20,275 research outputs found
Learning to Convolve: A Generalized Weight-Tying Approach
Recent work (Cohen & Welling, 2016) has shown that generalizations of
convolutions, based on group theory, provide powerful inductive biases for
learning. In these generalizations, filters are not only translated but can
also be rotated, flipped, etc. However, coming up with exact models of how to
rotate a 3 x 3 filter on a square pixel-grid is difficult. In this paper, we
learn how to transform filters for use in the group convolution, focussing on
roto-translation. For this, we learn a filter basis and all rotated versions of
that filter basis. Filters are then encoded by a set of rotation invariant
coefficients. To rotate a filter, we switch the basis. We demonstrate we can
produce feature maps with low sensitivity to input rotations, while achieving
high performance on MNIST and CIFAR-10.Comment: Accepted to ICML 201
Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism invariant context
Osterwalder and Schrader introduced a procedure to obtain a (Lorentzian)
Hamiltonian quantum theory starting from a measure on the space of (Euclidean)
histories of a scalar quantum field. In this paper, we extend that construction
to more general theories which do not refer to any background, space-time
metric (and in which the space of histories does not admit a natural linear
structure). Examples include certain gauge theories, topological field theories
and relativistic gravitational theories. The treatment is self-contained in the
sense that an a priori knowledge of the Osterwalder-Schrader theorem is not
assumed.Comment: Plain Latex, 25 p., references added, abstract and title changed
(originally :``Osterwalder Schrader Reconstruction and Diffeomorphism
Invariance''), introduction extended, one appendix with illustrative model
added, accepted by Class. Quantum Gra
Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure
We describe a method for exactly diagonalizing clean -dimensional lattice
systems of independent fermions subject to arbitrary boundary conditions in one
direction, as well as systems composed of two bulks meeting at a planar
interface. Our method builds on the generalized Bloch theorem [A. Alase et al.,
Phys. Rev. B 96, 195133 (2017)] and the fact that the bulk-boundary separation
of the Schrodinger equation is compatible with a partial Fourier transform
operation. Bulk equations may display unusual features because they are
relative eigenvalue problems for non-Hermitian, bulk-projected Hamiltonians.
Nonetheless, they admit a rich symmetry analysis that can simplify considerably
the structure of energy eigenstates, often allowing a solution in fully
analytical form. We illustrate our extension of the generalized Bloch theorem
to multicomponent systems by determining the exact Andreev bound states for a
simple SNS junction. We then analyze the Creutz ladder model, by way of a
conceptual bridge from one to higher dimensions. Upon introducing a new
Gaussian duality transformation that maps the Creutz ladder to a system of two
Majorana chains, we show how the model provides a first example of a
short-range chiral topological insulator hosting topological zero modes with a
power-law profile. Additional applications include the complete analytical
diagonalization of graphene ribbons with both zigzag-bearded and armchair
boundary conditions, and the analytical determination of the edge modes in a
chiral two-dimensional topological superconductor. Lastly, we revisit
the phenomenon of Majorana flat bands and anomalous bulk-boundary
correspondence in a two-band gapless -wave topological superconductor. We
analyze the equilibrium Josephson response of the system, showing how the
presence of Majorana flat bands implies a substantial enhancement in the
-periodic supercurrent.Comment: 20+9 pages, 10 figure
Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma models
We present a detailed study of quantized noncompact, nonlinear SO(1,N)
sigma-models in arbitrary space-time dimensions D \geq 2, with the focus on
issues of spontaneous symmetry breaking of boost and rotation elements of the
symmetry group. The models are defined on a lattice both in terms of a transfer
matrix and by an appropriately gauge-fixed Euclidean functional integral. The
main results in all dimensions \geq 2 are: (i) On a finite lattice the systems
have infinitely many nonnormalizable ground states transforming irreducibly
under a nontrivial representation of SO(1,N); (ii) the SO(1,N) symmetry is
spontaneously broken. For D =2 this shows that the systems evade the
Mermin-Wagner theorem. In this case in addition: (iii) Ward identities for the
Noether currents are derived to verify numerically the absence of explicit
symmetry breaking; (iv) numerical results are presented for the two-point
functions of the spin field and the Noether current as well as a new order
parameter; (v) in a large N saddle-point analysis the dynamically generated
squared mass is found to be negative and of order 1/(V \ln V) in the volume,
the 0-component of the spin field diverges as \sqrt{\ln V}, while SO(1,N)
invariant quantities remain finite.Comment: 60 pages, 12 Figures, AMS-Latex; v2: results on vacuum orbit and
spontaneous symmetry breaking extended to all dimension
Thermal Quantum Fields without Cut-offs in 1+1 Space-time Dimensions
We construct interacting quantum fields in 1+1 dimensional Minkowski space,
representing neutral scalar bosons at positive temperature. Our work is based
on prior work by Klein and Landau and Hoegh-KrohnComment: 48 page
Sketching for Large-Scale Learning of Mixture Models
Learning parameters from voluminous data can be prohibitive in terms of
memory and computational requirements. We propose a "compressive learning"
framework where we estimate model parameters from a sketch of the training
data. This sketch is a collection of generalized moments of the underlying
probability distribution of the data. It can be computed in a single pass on
the training set, and is easily computable on streams or distributed datasets.
The proposed framework shares similarities with compressive sensing, which aims
at drastically reducing the dimension of high-dimensional signals while
preserving the ability to reconstruct them. To perform the estimation task, we
derive an iterative algorithm analogous to sparse reconstruction algorithms in
the context of linear inverse problems. We exemplify our framework with the
compressive estimation of a Gaussian Mixture Model (GMM), providing heuristics
on the choice of the sketching procedure and theoretical guarantees of
reconstruction. We experimentally show on synthetic data that the proposed
algorithm yields results comparable to the classical Expectation-Maximization
(EM) technique while requiring significantly less memory and fewer computations
when the number of database elements is large. We further demonstrate the
potential of the approach on real large-scale data (over 10 8 training samples)
for the task of model-based speaker verification. Finally, we draw some
connections between the proposed framework and approximate Hilbert space
embedding of probability distributions using random features. We show that the
proposed sketching operator can be seen as an innovative method to design
translation-invariant kernels adapted to the analysis of GMMs. We also use this
theoretical framework to derive information preservation guarantees, in the
spirit of infinite-dimensional compressive sensing
Discrete Euler-Poincar\'{e} and Lie-Poisson Equations
In this paper, discrete analogues of Euler-Poincar\'{e} and Lie-Poisson
reduction theory are developed for systems on finite dimensional Lie groups
with Lagrangians that are -invariant. These discrete
equations provide ``reduced'' numerical algorithms which manifestly preserve
the symplectic structure. The manifold is used as an approximation
of , and a discrete Langragian is
construced in such a way that the -invariance property is preserved.
Reduction by results in new ``variational'' principle for the reduced
Lagrangian , and provides the discrete
Euler-Poincar\'{e} (DEP) equations. Reconstruction of these equations recovers
the discrete Euler-Lagrange equations developed in \cite{MPS,WM} which are
naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP
algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is
shown that when , the DEP and DLP algorithms for a particular
choice of the discrete Lagrangian are equivalent to the
Moser-Veselov scheme for the generalized rigid body. %As an application, a
reduced symplectic integrator for two dimensional %hydrodynamics is constructed
using the SU approximation to the volume %preserving diffeomorphism group
of
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