Manifold-adaptive dimension estimation (original) (raw)

Abstract

sparkles

AI

This paper presents a novel algorithm for estimating the unknown dimension of a data manifold from samples, proving it to be manifold-adaptive. The proposed method belongs to the nearest-neighbor framework, addressing the limitations of existing dimension-estimation techniques by focusing on finite-sample behavior and introducing a rigorous theoretical foundation. Empirical results demonstrate the algorithm's effectiveness across various datasets, indicating high accuracy and robustness in dimension estimation.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (13)

  1. Azuma, K. (1967). Weighted sums of certain dependent random variables. Tohoku Math. J., 19(2):357-367.
  2. Gine, E. and Koltchinskii, V. (2007). Empirical graph Laplacian approximation of Laplace-Beltrami operators: Large sample results. In Proc. of the 4 th Int. Conf. on High Dimensional Probability. to appear.
  3. Grassberger, P. and Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica D, 9:189-208.
  4. Hein, M. (2006). Uniform convergence of adaptive graph- based regularization. In COLT-2006, pages 50-64.
  5. Hein, M. and Audibert, J.-Y. (2005). Intrinsic dimension- ality estimation of submanifolds in Euclidean space. In ICML-2005, pages 289-296.
  6. Hein, M., Audibert, J.-Y., and von Luxburg, U. (2006). From graphs to manifolds -weak and strong pointwise consistency of graph Laplacians. submitted.
  7. Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13-30.
  8. Kegl, B. (2002). Intrinsic dimension estimation using pack- ing numbers. In NIPS-15, pages 681-688.
  9. Levina, E. and Bickel, P. J. (2005). Maximum likelihood estimation of intrinsic dimension. In NIPS-17, pages 777-784.
  10. McDiarmid, C. (1989). On the method of bounded differ- ences. Surveys in Combinatorics, pages 148-188.
  11. Pettis, K. W., Bailey, T. A., Jain, A. K., and Dubes, R. C. (1979). An intrinsic dimensionality estimator from near- neighbor information. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1:25-37.
  12. Stone, C. (1977). Consistent nonparametric regression. An- nals of Statistics, 5(4):595-620.
  13. Tenenbaum, J. B., de Silva, V., and Langford, J. C. (2000). A global geometric framework for nonlinear dimension- ality reduction. Science, 290:2319-2323.