Abstract: The learning with rounding (LWR) problem, introduced by Banerjee, Peikert and Rosen at EUROCRYPT ’12, is a variant of learning with errors (LWE), where one replaces random errors with deterministic rounding. The LWR problem was shown to be as hard as LWE for a setting of parameters where the modulus and modulus-to-error ratio are super-polynomial. In this work we resolve the main open problem and give a new reduction that works for a larger range of parameters, allowing for a polynomial modulus and modulus-to-error ratio. In particular, a smaller modulus gives us greater efficiency, and a smaller modulus-to-error ratio gives us greater security, which now follows from the worst-case hardness of GapSVP with polynomial (rather than super-polynomial) approximation factors.

Abstract: 5-axis high speed machine tools are widely used in industry. Most of the time, the tool path is described with linear segments (G1) which leads to tangency discontinuities between blocks. The aim of this paper is to smooth the tool path geometry using a 5-axis corner rounding method suitable for acceleration and jerk limited feedrate interpolation. Several methods have been developed in 3-axis but 5-axis corner rounding is still a challenge due to the difficulties linked to the smoothing of the orientation. The proposed corner rounding model allows to control precisely the contour and orientation tolerances in the Workpiece Coordinate System for 3 and 5-axis tool path. To smooth the tool tip position and the tool orientation in the corner, 5-axis tool paths are represented by two B-Spline curves. The main difficulty is the connection between the initial tool path and the newly inserted smoothing portion. To obtain a smooth connection of the orientation a parametrization spline is required to link the bottom and top B-Spline parameters. This algorithm is integrated to a feedrate interpolator which controls a 5-axis milling machine equipped with an Open CNC.

Abstract: Pade approximation is considered from the point of view of robust methods of numerical linear algebra, in particular, the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors, for which a MATLAB code is provided. The success of this algorithm suggests that there might be variants of Pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to $\infty$, unlike traditional Pade approximants, which converge only in measure or capacity.

Abstract: The learning with rounding (LWR) problem, introduced by Banerjee, Peikert and Rosen at EUROCRYPT ’12, is a variant of learning with errors (LWE), where one replaces random errors with deterministic rounding. The LWR problem was shown to be as hard as LWE for a setting of parameters where the modulus and modulus-to-error ratio are super-polynomial. In this work we resolve the main open problem and give a new reduction that works for a larger range of parameters, allowing for a polynomial modulus and modulus-to-error ratio. In particular, a smaller modulus gives us greater efficiency, and a smaller modulus-to-error ratio gives us greater security, which now follows from the worst-case hardness of GapSVP with polynomial (rather than super-polynomial) approximation factors. As a tool in the reduction, we show that there is a “lossy mode” for the LWR problem, in which LWR samples only reveal partial information about the secret. This property gives us several interesting new applications, including a proof that LWR remains secure with weakly random secrets of sufficient min-entropy, and very simple constructions of deterministic encryption, lossy trapdoor functions and reusable extractors. Our approach is inspired by a technique of Goldwasser et al. from ICS ’10, which implicitly showed the existence of a “lossy mode” for LWE. By refining this technique, we also improve on the parameters of that work to only requiring a polynomial (instead of super-polynomial) modulus and modulus-to-error ratio.

Abstract: @ are a coreset for the problem, consisting of sampled and rescaled rows of A and b; and s is independent of n and polynomial in d. Our results improve on the best previous algorithms when n > d, for all p e [1, ∞) except p = 2; in particular, they improve the O(nd1.376+) running time of Sohler and Woodruff (STOC, 2011) for p = 1, that uses asymptotically fast matrix multiplication, and the O(nd5 log n) time of Dasgupta et al. (SICOMP, 2009) for general p, that uses ellipsoidal rounding. We also provide a suite of improved results for finding well-conditioned bases via ellipsoidal rounding, illustrating tradeoffs between running time and conditioning quality, including a one-pass conditioning algorithm for general ep problems.To complement this theory, we provide a detailed empirical evaluation of implementations of our algorithms for p = 1, comparing them with several related algorithms. Among other things, our empirical results clearly show that, in the asymptotic regime, the theory is a very good guide to the practical performance of these algorithms. Our algorithms use our faster constructions of well-conditioned bases for ep spaces and, for p = 1, a fast subspace embedding of independent interest that we call the Fast Cauchy Transform: a matrix Π:

Abstract: The (real) Grothendieck constant , perform better than the ubiquitous random hyperplane technique. By establishing the usefulness of higher-dimensional rounding schemes, this fact has consequences in approximation algorithms. Specifically, it yields the best known polynomial-time approximation algorithm for the Frieze–Kannan Cut Norm problem, a generic and well-studied optimization problem with many applications.

