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Madhu Sudan - One of the best experts on this subject based on the ideXlab platform.
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sparse Affine Invariant linear codes are locally testable
Computational Complexity, 2017Co-Authors: Eli Bensasson, Noga Ronzewi, Madhu SudanAbstract:We show that sparse Affine-Invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field $${\mathbb{F}_q}$$Fq and an extension field $${\mathbb{F}_{q^n}}$$Fqn, a property is a set of functions mapping $${\mathbb{F}_{q^n}}$$Fqn to $${\mathbb{F}_q}$$Fq. The property is said to be Affine-Invariant if it is Invariant under Affine transformations of $${\mathbb{F}_{q^n}}$$Fqn, linear if it is an $${\mathbb{F}_q}$$Fq -vector space, and sparse if its size is polynomial in the domain size. Our work completes a line of work initiated by Grigorescu et al. (2009) and followed by Kaufman & Lovett (2011). The latter showed such a result for the case when q was prime. Extending to non-prime cases turns out to be non-trivial, and our proof involves some detours into additive combinatorics, as well as a new calculus for building property testers for Affine-Invariant linear properties.
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FOCS - Sparse Affine-Invariant Linear Codes Are Locally Testable
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, 2012Co-Authors: Eli Ben-sasson, Noga Ron-zewi, Madhu SudanAbstract:We show that sparse Affine-Invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field $\F_q$ and an extension field $\F_{q^n}$, a property is a set of functions mapping $\F_{q^n}$ to $\F_q$. The property is said to be Affine-Invariant if it is Invariant under Affine transformations of $\F_{q^n}$, and it is said to be sparse if its size is polynomial in the domain size. Our work completes a line of work initiated by Grigorescu et al. [RANDOM 2009] and followed by Kaufman and Lovett [FOCS 2011]. The latter showed such a result for the case when $q$ was prime. Extending to non-prime cases turns out to be non-trivial and our proof involves some detours into additive combinatorics, as well as a new calculus for building property testers for Affine-Invariant linear properties.
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Sparse Affine-Invariant Linear Codes Are Locally Testable
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, 2012Co-Authors: Eli Ben-sasson, Noga Ron-zewi, Madhu SudanAbstract:We show that sparse Affine-Invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field Fq and an extension field Fqn, a property is a set of functions mapping Fqn to Fq. The property is said to be Affine-Invariant if it is Invariant under Affine transformations of Fqn, and it is said to be sparse if its size is polynomial in the domain size. Our work completes a line of work initiated by Grigorescu et al. [RANDOM 2009] and followed by Kaufman and Lovett [FOCS 2011]. The latter showed such a result for the case when q was prime. Extending to non-prime cases turns out to be non-trivial and our proof involves some detours into additive combinatorics, as well as a new calculus for building property testers for Affine-Invariant linear properties.
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on sums of locally testable Affine Invariant properties
International Workshop and International Workshop on Approximation Randomization and Combinatorial Optimization. Algorithms and Techniques, 2011Co-Authors: Eli Bensasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, Madhu SudanAbstract:Affine-Invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine Invariant properties consider functions mapping a big field Fqn to the subfield Fq and include all properties that form an Fq-vector space and are Invariant under Affine transformations of the domain. Almost all the known locally testable Affine-Invariant properties have so-called "single-orbit characterizations" -- namely they are specified by a single local constraint on the property, and the "orbit" of this constraint, i.e., translations of this constraint induced by Affineinvariance. Single-orbit characterizations by a local constraint are also known to imply local testability. In this work we show that properties with single-orbit characterizations are closed under "summation". To complement this result, we also show that the property of being an n-variate low-degree polynomial over Fq has a single-orbit characterization (even when the domain is viewed as Fqn and so has very few Affine transformations). As a consequence we find that the sum of any sparse Affine-Invariant property (properties satisfied by qO(n)-functions) with the set of degree d multivariate polynomials over Fq has a single-orbit characterization (and is hence locally testable) when q is prime. We conclude with some intriguing questions/conjectures attempting to classify all locally testable Affine-Invariant properties.
