SVD compression, unitary transforms, and computational complexity

The search for fast unitary transforms and the need for data compression in linear systems are complementary issues. Compression requires the definition of a threshold dependent on the condition number, which is invariant over the unitary group. With respect to this threshold, it is shown that the SVD is the optimal tool. Considerations in connection with the Kronecker product and direct sum of unitary matrices show that the computational complexity of unitary transforms is entropy-like in nature, thereby indicating that the O(n log n) complexity unitary transforms are dense over the unitary group.