#### DMCA

## Data Interpolation An Efficient Sampling Alternative for Big Data Aggregation (2014)

### Citations

2869 |
The Design and Analysis of Computer I Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...r. One obvious candidate to construct approximating polynomial is interpolation at equidistant points. However, the sequence of interpolation polynomials does not converge uniformly to f for all f ∈ C=-=[0, 1]-=- due to Runge’s phenomenon [7]. Chebyshev interpolation (i.e., interpolate f using the points defined by the Chebyshev polynomial) minimizes Runge’s oscillation, but it is not suffice the polynomial f... |

441 |
Interpolation and approximation
- Davis
- 1975
(Show Context)
Citation Context ...ruct approximating polynomial is interpolation at equidistant points. However, the sequence of interpolation polynomials does not converge uniformly to f for all f ∈ C[0, 1] due to Runge’s phenomenon =-=[7]-=-. Chebyshev interpolation (i.e., interpolate f using the points defined by the Chebyshev polynomial) minimizes Runge’s oscillation, but it is not suffice the polynomial fitting problem presented above... |

273 | Decoding of Reed-Solomon codes beyond the error-correction bound
- Sudan
- 1997
(Show Context)
Citation Context ...rupted) data as a ratio of polynomials. Their solution holds for noise-free cases and a limited fraction of the corrupted data (δ = 0, ρ > 1/2). Almost 30 years later, Sudan’s list decoding algorithm =-=[19]-=- relaxed the Byzantine constraint (δ = 0, ρ can be less than 1/2) by using bivariate polynomial interpolation. Those concepts do not hold up well in the noisy case since they use the roots of the poly... |

132 |
Error correction for algebraic block codes
- Welsh, Berlekamp
- 1986
(Show Context)
Citation Context ...sts dealing with the noise-free case (i.e., δ = 0 and ρ < 1). In the next section, we present an algorithm that handles a combination of discrete noise and Byzantine data based on the Welch-Berlekamp =-=[22]-=- errorelimination method. Moreover, the fundamental Welch-Berlekamp algorithm treats only the onedimension case, where we suggest a means to deal with corrupted-noisy data appearing at one and multi-d... |

128 | Data-aggregation techniques in sensor networks: a survey
- Rajagopalan, Varshney
- 2006
(Show Context)
Citation Context ... there can be malicious inputs, i.e., part of the data may be corrupted. In contrast to distributed data aggregation where the resulting computation is a function such as COUNT, SUM and AVERAGE (e.g. =-=[9, 13, 16]-=-), in distributed data interpolation, our goal is to represent every value of the data by a single (abstracting) function. Our computational model consists of sampling the sensor network data and esti... |

105 |
M.: In-network aggregation techniques for wireless sensor networks: A survey
- Fasolo, Rossi, et al.
- 2007
(Show Context)
Citation Context ... there can be malicious inputs, i.e., part of the data may be corrupted. In contrast to distributed data aggregation where the resulting computation is a function such as COUNT, SUM and AVERAGE (e.g. =-=[9, 13, 16]-=-), in distributed data interpolation, our goal is to represent every value of the data by a single (abstracting) function. Our computational model consists of sampling the sensor network data and esti... |

87 | 1912/13], Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités - Bernstein |

48 | Computational aspects of polynomial interpolation in several variables - Boor, Ron - 1992 |

32 | The World’s Technological Capacity to Store - Hilbert, Lopez |

18 |
An Introduction to the Approximation of Functions (Blaisdell
- Rivlin
- 1969
(Show Context)
Citation Context ...onovskaya’s Theorem states that for functions that are twice differentiable, the rate of convergence is about 1/n, see Davis [7]). Considering other classical curve-fitting and approximation theories =-=[17]-=-, most research has used the `2 norm of noise, such as the method of least square errors. These attitudes not suffice the 2 adversarial noise we have assumed here. To our knowledge, only [2] referred ... |

