Ending index or upper bound: m_1

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…resembling “scientist,” “Germany,” and “famous person.” Writing $d_i^{(m_2)}$ for one feature direction recovered by a dictionary of size $m_2$ and $d_j^{(m_1)}$ for the directions recovered by a smaller dictionary of size $m_1$ , their finding is that $d_i^{(m_2)} \approx \sum_{j=1}^{m_1} g_j\, d_j^{(m_1)}, \qquad \lVert g \rVert_0 \ll m_1, \quad m_1 < m_2$. with g itself sparse. A direction that a large dictionary presents as one unit is recoverable as a sparse combination of a smaller dictionary’s units, which means the large dictionary’s “atoms” are not atomic. Put the incompleteness and the…

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