Abstract: The $k$-Facility Location problem is a generalization of the classical problems $k$-Median and Facility Location. The goal is to select a subset of at most $k$ facilities that minimizes the total cost of opened facilities and established connections between clients and opened facilities. We consider the hard-capacitated version of the problem, where a single facility may only serve a limited number of clients and creating multiple copies of a facility is not allowed. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard LP. It is well known that the standard LP (even in the case of uniform capacities and opening costs) has unbounded integrality gap if we only allow violating capacities by a factor smaller than $2$, or if we only allow violating the number of facilities by a factor smaller than $2$. In this paper, we present the first constant-factor approximation algorithms for the hard-capacitated variants of the problem. For uniform capacities, we obtain a $(2+\varepsilon)$-capacity violating algorithm with approximation ratio $O(1/\varepsilon^2)$; our result has not yet been improved. Then, for non-uniform capacities, we consider the case of $k$-Median, which is equivalent to $k$-Facility Location with uniform opening cost of the facilities. Here, we obtain a $(3+\varepsilon)$-capacity violating algorithm with approximation ratio $O(1/\varepsilon)$.

Abstract: Numerical algorithms lie at the heart of many safety-critical aerospace systems. The complexity and hybrid nature of these systems often requires the use of interactive theorem provers to verify that these algorithms are logically correct. Usually, proofs involving numerical computations are conducted in the infinitely precise realm of the field of real numbers. However, numerical computations in these algorithms are often implemented using floating point numbers. The use of a finite representation of real numbers introduces uncertainties as to whether the properties verified in the theoretical setting hold in practice. This short paper describes work in progress aimed at addressing these concerns. Given a formally proven algorithm, written in the Program Verification System (PVS), the Frama-C suite of tools is used to identify sufficient conditions and verify that under such conditions the rounding errors arising in a C implementation of the algorithm do not affect its correctness. The technique is illustrated using an algorithm for detecting loss of separation among aircraft.

Abstract: Pipage rounding is a dependent random sampling technique that has several interesting properties and diverse applications. One property that has been particularly useful is negative correlation of the resulting vector. Unfortunately negative correlation has its limitations, and there are some further desirable properties that do not seem to follow from existing techniques. In particular, recent concentration results for sums of independent random matrices are not known to extend to a negatively dependent setting. We introduce a simple but useful technique called concavity of pessimistic estimators. This technique allows us to show concentration of submodular functions and concentration of matrix sums under pipage rounding. The former result answers a question of Chekuri et al. (2009). To prove the latter result, we derive a new variant of Lieb's celebrated concavity theorem in matrix analysis. We provide numerous applications of these results. One is to spectrally-thin trees, a spectral analog of the thin trees that played a crucial role in the recent breakthrough on the asymmetric traveling salesman problem. We show a polynomial time algorithm that, given a graph where every edge has effective conductance at least $\kappa$, returns an $O(\kappa^{-1} \cdot \log n / \log \log n)$-spectrally-thin tree. There are further applications to rounding of semidefinite programs, to the column subset selection problem, and to a geometric question of extracting a nearly-orthonormal basis from an isotropic distribution.

