We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of practical interest can frequently be approximated through "cheap learning" with exponentially fewer parameters than generic ones. We explore how properties frequently encountered in physics such as symmetry, locality, compositionality, and polynomial log-probability translate into exceptionally simple neural networks. We further argue that when the statistical process generating the data is of a certain hierarchical form prevalent in physics and machine-learning, a deep neural network can be more efficient than a shallow one. We formalize these claims using information theory and discuss the relation to the renormalization group. We prove various "no-flattening theorems" showing when efficient linear deep networks cannot be accurately approximated by shallow ones without efficiency loss; for example, we show that n variables cannot be multiplied using fewer than 2 n neurons in a single hidden layer.

We have measured the weak-lensing signal as a function of rest-frame B-, V-, and R-band luminosity for a sample of ''isolated'' galaxies. These results are based on four-band photometry from the Red-Sequence Cluster Survey, enabling us to determine photometric redshifts for a large number of galaxies. We select a secure sample of lenses with photometric redshifts 0:2 < z < 0:4 and study the relation between the virial mass and baryonic contents. In addition, we discuss the implications of the derived photometric redshift distribution for published cosmic shear studies. The virial masses are derived from a fit to the observed lensing signal. For a galaxy with a fiducial luminosity of 10 10 h À2 L B, , we obtain a mass M vir ¼ 9:9 þ1:5 À1:3 ; 10 11 M . The virial mass as a function of luminosity is consistent with a power law /L 1.5 , with similar slopes for the three filters considered here. These findings are in excellent agreement with results from the Sloan Digital Sky Survey and semianalytic models of galaxy formation. We measure the fraction of mass in stars and the baryon fraction in galaxies by comparing the virial mass-to-light ratio to predicted stellar mass-to-light ratios. We find that star formation is inefficient in converting baryons into stars, with late-type galaxies converting $33% and early-type galaxies converting only $14% of baryons into stars. Our results imply that the progenitors of early-type galaxies must have low stellar mass fractions, suggestive of a high formation redshift.

This paper presents the results of a photometric redshift study of galaxies in the Hubble Deep Field (HDF). The method of determining redshifts from broadband colors is described, and the dangers inherent in using it to estimate redshifts, particularly at very high z, are discussed. In particular, the need for accurate high-z spectral energy distributions is illustrated. The validity of our photometric redshift technique is demonstrated both by direct verification with available HDF spectroscopic data and by comparisons of luminosity functions and luminosity densities with those obtained from z < 1 spectroscopic redshift surveys. Evolution of the galaxy population is studied over 0 ∼ < z < 4. Brightening is seen in both the luminosity function and the luminosity density out to z ≈ 3; this is followed by a decline in both at z > 3. A population of z < 0.5 star-forming dwarfs is observed to M F450W AB = −11. Our results are discussed in the context of recent developments in the understanding of galaxy evolution.

We present first results on the cooling properties derived from Chandra X-ray observations of 83 high-redshift (0.3 < z < 1.2) massive galaxy clusters selected by their Sunyaev-Zel'dovich signature in the South Pole Telescope data. We measure each cluster's central cooling time, central entropy, and mass deposition rate, and compare these properties to those for local cluster samples. We find no significant evolution from z ∼ 0 to z ∼ 1 in the distribution of these properties, suggesting that cooling in cluster cores is stable over long periods of time. We also find that the average cool core entropy profile in the inner ∼100 kpc has not changed dramatically since z ∼ 1, implying that feedback must be providing nearly constant energy injection to maintain the observed "entropy floor" at ∼10 keV cm 2 . While the cooling properties appear roughly constant over long periods of time, we observe strong evolution in the gas density profile, with the normalized central density (ρ g,0 /ρ crit ) increasing by an order of magnitude from z ∼ 1 to z ∼ 0. When using metrics defined by the inner surface brightness profile of clusters, we find an apparent lack of classical, cuspy, cool-core clusters at z > 0.75, consistent with earlier reports for clusters at z > 0.5 using similar definitions. Our measurements indicate that cool cores have been steadily growing over the 8 Gyr spanned by our sample, consistent with a constant, ∼150 M yr −1 cooling flow that is unable to cool below entropies of 10 keV cm 2 and, instead, accumulates in the cluster center. We estimate that cool cores began to

We consider a nearly-AdS 2 gravity theory on the two-sided wormhole geometry. We construct three gauge-invariant operators in NAdS 2 which move bulk matter relative to the dynamical boundaries. In a two-sided system, these operators satisfy an SL(2) algebra (up to non perturbative corrections). In a semiclassical limit, these generators act like SL(2) transformations of the boundary time, or conformal symmetries of the two sided boundary theory. These can be used to define an operator-state mapping. A particular large N and low temperature limit of the SYK model has precisely the same structure, and this construction of the exact generators also applies. We also discuss approximate, but simpler, constructions of the generators in the SYK model. These are closely related to the "size" operator and are connected to the maximal chaos behavior captured by out of time order correlators.