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limits on the rate of locally testable Affine Invariant codes
International Workshop and International Workshop on Approximation Randomization and Combinatorial Optimization. Algorithms and Techniques, 2011Co-Authors: Eli Bensasson, Madhu SudanAbstract:Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, "algebraic property testing" is not well-understood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) "Affine-Invariant properties", i.e., properties that are closed under summation, and under Affine transformations of the domain. Kaufman and Sudan showed that these two features (linearity of the property and its Affine-invariance) play a central role in the testability of many known algebraic properties. However their work does not give a complete characterization of the testability of Affine-Invariant properties, and several technical obstacles need to be overcome to obtain such a characterization. Indeed, their work left open the tantalizing possibility that locally testable codes of rate dramatically better than that of the family of Reed-Muller codes (the most popular form of locally testable codes, which also happen to be Affine-Invariant) could be found by systematically exploring the space of Affine-Invariant properties. In this work we rule out this possibility and show that general (linear) Affine-Invariant properties are contained in Reed-Muller codes that are testable with a slightly larger query complexity. A central impediment to proving such results was the limited understanding of the structural restrictions on Affine-Invariant properties imposed by the existence of local tests. We manage to overcome this limitation and present a clean restriction satisfied by Affine-Invariant properties that exhibit local tests. We do so by relating the problem to that of studying the set of solutions of a certain nice class of (uniform, homogenous, diagonal) systems of multivariate polynomial equations. Our main technical result completely characterizes (combinatorially) the set of zeroes, or algebraic set, of such systems.
Chun Hsiung Fang - One of the best experts on this subject based on the ideXlab platform.
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Synthesized Affine Invariant function for 2D shape recognition
Pattern Recognition, 2007Co-Authors: Chun Hsiung FangAbstract:By defining the weighted wavelet synthesis, the synthesized feature signals of an interesting shape are extracted to derive the innovative synthesized Affine Invariant function (SAIF). The synthesized feature signals hold the shape information with minimum loss by excluding simply the translation dependent and noise-contaminated bands. The SAIF is shown excellent in the invariance property and representative in describing the original shape for automated recognition. Experimental results demonstrate that automated shape recognition based on the SAIF achieves high correctness and significantly outperforms those using conventional wavelet Affine Invariant functions.
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Synthesized Affine Invariant function for 2D shape recognition
Pattern Recognition, 2007Co-Authors: Wei Song Lin, Chun Hsiung FangAbstract:By defining the weighted wavelet synthesis, the synthesized feature signals of an interesting shape are extracted to derive the innovative synthesized Affine Invariant function (SAIF). The synthesized feature signals hold the shape information with minimum loss by excluding simply the translation dependent and noise-contaminated bands. The SAIF is shown excellent in the invariance property and representative in describing the original shape for automated recognition. Experimental results demonstrate that automated shape recognition based on the SAIF achieves high correctness and significantly outperforms those using conventional wavelet Affine Invariant functions. © 2006 Pattern Recognition Society.
Liang Cheng - One of the best experts on this subject based on the ideXlab platform.
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remote sensing image matching by integrating Affine Invariant feature extraction and ransac
Computers & Electrical Engineering, 2012Co-Authors: Liang Cheng, Manchun Li, Yanming Chen, Kang YangAbstract:A new technical framework for remote sensing image matching by integrating Affine Invariant feature extraction and RANSAC is presented. The novelty of this framework is an automatic optimization strategy for Affine Invariant feature matching based on RANSAC. An automatic way to determine the distance threshold of RANSAC is proposed, which is a key problem to implement this RANSAC-based automatic optimization. Since Affine Invariant feature matching technology has been successfully applied to remote sensing image matching, we design an experiment to compare the proposed framework (with optimization) with the standard Affine Invariant feature matching (without optimization). By using three pairs with different types of imagery, the experimental results indicate that the proposed framework can always get higher correctness of image matching in automatic way, compared to the standard Affine Invariant feature matching technology.
