Journal of Computational and Applied Mathematics
Smooth Analysis of the Condition Number and the Least Singular Value
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Matrix completion from a few entries
IEEE Transactions on Information Theory
Matrix Completion from Noisy Entries
The Journal of Machine Learning Research
COLT'05 Proceedings of the 18th annual conference on Learning Theory
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We compare the Euclidean operator norm of arandom matrix with the Euclidean norm of its rows and columns. Inthe first part of this paper, we show that if A is a randommatrix with i.i.d. zero mean entries, thenE∥A∥h ≤Kh (E maxi∥ai•∥h + E maxj∥aj•∥h), where K is a constant whichdoes not depend on the dimensions or distribution of A(h, however, does depend on the dimensions). In the secondpart we drop the assumption that the entries of A are i.i.d.We therefore consider the Euclidean operator norm of a randommatrix, A, obtained from a (non-random) matrix byrandomizing the signs of the matrix's entries. We show that in thiscase, the best inequality possible (up to a multiplicativeconstant) is E∥A∥h≤ (c log1/4 min {m,n})h (E maxi∥ai• ∥h+ E maxj∥aj•∥h) (m, n the dimensions ofthe matrix and c a constant independent of m,n).