Invariance in property testing
Author: 
Madhu SudanAuthors Info & Claims
Madhu SudanAuthors Info & ClaimsAbstract
Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For "global properties", i.e., properties that really depend somewhat on every piece of the data, one could ask how it can be tested by so few samples? We suggest that for "natural" properties, this should happen because the property is invariant under "nice" set of "relabellings" of the data. We refer to this set of relabellings as the "invariance class" of the property and advocate explicit identification of the invariance class of locally testable properties. Our hope is the explicit knowledge of the invariance class may lead to more general, broader, results.
After pointing out the invariance classes associated with some the basic classes of testable properties, we focus on "algebraic properties" which seem to be characterized by the fact that the properties are themselves vector spaces, while their domains are also vector spaces and the properties are invariant under affine transformations of the domain. We survey recent results (obtained with Tali Kaufman, Elena Grigorescu and Eli Ben-Sasson) that give broad conditions that are sufficient for local testability among this class of properties, and some structural theorems that attempt to describe which properties exhibit the sufficient conditions.
References
[1]
Alon, N., Andoni, A., Kaufman, T., Matulef, K., Rubinfeld, R., Xie, N.: Testing k-wise and almost k-wise independence. In: Johnson, D.S., Feige, U. (eds.) STOC, pp. 496-505. ACM, New York (2007)
[2]
N. Alon, E. Fischer, I. Newman, A. Shapira. A combinatorial characterization of the testable graph properties: it's all about regularity. In: Kleinberg {42}, pp. 251- 260
[3]
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing reed-muller codes. IEEE Transactions on Information Theory 51(11), 4032-4039 (2005)
[4]
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501-555 (1998)
[5]
Babai, L., Fortnow, L., Levin, L.A., Szegedy, M.: Checking computations in polylogarithmic time. In: Proceedings of the 23rd ACM Symposium on the Theory of Computing, pp. 21-32. ACM Press, New York (1991)
[6]
Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has twoprover interactive protocols. Computational Complexity 1(1), 3-40 (1991)
[7]
Babai, L., Shpilka, A., Stefankovic, D.: Locally testable cyclic codes. IEEE Transactions on Information Theory 51(8), 2849-2858 (2005)
[8]
Batu, T., Dasgupta, S., Kumar, R., Rubinfeld, R.: The complexity of approximating the entropy. SIAM J. Comput. 35(1), 132-150 (2005)
[9]
Batu, T., Fortnow, L., Fischer, E., Kumar, R., Rubinfeld, R., White, P.: Testing random variables for independence and identity. In: FOCS, pp. 442-451 (2001)
[10]
Batu, T., Fortnow, L., Rubinfeld, R., Smith, W.D., White, P.: Testing that distributions are close. In: FOCS, pp. 259-269 (2000)
[11]
Batu, T., Kumar, R., Rubinfeld, R.: Sublinear algorithms for testing monotone and unimodal distributions. In: Babai, L. (ed.) STOC, pp. 381-390. ACM, New York (2004)
[12]
Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCP's and nonapproximability -- towards tight results. SIAM Journal on Computing 27(3), 804-915 (1998)
[13]
Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.P.: Robust pcps of proximity, shorter pcps, and applications to coding. SIAM Journal on Computing 36(4), 889-974 (2006)
[14]
Ben-Sasson, E., Harsha, P., Raskhodnikova, S.: Some 3CNF properties are hard to test. SIAM Journal on Computing 35, 1-21 (2005); Preliminary version in Proc. STOC 2003 (2003)
[15]
Ben-Sasson, E., Sudan, M.: Limits on the rate of locally testable affine-invariant codes (November 2009) (manuscript)
[16]
Bhattacharyya, A., Chen, V., Sudan, M., Xie, N.: Testing linear-invariant nonlinear properties. In: Albers, S., Marion, J.-Y. (eds.) STACS. LIPIcs, vol. 3, pp. 135-146. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2009)
[17]
Bhattacharyya, A., Xie, N.: Lower bounds for testing triangle-freeness in boolean functions. In: SODA 2010: Proceedings of the twenty-first Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 87-98. Society for Industrial and Applied Mathematics, Philadelphia (2010)
[18]
Blais, E.: Testing juntas nearly optimally. In: Mitzenmacher {47}, pp. 151-158
[19]
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences 47(3), 549-595 (1993)
[20]
Borgs, C., Chayes, J.T., Lovász, L., Sós, V.T., Szegedy, B., Vesztergombi, K.: Graph limits and parameter testing. In: Kleinberg {42}, pp. 261-270
[21]
Diakonikolas, I., Lee, H.K., Matulef, K., Onak, K., Rubinfeld, R., Servedio, R.A., Wan, A.: Testing for concise representations. In: FOCS, pp. 549-558. IEEE Computer Society, Los Alamitos (2007)
[22]
Dinur, I., Reingold, O.: Assignment testers: Towards a combinatorial proof of the PCP-theorem. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 155-164. IEEE Press, Los Alamitos (2004)
[23]
Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., Samorodnitsky, A.: Improved testing algorithms for monotonicity. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds.) RANDOM 1999 and APPROX 1999. LNCS, vol. 1671, pp. 97-108. Springer, Heidelberg (1999)
[24]
Ergün, F., Kannan, S., Kumar, R., Rubinfeld, R., Viswanathan, M.: Spot-checkers. J. Comput. Syst. Sci. 60(3), 717-751 (2000)
[25]