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A new method for remote sensing image matching by integrating Affine Invariant feature extraction and RANSAC
2010 3rd International Congress on Image and Signal Processing, 2010Co-Authors: Liang Cheng, Hao Hu, Yecheng Wang, Manchun LiAbstract:A new method on remote sensing image matching by integrating Affine Invariant feature extraction and RANSAC is presented. The novelty of this method is a strategy on automatic optimization for Affine Invariant feature matching based on RANSAC. An automatic way to determine the distance threshold of RANSAC is proposed, which is a key problem to implement this RANSAC-based automatic optimization. Since Affine Invariant feature matching technology has been successfully applied to remote sensing image matching, we design an experiment to compare the proposed method (with optimization) with the standard Affine Invariant feature matching (without optimization). By using two stereo pairs with different types of imagery, the experiment indicates that the proposed method can always get much matching score compared to the standard Affine Invariant feature matching method.
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Robust Affine Invariant Feature Extraction for Image Matching
IEEE Geoscience and Remote Sensing Letters, 2008Co-Authors: Liang Cheng, Jianya Gong, Xiaoxia YangAbstract:A new approach is presented to extract more robust Affine Invariant features for image matching. The novelty of our approach is a hierarchical filtering strategy for Affine Invariant feature detection, which is based on information entropy and spatial dispersion quality constraints. The concept of spatial dispersion quality is introduced to quantify the spatial distribution of features. Moreover, an integrated algorithm combined by the filtering strategy, maximally stable extremal region (MSER) and scale Invariant feature transform, is introduced for Affine Invariant feature extraction. Since Mikolajczyk et al. identified that MSER is the best detector in many cases, we design an experiment to compare our approach (ED-MSER) with the standard MSER. By using two stereo pairs and an image sequence with different types of imagery, the experiment indicates that ED-MSER can always get much higher repeatability and matching score compared to the standard MSER and other algorithms, thus benefiting the subsequent image matching and many other applications.
A. Enis Cetin - One of the best experts on this subject based on the ideXlab platform.
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Computationally efficient wavelet Affine Invariant functions for shape recognition
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004Co-Authors: Erdem Bala, A. Enis CetinAbstract:An Affine Invariant function for object recognition is constructed from wavelet coefficients of the object boundary. In previous works, undecimated dyadic wavelet transform was used to construct Affine Invariant functions. In this paper, an algorithm based on decimated wavelet transform is developed to compute an Affine Invariant function. As a result computational complexity is reduced without decreasing recognition performance. Experimental results are presented.
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Computationally efficient wavelet Affine Invariant functions for 2D object recognition
Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429), 2003Co-Authors: Erdem Bala, A. Enis CetinAbstract:In this paper, an Affine Invariant function is presented for object recognition from wavelet coefficients of the object boundary. In previous works, undecimated wavelet transform was used for Affine Invariant functions. In this paper, an algorithm based on decimated wavelet transform is developed to compute the Affine Invariant function. As a result, computational complexity is significantly reduced without decreasing recognition performance. Experimental results are presented.
Eli Ben-sasson - One of the best experts on this subject based on the ideXlab platform.
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FOCS - Sparse Affine-Invariant Linear Codes Are Locally Testable
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, 2012Co-Authors: Eli Ben-sasson, Noga Ron-zewi, Madhu SudanAbstract:We show that sparse Affine-Invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field $\F_q$ and an extension field $\F_{q^n}$, a property is a set of functions mapping $\F_{q^n}$ to $\F_q$. The property is said to be Affine-Invariant if it is Invariant under Affine transformations of $\F_{q^n}$, and it is said to be sparse if its size is polynomial in the domain size. Our work completes a line of work initiated by Grigorescu et al. [RANDOM 2009] and followed by Kaufman and Lovett [FOCS 2011]. The latter showed such a result for the case when $q$ was prime. Extending to non-prime cases turns out to be non-trivial and our proof involves some detours into additive combinatorics, as well as a new calculus for building property testers for Affine-Invariant linear properties.
