Gaussian Mixture Identifiability from degree 6 Moments

We resolve most cases of identifiability from sixth-order moments for Gaussian mixtures on spaces of large dimensions. Our results imply that the parameters of a generic mixture of $m\leq\mathcal{O}(n^4)$ Gaussians on $\mathbb R^n$ can be uniquely recovered from the mixture moments of degree 6. The constant hidden in the $\mathcal{O}$-notation is optimal and equals the one in the upper bound from counting parameters. We give an argument that degree-4 moments never suffice in any nontrivial case, and we conduct some numerical experiments indicating that degree 5 is minimal for identifiability.Comment: 22 page

Third Powers of Quadratics are generically Identifiable up to quadratic Rank

We consider the inverse problem for the map $$(S^2 (\mathbb C^n))^{m} \to S^6(\mathbb C^n), (q_{1},\ldots, q_{m}) \mapsto \sum_{i=1}^m q_{i}^3, \qquad (n, m \in \mathbb N)$$ which captures the moment problem for mixtures of centered Gaussians in the smallest interesting degree. We show that for any $n\in \mathbb N$, this map is generically one-to-one (up to permutations of $q_1,\ldots, q_m$) as long as $m\le {n\choose 2} + 1$, thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most $6$ up to rank ${n\choose 2} + 1$. We rely on the study of tangent spaces of secant varieties and the contact locus.Comment: 14 pages. Code for the base case computations can be found on GitHu

DevOps Continuous Integration: Moving Germany’s Federal Employment Agency Test System into Embedded In-Memory Technology

This paper describes the development of a continuous integration database test architecture for a highly important and large software application in the public sector in Germany. We apply action design research and draw from two emerging areas of research – DevOps continuous integration practices and in-memory database development – to define the problem, design, build and implement the solution, analyze challenges encountered, and make adjustments. The result is the transformation of a large test environment originally based on Oracle databases into a flexible and fast embedded in-memory architecture. The main challenges involved overcoming the differences between the SQL specifications supported by the development and production systems and optimizing the test runtime performance. The paper contributes to theory and practice by presenting one of the first studies showing a real-world implementation of a successful database test architecture that enables continuous integration, and identifying technical design principles for database test architectures in general

Gaussian mixture identifiability from degree 6 moments

We resolve most cases of identifiability from sixth-order moments for Gaussian mixtures on spaces of large dimensions. Our results imply that the parameters of a generic mixture of m~Θ(n^4) Gaussians on ℝ^n can be uniquely recovered from the mixture moments of degree 6. The constant hidden in the O-notation is optimal and equals the one in the upper bound from counting parameters. We give an argument that degree-4 moments never suffice in any nontrivial case, and we conduct some numerical experiments indicating that degree 5 is minimal for identifiability

Unique powers-of-forms decompositions from simple Gram spectrahedra

We consider simultaneous Waring decompositions: Given forms of degrees k*d, (d=2,3), which admit a representation as d-th power sums of k-forms, when is it possible to reconstruct the individual terms from the power sums? Such powers-of-forms decompositions model the moment problem for mixtures of centered Gaussians. The novel approach of this paper is to use semidefinite programming in order to perform a reduction to tensor decomposition. The proposed method works on typical parameter sets at least as long as m≤n−1, where m is the rank of the decomposition and n is the number of variables. While provably not tight, this analysis still gives the currently best known rank threshold for decomposing third order powers-of-forms, improving on previous work in both asymptotics and constant factors. Our algorithm can produce proofs of uniqueness for specific decompositions. A numerical study is conducted on Gaussian random trace-free quadratics, giving evidence that the success probability converges to 1 in an average case setting, as long as m=n and n→∞. Some evidence is given that the algorithm also succeeds on instances of rank quadratic in the dimension

Effect of molecular architecture on the crystalline structure and stiffness of iPP homopolymers: modeling based on annealing experiments

Five PP homopolymers were selected and their molecular structure was thoroughly characterized to determine the effect of molecular architecture on their annealing behavior and on the ultimate stiffness achieved by heat treatment. Molecular mass and its distribution were characterized by rheological measurements, while chain regularity was determined by calorimetry, by the stepwise isothermal segregation technique (SIST). The samples were annealed in two different ways. Tensile bars were treated in an oven at 165 °C for increasing times to determine changes in stiffness. Various defects developed during the annealing of tensile specimens that did not allow the reliable determination of modulus by direct measurement. On the other hand, the second approach, the annealing of small samples in a DSC cell clearly showed the changes occurring in crystalline structure and also the effect of nucleation and molecular architecture on them. The large molecular weight fraction used to facilitate nucleation hinders crystal perfection, while the presence of a heterogeneous nucleating agent increases overall crystallinity, but does not influence recrystallization during annealing. Melting traces were transformed into lamella thickness distributions, from which average lamella thickness was derived. Lamella thickness and crystallinity, the independent variables of the empirical equation used for the calculation of modulus, were extrapolated to infinite annealing time to predict maximum stiffness. The value obtained, 3.5 GPa, is very far from the theoretically predicted 40 GPa of oriented crystals, which cannot be achieved under practical conditions

Identifiability for mixtures of centered Gaussians and sums of powers of quadratics

We consider the inverse problem for the polynomial \nmap that sends an -tuple of quadratic forms in \nvariables to the sum of their th powers. This map cap- \ntures the moment problem for mixtures of centered \n-variate Gaussians. In the first nontrivial case = 3, \nwe show that for any \xe2\x88\x88 \xe2\x84\x95, this map is generically one- \nto-one (up to permutations of 1 , \xe2\x80\xa6 , and third roots \nof unity) in two ranges: \xe2\xa9\xbd ( \n2 \n) + 1 for < 16 and \xe2\xa9\xbd \n(+5 \n6 \n)\xe2\x88\x95(+1 \n2 \n) \xe2\x88\x92 (+1 \n2 \n) \xe2\x88\x92 1 for \xe2\xa9\xbe 16, thus proving generic \nidentifiability for mixtures of centered Gaussians from \ntheir (exact) moments of degree at most 6. The first \nresult is obtained by the explicit geometry of the tan- \ngential contact locus of the variety of sums of cubes of \nquadratic forms, as described by Chiantini and Ottaviani \n[SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 1018\xe2\x80\x93 \n1037], while the second result is accomplished using the \nlink between secant nondefectivity with identifiability, \nproved by Casarotti and Mella [J. Eur. Math. Soc. (JEMS) \n(2022)]. The latter approach also generalizes to sums of \nth powers of -forms for \xe2\xa9\xbe 3 and \xe2\xa9\xbe 2

Effect of the molecular structure of the polymer and nucleation on the optical properties of polypropylene homo- and copolymers.

Two soluble nucleating agents were used to modify the optical properties of nine PP homo- and random copolymers. The ethylene content of the polymers changed between 0 and 5.3 wt%. Chain regularity was characterized by the stepwise isothermal segregation technique (SIST), while optical properties by the measurement of the haze of injection molded samples. Crystallization and melting characteristics were determined by differential scanning calorimetry (DSC). The analysis of the results proved that lamella thickness and change in crystallinity influence haze only slightly. A model was introduced which describes quantitatively the dependence of nucleation efficiency and haze on the concentration of the nucleating agent. The model assumes that the same factors influence the peak temperature of crystallization and optical properties. The analysis of the results proved that the assumption is valid under the same crystallization conditions. The parameters of the model depend on the molecular architecture of the polymer. Chain regularity determines supermolecular structure and thus the dependence of optical properties on nucleation