# Haut PDF An Analysis of the gaussian algorithm for lattice reduction ### An Analysis of the gaussian algorithm for lattice reduction

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r] ### An Average-case Analysis of the Gaussian Algorithm for Lattice Reduction

An Average-case Analysis of the Gaussian Algorithm for Lattice Reduction Hervé Daudé, Philippe Flajolet, Brigitte Vallée.. To cite this version: Hervé Daudé, Philippe Flajolet, Brigitte [r] ### The lattice reduction algorithm of Gauss : an average case analysis

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r] ### An improved analysis of the Mömke-Svensson algorithm for graph-TSP on subquartic graphs

Theorem 4. When OP T LP (G) = (1 + )n and G is subquartic, there is a feasible circulation for C(G, T ) with cost at most n/8 + 2n. Consider a vertex v ∈ T exp . If both incoming back edges had f -value 1/2, then this vertex would not contribute anything to the cost of the circulation. Thus, on a high level, our goal is to find f -values that are as close to half as possible, while at the same time not creating any additional unsatisfied vertices. The f -value therefore corresponds to a decreased x-value if the x-value is high, and an increased x-value if the x-value is low. A set of f -values corresponding to decreased x-values may pose a problem if they correspond to the set of back edges that cover an LP-unsatisfied vertex. However, we note that in Section 4.1 , we only used (j)/2 to satisfy an LP-unsatisfied vertex j. We can actually use at least (j). This observation allows us to decrease the x-values. We use the rules depicted in Figure 1 to determine the values f : B(T ) → [0, 1].
En savoir plus ### An algorithm for U-Pb isotope dilution data reduction and uncertainty propagation

1. Introduction [ 2 ] U ‐Pb geochronology by isotope dilution ther- mal ionization mass spectrometry (ID‐TIMS) has become the gold standard for calibrating geologic time due to precisely determined uranium decay constants, high‐precision measurement methods, and an internal check for open ‐system behavior provided by the dual decay of 235 U and 238 U. Pre- cise, accurate ID ‐TIMS dates have been used to test and calibrate detailed tectonic models [e.g., Schoene et al., 2008], determine the timing and tempo of mass extinctions and ecological recovery [Bowring et al., 1998], calibrate a global geologic timescale [Davydov et al., 2010], and establish a precise chronology for the early solar system [Amelin et al., 2002]. These results rely on analysis and interpre- tation of precisely measured data, for which correct and transparent data reduction and error propagation are imperative.
En savoir plus ### An analysis of covariance parameters in Gaussian Process-based optimization

We wish to find the global minimum of a function f , min x∈D f (x), where the search space D = [LB, U B] d is a compact subset of R d . We assume that f is an expensive-to-compute black-box function. Subsequently, optimization can only be attempted for a low number of function evaluations. The Efficient Global Optimization (EGO) algorithm [3, 4] has become standard for optimizing such expensive unconstrained continuous problems. Its efficiency stems from an embedded conditional Gaussian process (GP), also known as kriging, which acts as a surrogate for the objective function. Certainly, other surrogate techniques can be employed instead of GPs. For example,  proposes a variant of EGO in which a quadratic regression model serves as a surrogate. However, it is shown by some of their examples that the standard EGO performs better than this variant.
En savoir plus ### Perturbation Analysis of the QR Factor R in the Context of LLL Lattice Basis Reduction

perturbed reduced basis remains reduced, possibly with respect to weaker reduc- tion parameters), we introduce a new notion of LLL-reduction (Definition 5.3). Matrices reduced in this new sense satisfy essentially the same properties as those satisfied by matrices reduced in the classical sense. But the new notion of reduction is more natural with respect to column-wise perturbations, as the perturbation of a reduced basis remains reduced (this is not the case with the classical notion of reduction). Another important ingredient of the main result, that may be of inde- pendent interest, is the improvement of the perturbation analyses of  and  for general full column rank matrices (section 2). More precisely, all our bounds are fully rigorous, in the sense that no higher order error term is neglicted, and explicit constant factors are provided. Explicit and rigorous bounds are invaluable for guaranteeing computational accuracy: one can choose a precision that will be known in advance to provide a certain degree of accuracy in the result. In [1, §6], a rigorous error bound was proved. A (much) smaller bound was given in [1, §8], but it is a first-order bound, i.e., high-order terms were neglected. Our rigorous bound is close to this improved bound. Our approach to deriving this rigorous bound is new and has been extended to the perturbation analysis of some other important matrix factorizations . Finally, we give explicit constants in the back- ward stability analysis of Householder’s algorithm from [8, §19], which, along with the perturbation analysis, provides fully rigorous and explicit error bounds for the computed R-factor of a LLL-reduced matrix.
En savoir plus ### Analysis and interpretation of glide characteristics in pursuit of an algorithm for recognition

