In this effort we propose a new deep architecture utilizing residual blocks inspired by implicit discretization schemes. As opposed to the standard feed-forward networks, the outputs of the proposed implicit residual blocks are defined as the fixed points of the appropriately chosen nonlinear transformations. We show that this choice leads to improved stability of both forward and backward propagations, has a favorable impact on the generalization power of the network and allows for higher learning rates. In addition, we consider a reformulation of ResNet which does not introduce new parameters and can potentially lead to a reduction in the number of required layers due to improved forward stability and robustness. Finally, we derive the memory efficient reversible training algorithm and provide numerical results in support of our findings.
The authors of @cite_13 concentrated on the well-posedness of the learning problem for ODE-constrained control and emphasized the importance of stability in the design of deep architectures. For instance, the solution of a homogeneous linear ODE with constant coefficients is given by where @math is the eigen-decomposition of a matrix @math , and @math is the diagonal matrix with corresponding eigenvalues. Similar equation holds for the backpropagation of gradients. To guarantee the efficient propagation of information through the network, one must ensure that the elements of @math have magnitudes close to one. This condition, of course, is satisfied when all eigenvalues of the matrix @math are imaginary with real parts close to zero. In order to preserve this property, the authors of @cite_13 proposed several time continuous architectures of the form When @math , @math , the equations above provide an example of a conservative Hamiltonian system with the total energy @math .
@cite_13
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We consider a discrete elliptic equation with random coefficients @math , which (to fix ideas) are identically distributed and independent from grid point to grid point @math . On scales large w. r. t. the grid size (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients @math are characterized by @math where the random field @math is the unique stationary solution of the ''corrector problem'' @math and @math denotes the ensemble average. It is known (''by ergodicity'') that the above ensemble average of the energy density @math , which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of @math on length scales @math is estimated as follows: @math where the averaging function (i. e. @math , @math ) has to be smooth in the sense that @math . In two space dimensions (i. e. @math ), there is a logarithmic correction. In other words, smooth averages of the energy density @math behave like as if @math would be independent from grid point to grid point (which it is not for @math ). This result is of practical significance, since it allows to estimate the error when numerically computing @math . We continue the study of the Hermitian random matrix ensemble with external source Open image in new window where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a=1, since the eigenvalues of M accumulate on two intervals for a>1, and on one interval for 0 1 was treated in Part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0<a<1. As in Part I we apply the Deift Zhou steepest descent analysis to a 3×3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems. We consider the problem of robustly recovering a @math -sparse coefficient vector from the Fourier series that it generates, restricted to the interval @math . The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of @math ) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm @math , we show upper and lower bounds on the minimax error rate that both scale like @math , providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity @math , or equivalently from comparing @math to the so-called @math -spark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis. We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy @math in the localized phase. Assume the density of states function is not too flat near @math . Restrict it to some large cube @math . Consider now @math , a small energy interval centered at @math that asymptotically contains infintely many eigenvalues when the volume of the cube @math grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive * uniform Poisson behavior of the locally unfolded eigenvalues, * a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales. * the distribution of the unfolded level spacings, locally and globally, * the distribution of the unfolded localization centers, locally and globally.