Modern statistical problems often involve such linear inequality constraints on model parameters. Ignoring natural parameter constraints usually results in less efficient statistical procedures. To this end, we define a notion of `sparsity' for such restricted sets using lower-dimensional features. We allow our framework to be flexible so that the number of restrictions may be higher than the number of parameters. One such situation arise in estimation of monotone curve using a non parametric approach e.g. splines. We show that the proposed notion of sparsity agrees with the usual notion of sparsity in the unrestricted case and proves the validity of the proposed definition as a measure of sparsity. The proposed sparsity measure also allows us to generalize popular priors for sparse vector estimation to the constrained case.

翻译：现代统计问题常常涉及模型参数上的线性不等式约束。忽略自然的参数约束通常会导致统计方法的效率下降。为此，我们利用低维特征定义了这类受限集合中的“稀疏性”概念。我们的框架具有灵活性，允许约束数量多于参数数量。例如，在利用非参数方法（如样条）估计单调曲线时就会出现这种情况。我们证明，所提出的稀疏性概念在无约束情形下与通常的稀疏性定义一致，并验证了该定义作为稀疏性度量的有效性。此外，该稀疏性度量还使我们能够将流行的稀疏向量估计先验推广到带约束的情形。