Abstract: We design new approximation algorithms for the Multiway Cut problem, improving the previously known factor of 1.32388 [Buchbinder et al., 2013]. We proceed in three steps. First, we analyze the rounding scheme of Buchbinder et al., 2013 and design a modification that improves the approximation to (3+sqrt(5))/4 (approximately 1.309017). We also present a tight example showing that this is the best approximation one can achieve with the type of cuts considered by Buchbinder et al., 2013: (1) partitioning by exponential clocks, and (2) single-coordinate cuts with equal thresholds. Then, we prove that this factor can be improved by introducing a new rounding scheme: (3) single-coordinate cuts with descending thresholds. By combining these three schemes, we design an algorithm that achieves a factor of (10 + 4 sqrt(3))/13 (approximately 1.30217). This is the best approximation factor that we are able to verify by hand. Finally, we show that by combining these three rounding schemes with the scheme of independent thresholds from Karger et al., 2004, the approximation factor can be further improved to 1.2965. This approximation factor has been verified only by computer.

Abstract: A round-for-reround mode (preferably in a BID encoded Decimal format) of a floating point instruction prepares a result for later rounding to a variable number of digits by detecting that the least significant digit may be a 0, and if so changing it to 1 when the trailing digits are not all 0 A subsequent reround instruction is then able to round the result to any number of digits at least 2 fewer than the number of digits of the result An optional embodiment saves a tag indicating the fact that the low order digit of the result is 0 or 5 if the trailing bits are non-zero in a tag field rather than modify the result Another optional embodiment also saves a half-way-and-above indicator when the trailing digits represent a decimal with a most significant digit having a value of 5 An optional subsequent reround instruction is able to round the result to any number of digits fewer or equal to the number of digits of the result using the saved tags

Abstract: This study presents hardware architectures performing correctly rounded Floating-Point (FP) multioperand addition and dot-product computation, both of which are widely used in various fields, such as scientific computing, digital signal processing, and 3D graphic applications. A novel realignment method is proposed to solve the catastrophic cancellation and multi-sticky bits. Only one rounding operation is performed in both of the proposed FP multi-operand adder and dot-product computation unit. Implementation results show that our architectures not only can produce correctly rounded results, whose errors are less than 0.5 ULP (Unit in the Last Place), but also have reduced delay comparing with the traditional network architecture, which uses 2-operand FP adders and multipliers to perform multi-operand addition and dot-product computation.

Abstract: We develop a panel data model of expectations of a continuous outcome variable elicited on a percentage-chance scale. The model explains the location and dispersion of the subjective distributions by socio-economic covariates and unobserved factors. Moreover, it accounts explicitly for non-response, non-informative focal answers, and recall and rounding errors. We apply the model to the expected retirement income replacement rate of Dutch wage workers. We find that incorporating these features of the answering process increases the size and significance of relationships with covariates. The estimates indicate substantial rounding but few focal answers. Respondents tend to stick to a certain answering strategy: non-response, rounding and especially non-informative focal answers are characterized by substantial unobserved heterogeneity across individuals.

Abstract: Many problems in machine learning can be solved by rounding the solution of an appropriate linear program (LP) This paper shows that we can recover solutions of comparable quality by rounding an approximate LP solution instead of the exact one These approximate LP solutions can be computed efficiently by applying a parallel stochastic-coordinate-descent method to a quadratic-penalty formulation of the LP We derive worst-case runtime and solution quality guarantees of this scheme using novel perturbation and convergence analysis Our experiments demonstrate that on such combinatorial problems as vertex cover, independent set and multiway-cut, our approximate rounding scheme is up to an order of magnitude faster than Cplex (a commercial LP solver) while producing solutions of similar quality

Abstract: In this paper, we apply a hierarchical identification principle to identify simultaneously the unknown time delay and dynamic parameters of discrete-time delay systems. In our approach, we separate the nonlinear cost function into two simple cost functions and present a gradient iterative algorithm for estimating directly the time delay and the parameters. The cost function used to estimate the time delay is an integer nonlinear programming problem (IP) and the solution of the integer version of a nonlinear optimization problem is obtained by rounding the solution for its continuous relaxation (CP). Furthermore, we give an appropriate choice of the convergence factor. Finally, the effectiveness of this method has been illustrated through simulation.

