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In many classification problems of interest, it is desirable to not only classify accurately but also to have access to the "raw data" that was used to do the classification. This naturally leads to the concept of joint classification and compression under system communication (or bandwidth) constraints. A typical system involves a complexity-constrained remote sensing unit and a central processing unit. In this paper, we will address the case of a single remote sensing unit (encoder) and a central processing unit (decoder) and a finite bit rate constraint to abstract the bandwidth-limited channel between the encoder and decoder. The goal is to spend this bit budget in the optimal sense, in terms of classification performance (minimize probability of classification error) as well as to enable reconstruction of the raw data with maximum fidelity (in the rate-distortion sense).