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Abstract: Employing large intelligent surfaces (LISs) is a promising solution for improving the coverage and rate of future wireless systems. These surfaces comprise massive numbers of nearly-passive elements that interact with the incident signals, for example by reflecting them, in a smart way that improves the wireless system performance. Prior work focused on the design of the LIS reflection matrices assuming full channel knowledge. Estimating these channels at the LIS, however, is a key challenging problem. With the massive number of LIS elements, channel estimation or reflection beam training will be associated with (i) huge training overhead if all the LIS elements are passive (not connected to a baseband) or with (ii) prohibitive hardware complexity and power consumption if all the elements are connected to the baseband through a fully-digital or hybrid analog/digital architecture. This paper proposes efficient solutions for these problems by leveraging tools from compressive sensing and deep learning. First, a novel LIS architecture based on sparse channel sensors is proposed. In this architecture, all the LIS elements are passive except for a few elements that are active (connected to the baseband). We then develop two solutions that design the LIS reflection matrices with negligible training overhead. In the first approach, we leverage compressive sensing tools to construct the channels at all the LIS elements from the channels seen only at the active elements. In the second approach, we develop a deep-learning based solution where the LIS learns how to interact with the incident signal given the channels at the active elements, which represent the state of the environment and transmitter/receiver locations. We show that the achievable rates of the proposed solutions approach the upper bound, which assumes perfect channel knowledge, with negligible training overhead and with only a few active elements, making them promising for future LIS systems.