Abstract
Lett≥1 and letn, M be natural numbers,n<M. Leta=(a i,j ) be ann xM matrix whose rows are orthonormal. Suppose that the ℓ2-norms of the columns ofA are uniformly bounded. Namely, for allj \(\sqrt {\frac{M}{n}} \cdot \left( {\sum\limits_{i = 1}^n {a_{i,j}^2 } } \right)^{1/2} \leqslant t.\)
Using majorizing measure estimates we prove that for every ε>0 there exists, a setI ⊃ {1,…,M} of cardinality at most\(C \cdot \frac{{t^2 }}{{\varepsilon ^2 }} \cdot n \cdot log\frac{{nt^2 }}{{\varepsilon ^2 }}\) such that the matrix\(\sqrt {M/\left| I \right|} \cdot A_I^T \), whereA I =(a i,j ) j∈I , acts as a (1+ε)-isomorphism from ℓ n2 into\(\ell _2^{\backslash I\backslash } \).
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Research supported in part by a grant of the US-Israel BSF. Part of this research was performed when the author held a postdoctoral position at MSRI. Research at MSRI was supported in part by NSF grant DMS-9022140.
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Rudelson, M. Almost orthogonal submatrices of an orthogonal matrix. Isr. J. Math. 111, 143–155 (1999). https://doi.org/10.1007/BF02810682
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DOI: https://doi.org/10.1007/BF02810682