We analyze the orthogonal greedy algorithm when applied to dictionaries $\mathbb{D}$ whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $\mathbb{D}$ decays at a rate of $O(n^{-\frac{1}{2}-\alpha})$ for $\alpha > 0$, then the orthogonal greedy algorithm converges at the same rate on the variation space of $\mathbb{D}$. This improves upon the well-known $O(n^{-\frac{1}{2}})$ convergence rate of the orthogonal greedy algorithm in many cases, most notably for dictionaries corresponding to shallow neural networks. These results hold under no additional assumptions on the dictionary beyond the decay rate of the entropy of its convex hull. In addition, they are robust to noise in the target function and can be extended to convergence rates on the interpolation spaces of the variation norm. Finally, we show that these improved rates are sharp and prove a negative result showing that the iterates generated by the orthogonal greedy algorithm cannot in general be bounded in the variation norm of $\mathbb{D}$.

翻译：我们分析在应用到词典 $\ mathbb{D} $ 时的正方贪婪算法 $\ mathb{D} 美元, 其 convex 船体的软质质体有小的 entropy 。 我们显示,如果 $\\ mathb{D} 美元 的 convex 船体的锥体母体的 公吨酶酶性衰减率以 $(n ⁇ -\ frabb{D} $ 美元, 美元 则正方圆形贪婪算法以美元 $( mathbb{D} 美元 ) 的相同速率 。 我们显示, 在许多案例中, 最明显的是, 与浅色神经网络相对应的词典典, 其结果在字典上除了 其锥体船体的酶状体衰减率之外, 没有附加的假设。 此外, 它们对于目标功能中的噪声力非常强大, 并且可以扩展到 差异规范内部空间的趋同率的趋同率率。 最后, 我们显示, 这些腐变的调率无法通过一般的变制结果显示, 。