Circuit Complexity of Visual Search
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We study computational hardness of feature and conjunction search through the lens of circuit complexity. Let x = (x_1, ... , x_n) (resp., y = (y_1, ... , y_n)) be Boolean variables each of which takes the value one if and only if a neuron at place i detects a feature (resp., another feature). We then simply formulate the feature and conjunction search as Boolean functions FTR_n(x) = ⋁_i=1^n x_i and CONJ_n(x, y) = ⋁_i=1^n x_i ∧ y_i, respectively. We employ a threshold circuit or a discretized circuit (such as a sigmoid circuit or a ReLU circuit with discretization) as our models of neural networks, and consider the following four computational resources: [i] the number of neurons (size), [ii] the number of levels (depth), [iii] the number of active neurons outputting non-zero values (energy), and [iv] synaptic weight resolution (weight). We first prove that any threshold circuit C of size s, depth d, energy e and weight w satisfies log rk(M_C) ≤ ed (log s + log w + log n), where rk(M_C) is the rank of the communication matrix M_C of a 2n-variable Boolean function that C computes. Since CONJ_n has rank 2^n, we have n ≤ ed (log s + log w + log n). Thus, an exponential lower bound on the size of even sublinear-depth threshold circuits exists if the energy and weight are sufficiently small. Since FTR_n is computable independently of n, our result suggests that computational capacity for the feature and conjunction search are different. We also show that the inequality is tight up to a constant factor if ed = o(n/ log n). We next show that a similar inequality holds for any discretized circuit. Thus, if we regard the number of gates outputting non-zero values as a measure for sparse activity, our results suggest that larger depth helps neural networks to acquire sparse activity.
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