Alexander Petukhov

Fast implementation of orthogonal greedy algorithm for tight wavelet frames

We introduce and study a fast implementation of orthogonal greedy algorithm for wavelet frames. The results of numerical simulation for N-term approximations of standard images by tight wavelet frames showed that to provide the same PSNR the popular 9/7 biorthogonal wavelet bases require 38-42% more terms of the expansion. This is equivalent to 2-4 dB advantage of the wavelet frames in PSNR over the 9/7 bases for a fixed number of decomposition terms.

Fast Implementation of ℓ1-Greedy Algorithm

We present an algorithm for finding sparse solutions of the system of linear equations A x = b with the rectangular matrix A of size n ×N, where n < N. The algorithm basic constructive block is one iteration of the standard interior-point linear programming algorithm. To find the sparse representation we modify (reweight) each iteration in the spirit of Petukhov (Fast implementation of orthogonal greedy algorithm for tight wavelet frames. Signal Process. 86, 471–479 (2006)). However, the weights are selected according to the l 1-greedy strategy developed in Kozlov and Petukhov (Sparse solutions for underdetermined systems of linear equations. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 1243–1259. Springer, Berlin (2010)). Our algorithm combines computational complexity close to plain l 1-minimization with the significantly higher efficiency of the sparse representations recovery than the reweighted l 1-minimization (Candes et al.: Enhancing sparsity by reweighted l 1 minimization. J. Fourier Anal. Appl. 14, 877–905 (2008) (special issue on sparsity)), approaching the capacity of the l 1-greedy algorithm.

Greedy algorithm for subspace clustering from corrupted and incomplete data

We describe the Fast Greedy Sparse Subspace Clustering (FGSSC) algorithm providing an efficient method for clustering data belonging to a few low-dimensional linear or affine subspaces. FGSSC is a modification of the SSC algorithm. The main difference of our algorithm from predecessors is its ability to work with noisy data having a high rate of erasures (missed entries at the known locations) and errors (corrupted entries at unknown locations). The algorithm has significant advantage over predecessor on synthetic models as well as for the Extended Yale B dataset of facial images. In particular, the face recognition misclassification rate turned out to be 6-20 times lower than for the SSC algorithm.

Error Resilient Neural Networks on Low-Dimensional Manifolds

We introduce an algorithm that improves Neural Network classification/registration of corrupted data belonging to low-dimensional manifolds. The algorithm combines ideas of the Orthogonal Greedy Algorithm with the standard gradient back-propagation engine incorporated in Neural Networks. Therefore, we call it the Greedient algorithm.