The recently introduced theory of Compressive Sensing (CS) enables a new method for
signal recovery from incomplete information (a reduced set of “compressive” linear measurements), based on the assumption that the
signal is sparse in some dictionary. Such compressive measurement schemes are desirable in practice for reducing the costs of
signal acquisition, storage, and
processing. However, the current CS framework considers only a certain task (
signal recovery) and only in a certain model setting (sparsity).We show that compressive measurements are in fact information scalable, allowing one to answer a
broad spectrum of questions about a signal when provided only with a reduced set of compressive measurements. These questions range from complete
signal recovery at one extreme down to a simple binary detection decision at the other. (Questions in between include, for example,
estimation and classification.) We provide techniques such as a “compressive
matched filter” for answering several of these questions given the available measurements, often without needing to first reconstruct the signal. In many cases, these techniques can succeed with far fewer measurements than would be required for full
signal recovery, and such techniques can also be computationally more efficient. Based on additional mathematical insight, we discuss information scalable algorithms in several model settings, including sparsity (as in CS), but also in parametric or manifold-based settings and in model-free settings for generic statements of detection, classification, and
estimation problems.