Abstracting Fairness: Oracles, Metrics, and Interpretability

It is well understood that classification algorithms, for example, for deciding on loan applications, cannot be evaluated for fairness without taking context into account. We examine what can be learned from a fairness oracle equipped with an underlying understanding of ``true'' fairness. The oracle takes as input a (context, classifier) pair satisfying an arbitrary fairness definition, and accepts or rejects the pair according to whether the classifier satisfies the underlying fairness truth. Our principal conceptual result is an extraction procedure that learns the underlying truth; moreover, the procedure can learn an approximation to this truth given access to a weak form of the oracle. Since every ``truly fair'' classifier induces a coarse metric, in which those receiving the same decision are at distance zero from one another and those receiving different decisions are at distance one, this extraction process provides the basis for ensuring a rough form of metric fairness, also known as individual fairness. Our principal technical result is a higher fidelity extractor under a mild technical constraint on the weak oracle's conception of fairness. Our framework permits the scenario in which many classifiers, with differing outcomes, may all be considered fair. Our results have implications for interpretablity -- a highly desired but poorly defined property of classification systems that endeavors to permit a human arbiter to reject classifiers deemed to be ``unfair'' or illegitimately derived.Comment: 17 pages, 1 figur

Author Correction: Task-dependent representations of stimulus and choice in mouse parietal cortex.

In the original version of this Article, the Acknowledgements section was inadvertently omitted. This has now been corrected in both the PDF and HTML versions of the Article

New mechanism of generation of large-scale magnetic field in a sheared turbulent plasma

A review of recent studies on a new mechanism of generation of large-scale magnetic field in a sheared turbulent plasma is presented. This mechanism is associated with the shear-current effect which is related to the W x J-term in the mean electromotive force. This effect causes the generation of the large-scale magnetic field even in a nonrotating and nonhelical homogeneous sheared turbulent convection whereby the alpha effect vanishes. It is found that turbulent convection promotes the shear-current dynamo instability, i.e., the heat flux causes positive contribution to the shear-current effect. However, there is no dynamo action due to the shear-current effect for small hydrodynamic and magnetic Reynolds numbers even in a turbulent convection, if the spatial scaling for the turbulent correlation time is k^{-2}, where k is the small-scale wave number. We discuss here also the nonlinear mean-field dynamo due to the shear-current effect and take into account the transport of magnetic helicity as a dynamical nonlinearity. The magnetic helicity flux strongly affects the magnetic field dynamics in the nonlinear stage of the dynamo action. When the magnetic helicity flux is not small, the saturated level of the mean magnetic field is of the order of the equipartition field determined by the turbulent kinetic energy. The obtained results are important for elucidation of origin of the large-scale magnetic fields in astrophysical and cosmic sheared turbulent plasma.Comment: 7 pages, Planetory and Space Science, in pres

Three-coordinate iron(II) expanded ring N-heterocyclic carbene complexes

A sterically demanding seven-membered expanded ring N-heterocyclic carbene (NHC) ligand allows access to rare examples of three-coordinate iron(II)-NHC complexes incorporating only halide coligands of the general formula [Fe(NHC)X 2 ] (NHC = 7-DiPP; X = Br (1) Cl (2)). Reducing the steric influence of the ancillary NHC ligand through modulation of the N-aryl substituents leads to either four- or three-coordinate complexes of the general formula [Fe(NHC)Br 2 (THF)] (3) or [Fe(NHC)Br 2 ] (4) (NHC = 7-Mes), dependent upon the solvent of recrystallization. The further reduction of NHC steric influence results in four-coordinate geometries at iron in the form of the dimeric species [Fe(NHC)Br(μ-Br)] 2 (5) or [Fe(NHC)Br 2 (THF)] (6) (NHC = SDiPP), again dependent upon the solvent of recrystallization. Compounds 1-6 have been analyzed by 1 H NMR spectroscopy, X-ray crystallography, elemental microanalysis, Mössbauer spectroscopy (for 1 and 3-5), and Evans method magnetic susceptibility. In addition to these measurements the three-coordinate species 1 and 4 have been further analyzed by SQUID magnetometry and CASSCF calculations, which show significant magnetic anisotropy that is extremely sensitive to the coordination geometry

