Active Sensing and Learning

Consider the problem of estimating a signal from noisy samples. The conventional approach (e.g., Shannon-Nyquist sampling) is to sample at many locations in a non-adaptive and more-or-less uniform manner. For example, in a digital camera we collect samples at locations on a square lattice (each sample being a pixel). Under certain scenarios though there is an extra flexibility, and an alternative and more versatile approach is possible: choose the sample locations ‘on-the-fly’, depending on the information collected up to that time. This is what we call adaptive sampling, as opposed to the conventional approach, referred to here as passive sampling. Although intuitively very appealing, adaptive sampling is seldom used because of the difficulty of design and analysis of such feedback techniques, especially in complex settings. The topic of adaptive sampling, or active learning as it is sometimes called, has attracted significant attention from various research communities, in particular in the fields of computer science and statistics. A large body of work exists proposing algorithmic ideas and methods [1, 2, 3, 4, 5], but unfortunately there are few performance guarantees for many of those methods. Further most of those results take place in very special or restricted scenarios (e.g., absence of noise or uncertainty, yielding perfect decisions). Under the adaptive sampling framework there are a few interesting theoretical results, some of which are presented here, namely the pioneering work of [6] regarding the estimation of step functions, that was later rediscovered in [7] using different algorithmic ideas and tools. Building on some of those ideas, the work in [8, 9, 10, 11] provides performance guarantees for function estimation under noisy conditions, for several function classes that are particularly relevant to signal processing and analysis. In this chapter we provide an introduction to adaptive sampling techniques for signal estimation, both in parametric and non-parametric settings. Note that the scenarios we consider do not have the Markovian structure inherent to the Markov Decision Processes (MDPs), that are the topic of many chapters in this book, and that the ‘actions’ (sample locations, or whether or not to collect a sample) do not affect the environment. Another major difference between the active learning problems considered and the MDPs is that, in the former, the set of possible actions/sample locations is generally uncountable.