The allocation of time to crime: A simple diagrammatical exposition

In his seminal article on the allocation of time to crime, Isaac Ehrlich (1973) derives five interesting theoretical results. He uses a state-preference diagram to derive one result, retreating to mathematics for deriving the remaining four results. This note shows that all five results can easily be derived from an alternative and simpler diagrammatical exposition that involves intersection of curves rather than tangency between curves.

Auditing ghosts by prosperity signals

Ghosts are economic agents who evade taxes by failing to file a return. Knowing nothing about them, the tax agency is unable to track them down through audit strategies which are based on reported income. The present paper develops a simple model of the audit decision for a ghost-busting tax agency which bases its audit strategy on signals of prosperous living, such as ownership of high-quality housing. Ghosts have a preference for high-quality housing, but may opt to own houses of a lower quality so as to escape detection. The paper compares the optimal audit rules and net tax collection under signal and blind auditing of the non-declaring population, deriving conditions under which each strategy will dominate the other.

The generalized Lasso with non-linear observations

We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that non-linear observations may be treated as noisy linear observations, and thus the signal may be estimated using the generalized Lasso. This is appealing because of the abundance of efficient, specialized solvers for this program. Just as noise may be diminished by projecting onto the lower dimensional space, the error from modeling non-linear observations with linear observations will be greatly reduced when using the signal structure in the reconstruction. We allow general signal structure, only assuming that the signal belongs to some set K in R^n. We consider the single-index model of non-linearity. Our theory allows the non-linearity to be discontinuous, not one-to-one and even unknown. We assume a random Gaussian model for the measurement matrix, but allow the rows to have an unknown covariance matrix. As special cases of our results, we recover near-optimal theory for noisy linear observations, and also give the first theoretical accuracy guarantee for 1-bit compressed sensing with unknown covariance matrix of the measurement vectors.Comment: 21 page

Effective Field Theory Amplitudes the On-Shell Way: Scalar and Vector Couplings to Gluons

We use on-shell methods to calculate tree-level effective field theory (EFT) amplitudes, with no reference to the EFT operators. Lorentz symmetry, unitarity and Bose statistics determine the allowed kinematical structures. As a by-product, the number of independent EFT operators simply follows from the set of polynomials in the Mandelstam invariants, subject to kinematical constraints. We demonstrate this approach by calculating several amplitudes with a massive, SM-singlet, scalar ($h$) or vector ($Z^\prime$) particle coupled to gluons. Specifically, we calculate $hggg$, $hhgg$ and $Z^\prime ggg$ amplitudes, which are relevant for the LHC production and three-gluon decays of the massive particle. We then use the results to derive the massless-$Z^\prime$ amplitudes, and show how the massive amplitudes decompose into the massless-vector plus scalar amplitudes. Amplitudes with the gluons replaced by photons are straightforwardly obtained from the above.Comment: 25 pages, 3 figures v2: references added, typos fixed and equation added in appendix, to appear in JHE

One-bit compressed sensing by linear programming

We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x in R^n from the signs of O(s log^2(n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our result is universal in the sense that with high probability, one measurement scheme will successfully recover all sparse vectors simultaneously. The argument is based on solving an equivalent geometric problem on random hyperplane tessellations.Comment: 15 pages, 1 figure, to appear in CPAM. Small changes based on referee comment