Aggressive quantization of neural networks can expose activation functions as bottlenecks for efficient hardware acceleration. Implementing quantized activations as multi-threshold operations has been shown to be a promising optimization, allowing for efficient hardware implementation of few-bit activations. However, current implementations are limited to monotonically increasing functions, quantization schemes with one-dimensional (per-channel) granularity and a small set of predefined activation-quantizer combinations. We formalize the multi-threshold representation to extend it to non-monotonic functions and fine-granular parameters, i.e., multi-dimensional thresholds. For automated conversion of quantized function compositions into threshold representation, we propose a systematic approach via algebraic rules integrated into the ONNX-based compiler infrastructure. We experimentally validate our solution on a selection of monotonic and non-monotonic functions by extending the binary search based multi-threshold operator that is integrated into the FINN framework for end-to-end synthesis of custom-tailored FPGA accelerators. The cost of hardware support for non-monotonicity and fine-granularity is characterized in comparison to an alternative approach utilizing existing operators via graph-level decomposition. We compare our method to prior work as well as the approach taken by the hls4ml framework. As an outlook, we demonstrate the prospects of using the multi-threshold representation during the training of a neural network as a form of quantization aware training.

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