论文标题
扩展表征的表现力
Expressivity of expand-and-sparsify representations
论文作者
论文摘要
几种生物的感觉系统中出现了一种简单的稀疏编码机制：对于粗略的近似，输入$ x \ in \ r^d $被映射到更高的尺寸$ m \ gg d $，通过随机线性转换率，然后由赢家 - $ k $ z $ z $ z $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k k. \ {0,1 \}^m $。我们研究此表示的好处，以便随后的学习。 我们首先显示一个通用近似属性，只要$ m $足够大，$ z $的$ x $的任意连续函数均与$ z $的线性函数近似。这可以解释为说$ z $将信息拆开以$ x $，使其更容易访问。可以明确指定线性函数，并且易于学习，并且我们为输入尺寸$ d $的函数以及目标函数的平滑度所需的大小$ m $提供了界限。接下来，我们考虑表示表示是否适应输入空间中的多种结构。这高度依赖于稀疏的特定方法：我们表明，在获胜者接收机制下未获得适应性，而是在轻微的变体下保持。最后，我们考虑到随机的表示空间，但已与数据分布进行调整，并在此设置中给出了有利的近似范围。
A simple sparse coding mechanism appears in the sensory systems of several organisms: to a coarse approximation, an input $x \in \R^d$ is mapped to much higher dimension $m \gg d$ by a random linear transformation, and is then sparsified by a winner-take-all process in which only the positions of the top $k$ values are retained, yielding a $k$-sparse vector $z \in \{0,1\}^m$. We study the benefits of this representation for subsequent learning. We first show a universal approximation property, that arbitrary continuous functions of $x$ are well approximated by linear functions of $z$, provided $m$ is large enough. This can be interpreted as saying that $z$ unpacks the information in $x$ and makes it more readily accessible. The linear functions can be specified explicitly and are easy to learn, and we give bounds on how large $m$ needs to be as a function of the input dimension $d$ and the smoothness of the target function. Next, we consider whether the representation is adaptive to manifold structure in the input space. This is highly dependent on the specific method of sparsification: we show that adaptivity is not obtained under the winner-take-all mechanism, but does hold under a slight variant. Finally we consider mappings to the representation space that are random but are attuned to the data distribution, and we give favorable approximation bounds in this setting.
