Information-Theoretic Lower Bounds for Compressive Sensing with Generative Models
This provides fundamental limits for compressive sensing with generative models, which is important for researchers in signal processing and machine learning, though it is incremental as it builds directly on prior upper bounds.
The paper tackles the problem of determining the minimum number of measurements needed for compressive sensing when using generative models, establishing algorithm-independent lower bounds that show the scaling laws from prior work are optimal or near-optimal, with results including Ω(k log L) for Lipschitz models and Ω(kd log w / log n) for ReLU networks.
It has recently been shown that for compressive sensing, significantly fewer measurements may be required if the sparsity assumption is replaced by the assumption the unknown vector lies near the range of a suitably-chosen generative model. In particular, in (Bora {\em et al.}, 2017) it was shown roughly $O(k\log L)$ random Gaussian measurements suffice for accurate recovery when the generative model is an $L$-Lipschitz function with bounded $k$-dimensional inputs, and $O(kd \log w)$ measurements suffice when the generative model is a $k$-input ReLU network with depth $d$ and width $w$. In this paper, we establish corresponding algorithm-independent lower bounds on the sample complexity using tools from minimax statistical analysis. In accordance with the above upper bounds, our results are summarized as follows: (i) We construct an $L$-Lipschitz generative model capable of generating group-sparse signals, and show that the resulting necessary number of measurements is $Ω(k \log L)$; (ii) Using similar ideas, we construct ReLU networks with high depth and/or high depth for which the necessary number of measurements scales as $Ω\big( kd \frac{\log w}{\log n}\big)$ (with output dimension $n$), and in some cases $Ω(kd \log w)$. As a result, we establish that the scaling laws derived in (Bora {\em et al.}, 2017) are optimal or near-optimal in the absence of further assumptions.