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Sparse Affine-Invariant Linear Codes Are Locally Testable
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, 2012Co-Authors: Eli Ben-sasson, Noga Ron-zewi, Madhu SudanAbstract:We show that sparse Affine-Invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field Fq and an extension field Fqn, a property is a set of functions mapping Fqn to Fq. The property is said to be Affine-Invariant if it is Invariant under Affine transformations of Fqn, and it is said to be sparse if its size is polynomial in the domain size. Our work completes a line of work initiated by Grigorescu et al. [RANDOM 2009] and followed by Kaufman and Lovett [FOCS 2011]. The latter showed such a result for the case when q was prime. Extending to non-prime cases turns out to be non-trivial and our proof involves some detours into additive combinatorics, as well as a new calculus for building property testers for Affine-Invariant linear properties.
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APPROX-RANDOM - On sums of locally testable Affine Invariant properties
Approximation Randomization and Combinatorial Optimization. Algorithms and Techniques, 2011Co-Authors: Eli Ben-sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, Madhu SudanAbstract:Affine-Invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine Invariant properties consider functions mapping a big field Fqn to the subfield Fq and include all properties that form an Fq-vector space and are Invariant under Affine transformations of the domain. Almost all the known locally testable Affine-Invariant properties have so-called "single-orbit characterizations" -- namely they are specified by a single local constraint on the property, and the "orbit" of this constraint, i.e., translations of this constraint induced by Affineinvariance. Single-orbit characterizations by a local constraint are also known to imply local testability. In this work we show that properties with single-orbit characterizations are closed under "summation". To complement this result, we also show that the property of being an n-variate low-degree polynomial over Fq has a single-orbit characterization (even when the domain is viewed as Fqn and so has very few Affine transformations). As a consequence we find that the sum of any sparse Affine-Invariant property (properties satisfied by qO(n)-functions) with the set of degree d multivariate polynomials over Fq has a single-orbit characterization (and is hence locally testable) when q is prime. We conclude with some intriguing questions/conjectures attempting to classify all locally testable Affine-Invariant properties.
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APPROX-RANDOM - Limits on the rate of locally testable Affine-Invariant codes
Approximation Randomization and Combinatorial Optimization. Algorithms and Techniques, 2011Co-Authors: Eli Ben-sasson, Madhu SudanAbstract:Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, "algebraic property testing" is not well-understood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) "Affine-Invariant properties", i.e., properties that are closed under summation, and under Affine transformations of the domain. Kaufman and Sudan showed that these two features (linearity of the property and its Affine-invariance) play a central role in the testability of many known algebraic properties. However their work does not give a complete characterization of the testability of Affine-Invariant properties, and several technical obstacles need to be overcome to obtain such a characterization. Indeed, their work left open the tantalizing possibility that locally testable codes of rate dramatically better than that of the family of Reed-Muller codes (the most popular form of locally testable codes, which also happen to be Affine-Invariant) could be found by systematically exploring the space of Affine-Invariant properties. In this work we rule out this possibility and show that general (linear) Affine-Invariant properties are contained in Reed-Muller codes that are testable with a slightly larger query complexity. A central impediment to proving such results was the limited understanding of the structural restrictions on Affine-Invariant properties imposed by the existence of local tests. We manage to overcome this limitation and present a clean restriction satisfied by Affine-Invariant properties that exhibit local tests. We do so by relating the problem to that of studying the set of solutions of a certain nice class of (uniform, homogenous, diagonal) systems of multivariate polynomial equations. Our main technical result completely characterizes (combinatorially) the set of zeroes, or algebraic set, of such systems.