English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Given an overcomplete dictionary $A$ and a signal $b = Ac^*$ for some sparsevector $c^*$ whose nonzero entries correspond to linearly independent columnsof $A$, classical sparse signal recovery theory considers the problem ofwhether $c^*$ can be recovered as the unique sparsest solution to $b = A c$. Itis now well-understood that such recovery is possible by practical algorithmswhen the dictionary $A$ is incoherent or restricted isometric. In this paper,we consider the more general case where $b$ lies in a subspace $\mathcal{S}_0$spanned by a subset of linearly dependent columns of $A$, and the remainingcolumns are outside of the subspace. In this case, the sparsest representationmay not be unique, and the dictionary may not be incoherent or restrictedisometric. The goal is to have the representation $c$ correctly identify thesubspace, i.e. the nonzero entries of $c$ should correspond to columns of $A$that are in the subspace $\mathcal{S}_0$. Such a representation $c$ is calledsubspace-preserving, a key concept that has found important applications forlearning low-dimensional structures in high-dimensional data. We presentvarious geometric conditions that guarantee subspace-preserving recovery. Amongthem, the major results are characterized by the covering radius and theangular distance, which capture the distribution of points in the subspace andthe similarity between points in the subspace and points outside the subspace,respectively. Importantly, these conditions do not require the dictionary to beincoherent or restricted isometric. By establishing that thesubspace-preserving recovery problem and the classical sparse signal recoveryproblem are equivalent under common assumptions on the latter, we show thatseveral of our proposed conditions are generalizations of some well-knownconditions in the sparse signal recovery literature.

Basis Pursuit and Orthogonal Matching Pursuit for Subspace-preserving Recovery: Theoretical Analysis