Reducing Computational and Statistical Complexity in Machine Learning Through Cardinality Sparsity

High-dimensional data has become ubiquitous across the sciences but causes computational and statistical challenges. A common approach for dealing with these challenges is sparsity. In this paper, we introduce a new concept of sparsity, called cardinality sparsity. Broadly speaking, we call a tensor sparse if it contains only a small number of unique values. We show that cardinality sparsity can improve deep learning and tensor regression both statistically and computationally. On the way, we generalize recent statistical theories in those fields

Unfolding simple chains inside circles

It is an open problem to determined whether a polygonal chain can be straightened inside a confi ning region if its links are not allowed to cross. In this paper we propose a special case: whether a polygonal chain can be straightened inside a circle without allowing its links to cross. We prove that this is possible if the straightened confi guration can fi t within circle. Then we show that these simple chains have just one equivalence class of confi gurations