基于关联几何的词典学习
An Incidence Geometry approach to Dictionary Learning
摘要：
我们研究字典学习（又名稀疏编码）的问题，获得一个稀疏表示的数据点，通过学习\emph{dictionary vectors}上的数据点可以写为稀疏的线性组合。我们认为这个问题从几何的角度来看，作为一个子空间安排的生成集，并专注于了解的情况下，当指定的子空间安排的基础超图。对于这个拟合字典学习问题，我们完全描述了相关子空间排列的组合学（即\它们的基本hypergraphs）。具体地，证明了一类几何关联系统的组合刚性型定理。该定理刻画了子空间排列的超图一般产生（a）至少一个字典（b）局部唯一字典（即，至多有限数量的隔离字典）。我们不知道组合刚性技术在字典学习甚至机器学习中的应用。我们还提供了与字典学习相关的问题的系统分类以及各种算法，它们的假设和性能。
英文摘要：
We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a sparse representation of data points, by learning \emph{dictionary vectors} upon which the data points can be written as sparse linear combinations. We view this problem from a geometry perspective as the spanning set of a subspace arrangement, and focus on understanding the case when the underlying hypergraph of the subspace arrangement is specified. For this Fitted Dictionary Learning problem, we completely characterize the combinatorics of the associated subspace arrangements (i.e.\ their underlying hypergraphs). Specifically, a combinatorial rigidity-type theorem is proven for a type of geometric incidence system. The theorem characterizes the hypergraphs of subspace arrangements that generically yield (a) at least one dictionary (b) a locally unique dictionary (i.e.\ at most a finite number of isolated dictionaries) of the specified size. We are unaware of prior application of combinatorial rigidity techniques in the setting of Dictionary Learning, or even in machine learning. We also provide a systematic classification of problems related to Dictionary Learning together with various algorithms, their assumptions and performance.
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