A statistical estimation algorithm of the weight distribution of a linear code is shown, based on using its generator matrix as a compression function on random bit strings.
The Lee-Brickell algorithm, as well as others, can be interpreted in the context of information set decoding, an equivalent definition of which can be given in terms of the generator matrix or the parity check matrix - see for instance @cite_14 and @cite_11 . with @math the @math th column of @math .
@cite_14 @cite_11
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Designing good error correcting codes whose generator matrix has a support constraint, i.e., one for which only certain entries of the generator matrix are allowed to be non-zero, has found many recent applications, including in distributed coding and storage, multiple access networks, and weakly secure data exchange. The dual problem, where the parity check matrix has a support constraint, comes up in the design of locally repairable codes. The central problem here is to design codes with the largest possible minimum distance, subject to the given support constraint on the generator matrix. An upper bound on the minimum distance can be obtained through a set of singleton bounds, which can be alternatively thought of as a cut-set bound. Furthermore, it is well known that, if the field size is large enough, any random generator matrix obeying the support constraint will achieve the maximum minimum distance with high probability. Since random codes are not easy to decode, structured codes with efficient decoders, e.g., Reed-Solomon codes, are much more desirable. The GM-MDS conjecture of states that the maximum minimum distance over all codes satisfying the generator matrix support constraint can be obtained by a Reed Solomon code. If true, this would have significant consequences. The conjecture has been proven for several special case: when the dimension of the code k is less than or equal to five, when the number of distinct support sets on the rows of the generator matrix m, say, is less than or equal to three, or when the generator matrix is sparsest and balanced. In this paper, we report on further progress on the GM-MDS conjecture. In particular, we show that the conjecture is true for all m less than equal to six. This generalizes all previous known results (except for the sparsest and balanced case, which is a very special support constraint). Let M be a random (alpha n) x n matrix of rank r<<n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(rn) observed entries with relative root mean square error RMSE <= C(rn |E|)^0.5 . Further, if r=O(1), M can be reconstructed exactly from |E| = O(n log(n)) entries. These results apply beyond random matrices to general low-rank incoherent matrices. This settles (in the case of bounded rank) a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The complexity of our algorithm is O(|E|r log(n)), which opens the way to its use for massive data sets. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices. Given a matrix A e ℝ m ×n of rank r, and an integer k < r, the top k singular vectors provide the best rank-k approximation to A. When the columns of A have specific meaning, it is desirable to find (provably) "good" approximations to A k which use only a small number of columns in A. Proposed solutions to this problem have thus far focused on randomized algorithms. Our main result is a simple greedy deterministic algorithm with guarantees on the performance and the number of columns chosen. Specifically, our greedy algorithm chooses c columns from A with @math such that @math where C gr is the matrix composed of the c columns, @math is the pseudo-inverse of C gr ( @math is the best reconstruction of A from C gr), and ¼(A) is a measure of the coherence in the normalized columns of A. The running time of the algorithm is O(SVD(A k) + mnc) where SVD(A k) is the running time complexity of computing the first k singular vectors of A. To the best of our knowledge, this is the first deterministic algorithm with performance guarantees on the number of columns and a (1 + e) approximation ratio in Frobenius norm. The algorithm is quite simple and intuitive and is obtained by combining a generalization of the well known sparse approximation problem from information theory with an existence result on the possibility of sparse approximation. Tightening the analysis along either of these two dimensions would yield improved results. We observe (Xi,Yi)i=1n where the Yi's are real valued outputs and the Xi's are m × T matrices. We observe a new entry X and we want to predict the output Y associated with it. We focus on the high-dimensional setting, where mT ≫ n. This includes the matrix completion problem with noise, as well as other problems. We consider linear prediction procedures based on different penalizations, involving a mixture of several norms: the nuclear norm, the Frobenius norm and the l1-norm. For these procedures, we prove sharp oracle inequalities, using a statistical learning theory point of view. A surprising fact in our results is that the rates of convergence do not depend on m and T directly. The analysis is conducted without the usually considered incoherency condition on the unknown matrix or restricted isometry condition on the sampling operator. Moreover, our results are the first to give for this problem an analysis of penalization (such as nuclear norm penalization) as a regularization algorithm: our oracle inequalities prove that these procedures have a prediction accuracy close to the deterministic oracle one, given that the reguralization parameters are well-chosen.