Highly Nonlinear Approximations for Sparse Signal Representation

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Orthogonality

Two vectors $ v$ and $ w$ in an inner product space are said to be orthogonal if $ \langle v , w \rangle =0.$ If, in addition, $ \Vert v \Vert=\Vert w\Vert=1$ they are orthonormal.

Two subspaces $ {\cal{V}}_1$ and $ {\cal{V}}_2$ are orthogonal if $ \langle v_1,v_2 \rangle =0$ for all $ v_1\in {\cal{V}}_1$ and $ v_2\in {\cal{V}}_1$. The sum of two orthogonal subspaces $ {\cal{V}}_1$ and $ {\cal{V}}_2$ is termed orthogonal sum and will be indicated as $ {\cal{V}}= {\cal{V}}_1 \oplus^\bot {\cal{V}}_2.$ The subspace $ {\cal{V}}_2$ is called the orthogonal complement of $ {\cal{V}}_1$ in $ {\cal{V}}$. Equivalently, $ {\cal{V}}_1$ is the orthogonal complement of $ {\cal{V}}_2$ in $ {\cal{V}}$.