multi-species generalizations of the one dimensional exclusion process with reflective boundaries. The Markov matrix governing the dynamics of the system splits into blocks (sectors) specified by the number of particles of each kind. We find matrices connecting the blocks in a matrix product form. The procedure (generalized matrix ansatz) to verify that a matrix intertwines blocks of the Markov matrix was introduced in the periodic boundary condition, which starts with a local relation [Arita et al, J. Phys. A 44, 335004 (2011)]. The solution to this relation for the reflective boundary condition is much simpler than that for the periodic boundary condition