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• W2789121304 abstract "We consider the problem of representing Boolean functions exactly by linear combinations (over $mathbb{R}$) of functions from some simple class ${cal C}$. In particular, given ${cal C}$ we are interested in finding low-complexity functions lacking sparse representations. When ${cal C}$ is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer, the problem becomes interesting when ${cal C}$ is overcomplete and the set of functions is not linearly independent. We focus on the cases where ${cal C}$ is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts. We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for semi-explicit Boolean functions. For example, we show there are functions in nondeterministic quasi-polynomial time that require super-polynomial size: $bullet$ Depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds. $bullet$ Depth-two neural networks with ReLU activation function. $bullet$ $mathbb{R}$-linear combinations of $O(1)$-degree $mathbb{F}_p$-polynomials, for every prime $p$ (related to problems regarding Higher-Order Uncertainty Principles). We also obtain a function in $E^{NP}$ requiring $2^{Omega(n)}$ linear combinations. $bullet$ $mathbb{R}$-linear combinations of $ACC circ THR$ circuits of polynomial size (further generalizing the recent lower bounds of Murray and the author). (The above is a shortened abstract. For the full abstract, see the paper.)" @default.
• W2789121304 created "2018-03-06" @default.
• W2789121304 creator A5063871533 @default.
• W2789121304 date "2018-02-26" @default.
• W2789121304 modified "2023-09-25" @default.
• W2789121304 title "Limits on representing Boolean functions by linear combinations of simple functions: thresholds, ReLUs, and low-degree polynomials" @default.
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• W2789121304 hasPublicationYear "2018" @default.
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