Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

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Mathematical Methods Of Statistics

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Methods of Numerical Integration

We present the design, implementation, and evaluation of CONGA, a network-based distributed congestion-aware load balancing mechanism for datacenters. CONGA exploits recent trends including the use of regular Clos topologies and overlays for network virtualization. It splits TCP flows into flowlets, estimates real-time congestion on fabric paths, and allocates flowlets to paths based on feedback from remote switches. This enables CONGA to efficiently balance load and seamlessly handle asymmetry, without requiring any TCP modifications. CONGA has been implemented in custom ASICs as part of a new datacenter fabric. In testbed experiments, CONGA has 5x better flow completion times than ECMP even with a single link failure and achieves 2-8x better throughput than MPTCP in Incast scenarios. Further, the Price of Anarchy for CONGA is provably small in Leaf-Spine topologies; hence CONGA is nearly as effective as a centralized scheduler while being able to react to congestion in microseconds. Our main thesis is that datacenter fabric load balancing is best done in the network, and requires global schemes such as CONGA to handle asymmetry.

/pdf/conga-distributed-congestion-aware-load-balancing-for-3c0czbo3yn.pdf

CONGA: distributed congestion-aware load balancing for datacenters

5G roadmap

Consider the problem of estimating the Shannon entropy of a distribution over $k$ elements from $n$ independent samples. We show that the minimax mean-square error is within the universal multiplicative constant factors of $\big ({k }/{n \log k}\big )^{2} + {\log ^{2} k}/{n}$ if $n$ exceeds a constant factor of $({k}/{\log k})$ ; otherwise, there exists no consistent estimator. This refines the recent result of Valiant and Valiant that the minimal sample size for consistent entropy estimation scales according to $\Theta ({k}/{\log k})$ . The apparatus of the best polynomial approximation plays a key role in both the construction of optimal estimators and, by a duality argument, the minimax lower bound.

Minimax Rates of Entropy Estimation on Large Alphabets via Best Polynomial Approximation

Clos-based networks including Fat-tree and VL2 are being built in data centers, but existing per-flow based routing causes low network utilization and long latency tail. In this paper, by studying the structural properties of Fat-tree and VL2, we propose a per-packet round-robin based routing algorithm called Digit-Reversal Bouncing (DRB). DRB achieves perfect packet interleaving. Our analysis and simulations show that, compared with random-based load-balancing algorithms, DRB results in smaller and bounded queues even when traffic load approaches 100%, and it uses smaller re-sequencing buffer for absorbing out-of-order packet arrivals. Our implementation demonstrates that our design can be readily implemented with commodity switches. Experiments on our testbed, a Fat-tree with 54 servers, confirm our analysis and simulations, and further show that our design handles network failures in 1-2 seconds and has the desirable graceful performance degradation property.

/pdf/per-packet-load-balanced-low-latency-routing-for-clos-based-5brg1qz6ey.pdf

Per-packet load-balanced, low-latency routing for clos-based data center networks

Big data analysis has found applications in many industries due to its ability to turn huge amounts of data into insights for informed business and operational decisions. Advanced data mining techniques have been applied in many sectors of supply chains in the food industry. However, the previous work has mainly focused on the analysis of instrument-generated data such as those from hyperspectral imaging, spectroscopy, and biometric receptors. The importance of digital text data in the food and nutrition has only recently gained attention due to advancements in big data analytics. The purpose of this review is to provide an overview of the data sources, computational methods, and applications of text data in the food industry. Text mining techniques such as word-level analysis (e.g., frequency analysis), word association analysis (e.g., network analysis), and advanced techniques (e.g., text classification, text clustering, topic modeling, information retrieval, and sentiment analysis) will be discussed. Applications of text data analysis will be illustrated with respect to food safety and food fraud surveillance, dietary pattern characterization, consumer-opinion mining, new-product development, food knowledge discovery, food supply-chain management, and online food services. The goal is to provide insights for intelligent decision-making to improve food production, food safety, and human nutrition.

https://onlinelibrary.wiley.com/doi/pdf/10.1111/1541-4337.12540

Utilization of text mining as a big data analysis tool for food science and nutrition.

We consider the problem of estimating the support size of a discrete distribution whose minimum nonzero mass is at least $\frac{1}{k}$. Under the independent sampling model, we show that the sample complexity, that is, the minimal sample size to achieve an additive error of $\varepsilon k$ with probability at least 0.1 is within universal constant factors of $\frac{k}{\log k}\log^{2}\frac{1}{\varepsilon }$, which improves the state-of-the-art result of $\frac{k}{\varepsilon^{2}\log k}$ in [In Advances in Neural Information Processing Systems (2013) 2157–2165]. Similar characterization of the minimax risk is also obtained. Our procedure is a linear estimator based on the Chebyshev polynomial and its approximation-theoretic properties, which can be evaluated in $O(n+\log^{2}k)$ time and attains the sample complexity within constant factors. The superiority of the proposed estimator in terms of accuracy, computational efficiency and scalability is demonstrated in a variety of synthetic and real datasets.

/pdf/chebyshev-polynomials-moment-matching-and-optimal-estimation-moyvcq5k4u.pdf

Chebyshev polynomials, moment matching, and optimal estimation of the unseen

The method of moments (Philos. Trans. R. Soc. Lond. Ser. A 185 (1894) 71–110) is one of the most widely used methods in statistics for parameter estimation, by means of solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or nonuniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analyses of the method of moments are mainly confined to asymptotic normality style of results established under strong assumptions. This paper considers estimating a $k$-component Gaussian location mixture with a common (possibly unknown) variance parameter. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space (via semidefinite programming) before solving the method of moments equations. Not only does this regularization ensure existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in the Wasserstein distance via polynomial interpolation and majorization techniques, we establish the statistical guarantees and adaptive optimality of the proposed procedure, as well as oracle inequality in misspecified models. These results can also be viewed as provable algorithms for generalized method of moments (Econometrica 50 (1982) 1029–1054) which involves nonconvex optimization and lacks theoretical guarantees.

Pengkun Yang

Papers

Minimax Rates of Entropy Estimation on Large Alphabets via Best Polynomial Approximation

Per-packet load-balanced, low-latency routing for clos-based data center networks

Utilization of text mining as a big data analysis tool for food science and nutrition.

Chebyshev polynomials, moment matching, and optimal estimation of the unseen

Optimal estimation of Gaussian mixtures via denoised method of moments