two events were not known, it was assumed that the two possibilities (glide versus non-glide) were equally likely. Normally, this may seem unusual, but in the anal- ysis given in the following chapters, it is noted that the test for a glide occurs by taking data starting thirty milliseconds from the point of the onset of voicing. In cases where the distance between the landmark and the following vowel was less than thirty milliseconds, the region in question was automatically labeled as a non-glide section. This is reasonable because the duration from a glide into a vowel is more than thirty milliseconds, with minimums being only as small as fifty milliseconds in con- tinuous speech. Thus, only regions where the duration from landmark to vowel was more than thirty milliseconds were considered in the hypothesis test. The remaining speech segments in question were modeled to have a 50% chance of being a glide, and 50% chance of being a non-glide. Albeit a rough estimate, setting the two apriori probabilities equal served as a reasonable assumption which would at the same time simplify computation. Next, the probability distribution had to be modeled. The variability of the parameters selected was modeled to be distributed in a Gaussian probability density function around its mean. This distribution was used as a simpli- fication for detection without specification of context, and can likely be improved if specific contexts are considered separately since the measurements will be localized around different means for different vowels which follow the glides. However, in this thesis, a general hypothesis test is performed on each new set of measurements with- out regard to specific context. The covariances of this distribution were determined by taking the covariance of the sampled data from the training set. Likewise, the means were the sample means of the training data.
En savoir plus ### The JCMT Gould Belt Survey: SCUBA-2 data reduction methods and Gaussian source recovery analysis

In Table 5 , we summarize the peak ﬂux ratios and size ratio data shown in Figures 7 and 8 so that accurate completeness can be estimated for future core-population studies. We emphasize that even for analyses of relatively compact sources using the DR2 external-mask reduction, extra attention should be paid to three factors. First, the population of sources near the completeness limit (peak ﬂuxes of 3–5 times the noise) likely have contributions from even fainter sources (peak ﬂuxes of 1–2 times the noise) that have been boosted to higher ﬂuxes through noise spikes, etc. If the true underlying source population is expected to increase with decreasing peak ﬂux, then this contribution of fainter sources could be signiﬁcant. Second, faint compact sources could be either intrinsically faint and compact or brighter and larger sources that are not fully recovered. Examination of the size distribution of the brighter sources in the map should help determine what the expected properties of the fainter sources are. Third, for analyses where the source-detection rate is important (e.g., applying correc- tions to an observed core mass function), the source recovery rates presented in Section 4.1 should not be blindly applied, as they do not include factors such as crowding or the limitations of core-ﬁnding algorithms running without prior knowledge on a map, both of which are expected to decrease the real observational detection rate. Furthermore, while the results presented here are uncontaminated by false-positive detections, such complications will need to be carefully considered when running source-identiﬁcation algorithms on real observations.
En savoir plus ### Stochastic Behavior Analysis of the Gaussian Kernel-Least-Mean-Square Algorithm

Suppose a design goal is to obtain an MSE which is less than a speciÞed value . The following procedure could be applied. 1) Set a coherence threshold and deÞne a set of kernel bandwidths to be tested. For the following design exam- ples, a set of values for was chosen as equally spaced points in (0, 1). Then, was chosen to yield reasonable values of for the chosen set of values. The value of is determined for each pair by training the dic- tionary with the input signal until its size stabilizes. The training is repeated several times. A value is de- termined for the th realization. The value of associated with the pair is the average of the dictionary sizes for all realizations, rounded to the nearest integer. This is the value of to be used in the theoretical model. 2) Using the system input , determine the desired output
En savoir plus ### Gaussian framework for interference reduction in live recordings

