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Split computing has emerged as a recent paradigm for implementation of DNN-based AI workloads, wherein a DNN model is split into two parts, one of which is executed on a mobile/client device and the other on an edge-server (or cloud). Data compression is applied to the intermediate tensor from the DNN that needs to be transmitted, addressing the challenge of optimizing the rate-accuracy-complexity trade-off. Existing split-computing approaches adopt ML-based data compression, but require that the parameters of either the entire DNN model, or a significant portion of it, be retrained for different compression levels. This incurs a high computational and storage burden: training a full DNN model from scratch is computationally demanding, maintaining multiple copies of the DNN parameters increases storage requirements, and switching the full set of weights during inference increases memory bandwidth. In this paper, we present an approach that addresses all these challenges. It involves the systematic design and training of bottleneck units - simple, low-cost neural networks - that can be inserted at the point of split. Our approach is remarkably lightweight, both during training and inference, highly effective and achieves excellent rate-distortion performance at a small fraction of the compute and storage overhead compared to existing methods.