Effect Of Undersampling On Non-Negative Blind Deconvolution With Autoregressive Filters

This paper considers the problem of blind deconvolution where the input signal is non-negative and sparse, and the unknown convolutional kernel is a first order autoregressive filter. Our objective is to understand if it is possible to recover both the signal and the kernel from downsampled measurements of their convolution. This work is motivated by the problem of neural spike deconvolution from calcium imaging, where it is desirable to recover spikes at a higher rate from uniformly undersampled measurements. Assuming that the signals are generated according to a Bernoulli model, we show that it is possible to uniquely identify both the signal and the kernel with high probability using only O(s) measurements, where s is the expected sparsity. The key idea is to exploit non-negative constraints on the input signal as well as the parametric structure of the kernel.