Abstract
A function f(x 1, ... , x d ), where each input is an integer from 1 to n and output is a real number, is Lipschitz if changing one of the inputs by 1 changes the output by at most 1. In other words, Lipschitz functions are not very sensitive to small changes in the input. Our main result is an efficient tester for the Lipschitz property of functions f: [n]d → δℤ, where δ ∈ (0,1] and δℤ is the set of integer multiples of δ.
The main tool in the analysis of our tester is a smoothing procedure that makes a function Lipschitz by modifying it at a few points. Its analysis is already nontrivial for the 1-dimensional version, which we call Bubble Smooth, in analogy to Bubble Sort. In one step, Bubble Smooth modifies two values that violate the Lipschitz property, i.e., differ by more than 1, by transferring δ units from the larger to the smaller. We define a transfer graph to keep track of the transfers, and use it to show that the ℓ1 distance between f and BubbleSmooth(f) is at most twice the ℓ1 distance from f to the nearest Lipschitz function. Bubble Smooth has other important properties, which allow us to obtain a dimension reduction, i.e., a reduction from testing functions on multidimensional domains to testing functions on the 1-dimensional domain, that incurs only a small multiplicative overhead in the running time and thus avoids the exponential dependence on the dimension.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
All omitted proofs appear in the full version [1].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Awasthi, P., Jha, M., Molinaro, M., Raskhodnikova, S.: Testing Lipschitz functions on hypergrid domains. Electronic Colloquium on Computational Complexity (ECCC) TR12-076 (2012)
Chakrabarty, D., Seshadhri, C.: Optimal bounds for monotonicity and Lipschitz testing over the hypercube. Electronic Colloquium on Computational Complexity (ECCC) TR12-030 (2012)
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-APPROX 1999. LNCS, vol. 1671, pp. 97–108. Springer, Heidelberg (1999)
Goldreich, O., Goldwasser, S., Lehman, E., Ron, D., Samorodnitsky, A.: Testing monotonicity. Combinatorica 20(3), 301–337 (2000)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)
Gromov, M.: Metric Structures for Riemannian and non-Riemannian Spaces (1999)
Jha, M., Raskhodnikova, S.: Testing and reconstruction of Lipschitz functions with applications to data privacy. In: IEEE FOCS, pp. 433–442 (2011) full version available at, http://eccc.hpi-web.de/report/2011/057/
Rubinfeld, R., Sudan, M.: Robust characterization of polynomials with applications to program testing. SIAM J. Comput. 25(2), 252–271 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Awasthi, P., Jha, M., Molinaro, M., Raskhodnikova, S. (2012). Testing Lipschitz Functions on Hypergrid Domains. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-32512-0_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32511-3
Online ISBN: 978-3-642-32512-0
eBook Packages: Computer ScienceComputer Science (R0)