Figure 2 shows the confidence ellipse of the scores obtained by each algorithm on each pair of scales. It shows how the EM and MM perform slightly better than KAMIR in both of its fashions. As in Figure 3, we see the benefits of the sparsity penalty as improving background suppression at the cost of introducing some artifacts. An interesting observation is that EM + S and MM + S appear closer to K and ˜ K than EM and MM. Regardless of the amount of noise that may affect the evaluation results, the EM method presented in this paper leads to slightly better results than state of the art. Close investigation reveals that its main difference with KAMIR lies in handling the uncertainty of the model through the posterior variance in (7). Then, the W-disjoint orthogonality penalty γ in (19) is seen as controlling the trade-off between isolation and distor- tion. The MM approach does not seem to perform sig- nificantly better than KAMIR algorithms, especially for the suppression of background. Still, adding a penalty γ brings it closer to EM, while having a significantly smaller computational complexity.
En savoir plus ### The algorithm for the analysis of combined chaotic-stochastic processes

ence equation like x n = X(t n ), the behavior can be highly irregular and extremely complex. In some cases the behavior is estimated like chaotic. In the first approximation, we can determine the chaocity by the property of the system to construct it’s trajectories in a bounded domain of the phase space. Properties of dynamical systems which generate chaotic solutions, have been widely discussed by the authors (results and references in the monographs [13, 14]). The simplest example is an one-dimensional dynami- cal system x n+1 = f (x n , µ) which generates chaotic solution for some func- tions f and values of parameter µ . In particular, for logistic function f such as x n+1 = µx n (1 − x n ), the plot of solution looks like white noise with some values µ > 3.6. So, the problem statement the nature of time series analysis nature is do the observed data have stochastic nature, or deterministic. Let B (t), 0 ≤ t ≤ 1 be a fractional Brownian motion with Hurst exponent H. Let’s consider the normalized increments
En savoir plus ### A Lattice Basis Reduction Approach for the Design of Finite Wordlength FIR Filters

We notice that lattice-based quantization gives results which are optimal (or the best known ones) in eight cases out of fifteen and the other seven cases are very close to the optimal ones. It outperforms telescoping rounding in twelve cases out of fifteen and yields the same (optimal) result in a thirteenth case: the use of the LLL option gives a better result in twelve cases and an identical (optimal) result in a thirteenth case, the use of the BKZ option gives a better result in eleven cases and the use of the HKZ option in nine cases. We remark, in particular, the good behavior of the LLL algorithm in all 15 test cases, further emphasizing the idea that the lattice bases we use are close to being reduced. Our approach seems to work particularly well when the gap between the minimax error and the naive rounding error is significant, which is the case where one is most interested in improvements over the naive rounding filter. Eventually, note that, in the C125/21 and D125/22 cases, our approach returns (in less than 8 seconds, see Subsection VI-D ) results that are better than the ones provided by (time-limited) MILP tools.
En savoir plus ### An algorithm reconstructing convex lattice sets

in most cases the bases need not to be chosen a priori, but much work should be done, and in particular we need an algorithm which generates uniformly convex lattice sets of a given size at random. Secondly what can we say about the reconstruction problem for any set of lattice directions not uniquely determining convex lattice sets ? Does there exist a polynomial algorithm in this case ? We could apply our algorithm until the reduction to a 2-SAT formula, but then we do not see any way to express the convexity by a formula whose satisfiability could be checked in polynomial time.
En savoir plus ### An Algorithm for Minimal Insertion in a Type Lattice

ODD L ( x ) = f a j a k x; 9 b;c 2 Inf ( x ) with a = b _ L c g will denote the set of LUB of elements in Inf ( x ), called lower odds. The duality principle, allows us to consider only ODD U ( x ). Odds give rise to auxiliary elements so they may be compared to generators in  and to canonical representatives in . The existing strategies for lattice insertion di er as to the way odds are detected and the number of auxiliaries per odd. Thus, the algorithm in  checks the GLB of all couples of elements in Sup ( x ) and inserts an auxiliary each time this GLB is incomparable to x . In this way, the same odd may provoke the generation of a set of auxiliaries. For example, the element g on Fig.1 is an odd since g = V
En savoir plus ### Stochastic Behavior Analysis of the Gaussian Kernel Least-Mean-Square Algorithm