Abstract: JInterval [1] was started in 2008 as a research project to develop a Java library for interval computations. The library is intended mainly for developers who create Java-based applied software. The design of the JInterval library was guided by the following requirements ordered by descending priority: 1. The library must be clear and easy to use. No matter how wonderful a software tool is, it will be hardly accepted by developers if it is not transparent and easy to use. 2. The library should provide flexibility in the choice of interval arithmetic for computations. The user must be able to choose interval arithmetic (classical, Kaucher, complex rectangular, complex circular, etc.) and to switch one arithmetic to another if they are compatible. Syntactic differences between the use of this or that arithmetic should be minimized. 3. The library should provide flexibility in extending its functionality. The library must be layered functionally. Four layers should be defined: interval arithmetic operators, elementary interval functions, interval vector and matrix operations, and, finally, high-level interval methods, such as solvers of equations, optimization procedures, etc. Architecture of the library must allow for extensions at every layer, starting from the bottom one. 4. The library should provide flexibility in choosing precision of interval endpoints and associated rounding policies. The choice of interval endpoints representation and the rounding mode could allow the user to tune accuracy and speed of computation depending on the problem he solves. 5. The library must be portable. Cross-platform portability of the library is one of its major strengths, being a key distinction over its closest competitors. To a large extent, this requirement is ensured by the choice of the Java technology built on the principle ”write once, run anywhere”. However, the design must adhere to certain restrictions on practical implementation of this requirement.

Abstract: This paper proposes a new discrete approach to solve network reconfiguration of power system. Like other problems of optimization, network reconfiguration is different. Here the positions of tie switches represent integer numbers, thus the Gaussian formulation and rounding off the number, in this case is not very useful. Evolutionary Programming (EP) based discrete optimization technique is proposed to find out the optimum switch(s) combination. Minimization of power losses is considered as a fitness function. The proposed technique is tested on standard 33-bus and 69-bus radial distribution network, having 5-tie switches. The result shows that the proposed technique achieve better fitness function in less computation time and less number of iterations.

Abstract: Floating Point (FP) addition, subtraction and multiplication are widely used in large set of scientific and signal processing computation. A high speed floating point double precision adder/subtractor and multiplier are implemented on a Virtex-6 FPGA. In addition, the proposed designs are compliant with IEEE-754 format and handles over flow, under flow, rounding and various exception conditions. The adder/subtractor and multiplier designs achieved the operating frequencies of 363.76 MHz and 414.714 MHz with an area of 660 and 648 slices respectively.

Abstract: The lexicographic bi-criteria combinatorial food packing problem to be discussed in this paper is described as follows. Given a set I = {i | i = 1, 2, . . . , n} of current n items (for example, n green peppers) with their weights wi and priorities γi, the problem asks to find a subset I′ (⊆ I) so that the total weight i∈I′ wi is no less than a specified target weight T for each package, and it is minimized as the primary objective, and further the total priority ∑ i∈I′ γi is maximized as the second objective. The problem has been known to be NP-hard, while it can be solved exactly in O(nT ) time if all the input data are assumed to be integral. In this paper, we design a heuristic algorithm for the problem by applying a data rounding technique to an O(nT ) time dynamic programming procedure. We also conduct numerical experiments to examine the empirical performance such as execution time and solution quality.

Abstract: This report reduces the total number of lifting steps in a three-dimensional (3D) double lifting discrete wavelet transform (DWT), which has been widely applied for analyzing volumetric medical images. The lifting steps are necessary components in a DWT. Since calculation in a lifting step must wait for a result of former step, cascading many lifting steps brings about increase of delay from input to output. We decrease the total number of lifting steps introducing 3D memory accessing for the implementation of low delay 3D DWT. We also maintain compatibility with the conventional 5/3 DWT defined by JPEG 2000 international standard for utilization of its software and hardware resources. Finally, the total number of lifting steps and rounding operations were reduced to 67 % and 33 %, respectively. It was observed that total amount of errors due to rounding operations in the lifting steps was also reduced.

Abstract: We study the Euler-Frobenius numbers, a generalization of the Eulerian numbers, and the probability distribution obtained by normalizing them. This distribution can be obtained by rounding a sum of independent uniform random variables; this is more or less implicit in various results and we try to explain this and various connections to other areas of mathematics, such as spline theory. The mean, variance and (some) higher cumulants of the distribution are calculated. Asymptotic results are given. We include a couple of applications to rounding errors and election methods.