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Cohen M.B.,MIT
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2016

We present a new analysis of sparse oblivious subspace embeddings, based on the "matrix Chernoff" technique. These are probability distributions over (relatively) sparse matrices such that for any d-dimensional subspace of Rn, the norms of all vectors in the subspace are simultaneously approximately preserved by the embedding with high probability-typically with parameters depending on d but not on n. The families of embedding matrices considered here are essentially the same as those in [NN13], but with better parameters (sparsity and embedding dimension). Because of this, this analysis essentially serves as a "drop-in replacement" for Nelson-Nguyen's, improving bounds on its many applications to problems such as as least squares regression and low-rank approximation. This new method is based on elementary tail bounds combined with matrix trace inequalities (Golden-Thompson or Lieb's theorem), and does not require combinatorics, unlike the Nelson-Nguyen approach. There are also variants of this method that are even simpler, at the cost of worse parameters. Furthermore, the bounds obtained are much tighter than previous ones, matching known lower bounds up to a single log(d) factor in embedding dimension (previous results had more log factors and also had suboptimal tradeoffs with sparsity). Source

Due to the increasing demand for fossil fuels and environmental threat due to pollution a number renewable sources of energy have been studied worldwide. In the present investigation influence of injection timing on the performance and emissions of a single cylinder, four stroke stationary, variable compression ratio, diesel engine was studied using waste cooking oil (WCO) as the biodiesel blended with diesel. The tests were performed at three different injection timings (24°, 27°, 30° CA BTDC) by changing the thickness of the advance shim. The experimental results showed that brake thermal efficiency for the advanced as well as the retarded injection timing was lesser than that for the normal injection timing (27° BTDC) for all sets of compression ratios. Smoke, un-burnt hydrocarbon (UBHC) emissions were reduced for advanced injection timings where as NOx emissions increased. Artificial Neural Networks (ANN) was used to predict the engine performance and emission characteristics of the engine. Separate models were developed for performance parameters as well as emission characteristics. To train the network, compression ratio, injection timing, blend percentage, percentage load, were used as the input parameters where as engine performance parameters like brake thermal efficiency (BTE), brake specific energy consumption (BSEC), exhaust gas temperature (Texh) were used as the output parameters for the performance model and engine exhaust emissions such as NOx, smoke and (UBHC) values were used as the output parameters for the emission model. ANN results showed that there is a good correlation between the ANN predicted values and the experimental values for various engine performance parameters and exhaust emission characteristics and the relative mean error values (MRE) were within 8%, which is acceptable. © 2010 Elsevier Ltd. Source

Rothvoss T.,MIT
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2012

Let•A be a matrix, c be any linear objective function and x be a fractional vector, say an LP solution to some discrete optimization problem. Then a recurring task in theoretical computer science (and in approximation algorithms in particular) is to obtain an integral vector y such that Ax ≈ Ay and c Ty exceeds c Tx by only a moderate factor. We give a new randomized rounding procedure for this task, provided that A has bounded Δ-approximate entropy. This property means that for uniformly chosen random signs χ(j) ∈ {±1} on any subset of the columns, the outcome Aχ can be approximately described using at most m/5 bits in expectation (with m being the number of selected columns). To achieve this result, we modify well-known techniques from the field of discrepancy theory, especially we rely on Beck's entropy method, which to the best of our knowledge has never been used before in the context of approximation algorithms. Our result can be made constructive using the Bansal framework based on semidefinite programming. We demonstrate the versatility of our procedure by rounding fractional solutions to column-based linear programs for some generalizations of BIN PACKING. For example we obtain a polynomial time OPT + O(log 2 OPT) approximation for BIN PACKING WITH REJECTION and the first AFPTAS for the TRAIN DELIVERY problem. Copyright © SIAM. Source

Nedic A.,Enterprise Systems | Bertsekas D.P.,MIT
Mathematical Programming | Year: 2010

In this paper, we study the influence of noise on subgradient methods for convex constrained optimization. The noise may be due to various sources, and is manifested in inexact computation of the subgradients and function values. Assuming that the noise is deterministic and bounded, we discuss the convergence properties for two cases: the case where the constraint set is compact, and the case where this set need not be compact but the objective function has a sharp set of minima (for example the function is polyhedral). In both cases, using several different stepsize rules, we prove convergence to the optimal value within some tolerance that is given explicitly in terms of the errors. In the first case, the tolerance is nonzero, but in the second case, the optimal value can be obtained exactly, provided the size of the error in the subgradient computation is below some threshold. We then extend these results to objective functions that are the sum of a large number of convex functions, in which case an incremental subgradient method can be used. © 2008 Springer-Verlag. Source

Seidel P.,MIT
Communications in Mathematical Physics | Year: 2010

We describe the behaviour of Fukaya categories under "suspension", which means passing from the fibre of a Lefschetz fibration to the double cover of the total space branched along that fibre. As an application, we consider the mirrors of canonical bundles of toric Fano surfaces. © 2009 Springer-Verlag. Source

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