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Register a new userBenchmarking Compressed Sensing, Super-Resolution, and Filter Diagonalization
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Signal processing techniques have been developed that use different strategies to bypass the Nyquist sampling theorem in order to recover more information than a traditional discrete Fourier transform. Here we examine three such methods: filter diagonalization, compressed sensing, and super-resolution. We apply them to a broad range of signal forms commonly found in science and engineering in order to discover when and how each method can be used most profitably. We find that filter diagonalization provides the best results for Lorentzian signals, while compressed sensing and super-resolution perform better for arbitrary signals.
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The 1-bit compressed sensing framework enables the recovery of a sparse vector x from the sign information of each entry of its linear transformation. Discarding the amplitude information can significantly reduce the amount of data, which is highly beneficial in practical applications. In this paper, we present a Bayesian approach to signal reconstruction for 1-bit compressed sensing, and analyze its typical performance using statistical mechanics. Utilizing the replica method, we show that the Bayesian approach enables better reconstruction than the L1-norm minimization approach, asymptotically saturating the performance obtained when the non-zero entries positions of the signal are known. We also test a message passing algorithm for signal reconstruction on the basis of belief propagation. The results of numerical experiments are consistent with those of the theoretical analysis.
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Despite of their success, the results of first-principles quantum mechanical calculations contain inherent numerical errors caused by various approximations. We propose here a neural-network algorithm to greatly reduce these inherent errors. As a demonstration, this combined quantum mechanical calculation and neural-network correction approach is applied to the evaluation of standard heat of formation $DelH$ and standard Gibbs energy of formation $DelG$ for 180 organic molecules at 298 K. A dramatic reduction of numerical errors is clearly shown with systematic deviations being eliminated. For examples, the root--mean--square deviation of the calculated $DelH$ ($DelG$) for the 180 molecules is reduced from 21.4 (22.3) kcal$cdotp$mol$^{-1}$ to 3.1 (3.3) kcal$cdotp$mol$^{-1}$ for B3LYP/6-311+G({it d,p}) and from 12.0 (12.9) kcal$cdotp$mol$^{-1}$ to 3.3 (3.4) kcal$cdotp$mol$^{-1}$ for B3LYP/6-311+G(3{it df},2{it p}) before and after the neural-network correction.
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