Fischer, E., Kindler, G., Ron, D., Safra, S., Samorodnitsky, A.: Testing juntas. J. Comput. Syst. Sci. 68(4), 753-787 (2004)
[26]
Friedl, K., Sudan, M.: Some improvements to total degree tests. In: Proceedings of the 3rd Annual Israel Symposium on Theory of Computing and Systems, Washington, DC, USA, January 4-6, pp. 190-198. IEEE Computer Society, Los Alamitos (1995), http://people.csail.mit.edu/madhu/papers/friedl.ps
[27]
Goldreich, O., Goldwasser, S., Lehman, E., Ron, D., Samorodnitsky, A.: Testing monotonicity. Combinatorica 20(3), 301-337 (2000)
[28]
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. JACM 45(4), 653-750 (1998)
[29]
Goldreich, O., Kaufman, T.: Proximity oblivious testing and the role of invariances (March 2010) (manuscript)
[30]
Goldreich, O., Ron, D.: On testing expansion in bounded-degree graphs. Electronic Colloquium on Computational Complexity (ECCC) 7(20) (2000)
[31]
Goldreich, O., Ron, D.: Property testing in bounded degree graphs. Algorithmica 32(2), 302-343 (2002)
[32]
Goldreich, O., Ron, D.: On proximity oblivious testing. In: Mitzenmacher {47}, pp. 141-150
[33]
Goldreich, O., Sheffet, O.: On the randomness complexity of property testing. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 509-524. Springer, Heidelberg (2007)
[34]
Gopalan, P., O'Donnell, R., Servedio, R.A., Shpilka, A., Wimmer, K.: Testing fourier dimensionality and sparsity. In: Albers, S.E., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 500-512. Springer, Heidelberg (2009)
[35]
Green, B.: A Szemerédi-type regularity lemma in abelian groups, with applications. Geometric and Functional Analysis 15(2), 340-376 (2005)
[36]
Grigorescu, E., Kaufman, T., Sudan, M.: 2-transitivity is insufficient for local testability. In: CCC 2008: Proceedings of the 23rd IEEE Conference on Computational Complexity, June 23-26. IEEE Computer Society, Los Alamitos (2008) (to appear)
[37]
Grigorescu, E., Kaufman, T., Sudan, M.: Succinct representation of codes with applications to testing. In: Dinur, I., Jansen, K., Naor, J., Rolim, J.D.P. (eds.) APPROX-RANDOM 2009. LNCS, vol. 5687, pp. 534-547. Springer, Heidelberg (2009)
[38]
Jutla, C.S., Patthak, A.C., Rudra, A., Zuckerman, D.: Testing low-degree polynomials over prime fields. In: FOCS 2004: Proceedings of the Forty-Fifth Annual IEEE Symposium on Foundations of Computer Science, pp. 423-432. IEEE Computer Society, Los Alamitos (2004)
[39]
Kaufman, T., Ron, D.: Testing polynomials over general fields. SIAM J. Comput. 36(3), 779-802 (2006)
[40]
Kaufman, T., Sudan, M.: Algebraic property testing: The role of invariance. Technical Report TR07-111, Electronic Colloquium on Computational Complexity, November 2 (2007); Extended abstract in Proc. 40th STOC (2008)
[41]
Kaufman, T., Wigderson, A.: Symmetric LDPC codes and local testing. In: Proceedings of ICS 2010 (January 2010)
[42]
Kleinberg, J.M. (ed.): Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23. ACM, New York (2006)
[43]
Král', D., Serra, O., Vena, L.: A removal lemma for systems of linear equations over finite fields. arxiv.org:0809.1846v1 {math.CO}
[44]
Král', D., Serra, O., Vena, L.: A combinatorial proof of the removal lemma for groups. Journal of Combinatorial Theory, Series A 116(4), 971-978 (2009)
[45]
Ladner, R.E., Dwork, C. (eds.): Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20. ACM, New York (2008)
[46]
Matulef, K., O'Donnell, R., Rubinfeld, R., Servedio, R.A.: Testing halfspaces. In: Mathieu, C. (ed.) SODA, pp. 256-264. SIAM, Philadelphia (2009)
[47]
Mitzenmacher, M. (ed.): Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31-June 2. ACM, New York (2009)
[48]
Onak, K., Sudan, M.: Learnability of general discrete distributions (March 2010) (manuscript)
[49]
Parnas, M., Ron, D., Samorodnitsky, A.: Testing basic boolean formulae. SIAM J. Discrete Math. 16(1), 20-46 (2002)
[50]
Raskhodnikova, S., Ron, D., Shpilka, A., Smith, A.: Strong lower bounds for approximating distribution support size and the distinct elements problem. SIAM J. Comput. 39(3), 813-842 (2009)
[51]
Rubinfeld, R.: Robust functional equations and their applications to program testing. SIAM Journal on Computing 28(6), 1972-1997 (1999)
[52]
Rubinfeld, R., Servedio, R.A.: Testing monotone high-dimensional distributions. Random Struct. Algorithms 34(1), 24-44 (2009)
[53]
Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252-271 (1996)
[54]
Shapira, A.: Green's conjecture and testing linear-invariant properties. In: Mitzenmacher {47}, pp. 159-166
[55]
Valiant, P.: Testing symmetric properties of distributions. In: Ladner and Dwork {45}, pp. 383-392
Index Terms
- Invariance in property testing
Index terms have been assigned to the content through auto-classification.
Recommendations
Invariance in property testing
Property testingProperty testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For "global properties", i.e., properties that ...
Algebraic property testing: the role of invariance
STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computingWe argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space Kn over a field K to a subfield ...
Comments
Information & Contributors
Information
Published In
Publisher
Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 01 January 2010
Qualifiers
- Chapter
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 12 Aug 2024