12 |
A Bernstein polynomial approach to density estimation
- Vitale
- 1975
(Show Context)
Citation Context ...uniformly to any continuous function f which is bounded on [0, 1]. The formal Berenstein polynomial samples the function f in an equidistant fashion. To handle a random sample data, we can use Vitale =-=[21]-=- results which consider that the datapoints S = x1, ..., xN are i.i.d observations drawn from an unknown density function f . The Bernstein polynomial estimate of f defined as B̃fn(x) = n+1 N n∑ i=0 µ... |

8 |
A generalization of the Stone-Weierstrass theorem
- Bishop
- 1961
(Show Context)
Citation Context ...a (Byzantine data, denoted by ρ) that can cause inaccurate sampling and, thus, lead to badly constructed polynomials. Given that the function f is continuous, by the Weierstrass approximation Theorem =-=[4]-=- we know that for any given > 0, there exists a polynomial p′ such that∥∥f − p′∥∥∞ < (2) This can tell us that our desired polynomial p exists (i.e., p′ = p and = δ, satisfying eq.1), and we can... |

7 |
Two-dimensional bernstein polynomial density estimation
- Tenbusch
- 1994
(Show Context)
Citation Context ...n+1 N n∑ i=0 µNin ( n i ) xi(1−x)n−i where µNin is the number of points (xi’s) appear in the interval [ in+1 , i+1 n+1 ]. Vitale [21] showed that ∥∥∥B̃fn(x)− f∥∥∥∞ ≤ for every given > 0. Tenbusch =-=[20]-=- extended Vitale’s idea to multidimensional densities, where there is need to note that those works hold only when the datapoints are i.i.d observations. Another reason not to use the Bernstien polyno... |

4 |
Multivariate Bernstein and Markov inequalities
- Ditzian
- 1992
(Show Context)
Citation Context ...ver the choice of S), any feasible solution p to the above LP is cδ-approximation of f . Proof. For our proof, we need Bernstein-Markov inequality which we state below. Theorem 3.2. (Bernstein-Markov =-=[8]-=-) For a polynomial Pd of total degree d, a direction ξ and a bounded convex set A ⊂ Rk ∥∥∥∥ ∂∂ξPd ∥∥∥∥ ∞ ≤ cAd2 ‖Pd‖∞ (7) where cA is independent of d (and dependent on the geometric structure of A). ... |

3 | i b o m . The P e e p t i o n of Visual Surfaces - unknown authors - 1955 |

3 | Bernstein polynomials for functions of two variables of class C(k - Kingsley - 1951 |

3 | How do your data grow - Lynch - 2008 |

2 |
Faugre, “A new efficient algorithm for computing Gröbner base
- C
- 1999
(Show Context)
Citation Context ...ated to the complexity change, when the error-locating polynomial is multivariate, Step 2 of Algorithm 2 is more challenging since it contains multivariate polynomial division. A related reference is =-=[10]-=- which is the most efficient implementation for the computation of Gröbner bases relies on linear algebra. Using Gröbner bases we can implement the division at close to O(NlogN) time, as done in Alg... |

2 |
A Simple Expression for Multivariate Lagrange
- Saniee
- 2008
(Show Context)
Citation Context ...4 (Time complexity). Given N = t + ( d+t+2 d+t ) data samples, we can reconstruct p(x, y) using O(Nω) running time. Proof. Generally, for m variate polynomial with degree d, there are ( d+m d ) terms =-=[18]-=-; thus, it is a necessary condition that we have t+ ( d+t+2 d+t ) distinct points for q and e to be uniquely defined. We have N linear equation in at most N variables, which we can solve e.g., by Gaus... |

1 |
Khot,“Fitting algebraic curves to noisy data”, STOC
- Arora, S
- 2002
(Show Context)
Citation Context ...theories [17], most research has used the `2 norm of noise, such as the method of least square errors. These attitudes not suffice the 2 adversarial noise we have assumed here. To our knowledge, only =-=[2]-=- referred the `∞ noise that fits our considered problem and we further relate [2] study. The polynomial fitting problem as stated in Definition 1.1 can also be studied by ErrorCorrecting Code Theory. ... |

1 | Approximation by spline functions - Guenther - 1989 |