Suppose a design goal is to obtain an MSE which is less than a specified value . The following procedure could be applied. 1) Set a coherence threshold and define a set of kernel bandwidths to be tested. For the following design exam- ples, a set of values for was chosen as equally spaced points in (0, 1). Then, was chosen to yield reasonable values of for the chosen set of values. The value of is determined for each pair by training the dic- tionary with the input signal until its size stabilizes. The training is repeated several times. A value is de- termined for the th realization. The value of associated with the pair is the average of the dictionary sizes for all realizations, rounded to the nearest integer. This is the value of to be used in the theoretical model. 2) Using the system input , determine the desired output
En savoir plus ### A POSTERIORI ANALYSIS OF AN ITERATIVE ALGORITHM FOR NAVIER-STOKES PROBLEM

Keywords: A posteriori error estimation, Navier-Stokes problem, iterative method. 1. Introduction The a posteriori analysis controls the overall discretization error of a problem by providing error indicators easy to compute. Once these error indicators are constructed, we prove their efficiency by bounding each indicator by the local error. This analysis was first introduced by I. Babuˇ ska , and developed by R. Verf¨ urth . The present work investigates a posteriori error estimates of the finite element discretization of the Navier-Stokes equations in polygonal domains. In fact, many works have been carried out in this field. In , C. Bernardi, F. Hecht and R. Verfürth considered a variational formulation of the three-dimensional Navier-Stokes equations with mixed boundary conditions and they proved that it admits a solution if the domain satisfies a suitable regularity assumption. In addition, they established the a priori and the a posteriori error estimates. As well, in , V. Ervin, W. Layton and J. Maubach present locally calculable a posteriori error estimators for the basic two-level discretization of the Navier- Stokes equations. In this work, we propose a finite element discretization of the Navier-Stokes equations relying on the Galerkin method. In order to solve the discrete problem we propose an iterative method. Therefore two sources of error appear, due to the discretization and the algorithm. Balancing these two errors leads to important computational savings. We apply this strategy on the following Navier-Stokes equations:
En savoir plus ### An adaptive model reduction strategy for post-buckling analysis of stiffened structures

4. Conclusion and Perspectives In the framework of model reduction techniques, a strategy has been developed that reduces the compu- tational cost of post-buckling simulations. This approach makes the most of an on the fly adaptive procedure and little knowledge on post-buckling phenomenon. More precisely, as stated in the semi-analytical meth- ods, the post-buckling equilibrium state of structures is taken to be a combination of the pre-buckling state (known) and a variation, which is decomposed into a buckling mode component and a higher order variation arising from an automatic completion procedure. This results in a fast algorithm for solving post-buckling problems, which does not require expensive pre-calculations and is positioned both as an alternative to POD-based model reduction and as a way to build POD snapshots for post-buckling analysis at lower cost. In this paper, the relevance of this approach has been demonstrated in the case of post-buckling analysis of plates. In spite of the limits of the home-made finite element research code, insight was gained into the computational performances of the strategy for an in-plane load reaching more than twice the buckling load
En savoir plus ### An adaptive Gaussian quadrature for the Voigt function

However, before exploring potential departures from Gaus- sianity, we need to adopt a robust enough numerical strategy in order to numerically evaluate integrals such as Eq. ( 4 ), a task which is notoriously di fficult even with Maxwell–Boltzmann VDFs. It is very easy to verify that, for instance, a standard Gauss–Hermite (GH) quadrature, even at high rank k, fails at properly computing a somewhat simpler expression like the Voigt 3 function given in Eq. ( 13 ). We display, in Fig. 1 , the com- parison between a GH integration and the new numerical scheme that is presented hereafter.
En savoir plus ### A polynomial reduction algorithm

Applying POLRED to P(X) we obtain thus showing that the fields generated by the roots of the polynomials given in [PMD] are isomorphic, and also that is a subfield. The fact that the same polynomial is obtained several times gives also some information on the Galois group of the Galois closure of the number field I, since it