Power-law corrections to entanglement entropy of horizons

We re-examine the idea that the origin of black-hole entropy may lie in the entanglement of quantum fields between inside and outside of the horizon. Motivated by the observation that certain modes of gravitational fluctuations in a black-hole background behave as scalar fields, we compute the entanglement entropy of such a field, by tracing over its degrees of freedom inside a sphere. We show that while this entropy is proportional to the area of the sphere when the field is in its ground state, a correction term proportional to a fractional power of area results when the field is in a superposition of ground and excited states. The area law is thus recovered for large areas. Further, we identify location of the degrees of freedom that give rise to the above entropy.Comment: 16 pages, 6 figures, to appear in Phys. Rev.

Growth rate of small-scale dynamo at low magnetic Prandtl numbers

In this study we discuss two key issues related to a small-scale dynamo instability at low magnetic Prandtl numbers and large magnetic Reynolds numbers, namely: (i) the scaling for the growth rate of small-scale dynamo instability in the vicinity of the dynamo threshold; (ii) the existence of the Golitsyn spectrum of magnetic fluctuations in small-scale dynamos. There are two different asymptotics for the small-scale dynamo growth rate: in the vicinity of the threshold of the excitation of the small-scale dynamo instability, $\lambda \propto \ln({\rm Rm}/ {\rm Rm}^{\rm cr})$, and when the magnetic Reynolds number is much larger than the threshold of the excitation of the small-scale dynamo instability, $\lambda \propto {\rm Rm}^{1/2}$, where ${\rm Rm}^{\rm cr}$ is the small-scale dynamo instability threshold in the magnetic Reynolds number ${\rm Rm}$. We demonstrated that the existence of the Golitsyn spectrum of magnetic fluctuations requires a finite correlation time of the random velocity field. On the other hand, the influence of the Golitsyn spectrum on the small-scale dynamo instability is minor. This is the reason why it is so difficult to observe this spectrum in direct numerical simulations for the small-scale dynamo with low magnetic Prandtl numbers.Comment: 14 pages, 1 figure, revised versio

Cross helicity and turbulent magnetic diffusivity in the solar convection zone

In a density-stratified turbulent medium the cross helicity is considered as a result of the interaction of the velocity fluctuations and a large-scale magnetic field. By means of a quasilinear theory and by numerical simulations we find the cross helicity and the mean vertical magnetic field anti-correlated. In the high-conductivity limit the ratio of the helicity and the mean magnetic field equals the ratio of the magnetic eddy diffusivity and the (known) density scale height. The result can be used to predict that the cross helicity at the solar surface exceeds the value of 1 Gauss km/s. Its sign is anti-correlated with that of the radial mean magnetic field. Alternatively, we can use our result to determine the value of the turbulent magnetic diffusivity from observations of the cross helicity.Comment: 9 pages, 2 figures, submitted to Solar Physic

Kinematic alpha effect in isotropic turbulence simulations

Using numerical simulations at moderate magnetic Reynolds numbers up to 220 it is shown that in the kinematic regime, isotropic helical turbulence leads to an alpha effect and a turbulent diffusivity whose values are independent of the magnetic Reynolds number, \Rm, provided \Rm exceeds unity. These turbulent coefficients are also consistent with expectations from the first order smoothing approximation. For small values of \Rm, alpha and turbulent diffusivity are proportional to \Rm. Over finite time intervals meaningful values of alpha and turbulent diffusivity can be obtained even when there is small-scale dynamo action that produces strong magnetic fluctuations. This suggests that small-scale dynamo-generated fields do not make a correlated contribution to the mean electromotive force.Comment: Accepted for publication in MNRAS Letter

Monge Distance between Quantum States

We define a metric in the space of quantum states taking the Monge distance between corresponding Husimi distributions (Q--functions). This quantity fulfills the axioms of a metric and satisfies the following semiclassical property: the distance between two coherent states is equal to the Euclidean distance between corresponding points in the classical phase space. We compute analytically distances between certain states (coherent, squeezed, Fock and thermal) and discuss a scheme for numerical computation of Monge distance for two arbitrary quantum states.Comment: 9 pages in LaTex - RevTex + 2 figures in ps. submitted to Phys. Rev.