2,128 research outputs found
Online Tensor Methods for Learning Latent Variable Models
We introduce an online tensor decomposition based approach for two latent
variable modeling problems namely, (1) community detection, in which we learn
the latent communities that the social actors in social networks belong to, and
(2) topic modeling, in which we infer hidden topics of text articles. We
consider decomposition of moment tensors using stochastic gradient descent. We
conduct optimization of multilinear operations in SGD and avoid directly
forming the tensors, to save computational and storage costs. We present
optimized algorithm in two platforms. Our GPU-based implementation exploits the
parallelism of SIMD architectures to allow for maximum speed-up by a careful
optimization of storage and data transfer, whereas our CPU-based implementation
uses efficient sparse matrix computations and is suitable for large sparse
datasets. For the community detection problem, we demonstrate accuracy and
computational efficiency on Facebook, Yelp and DBLP datasets, and for the topic
modeling problem, we also demonstrate good performance on the New York Times
dataset. We compare our results to the state-of-the-art algorithms such as the
variational method, and report a gain of accuracy and a gain of several orders
of magnitude in the execution time.Comment: JMLR 201
Second-order Democratic Aggregation
Aggregated second-order features extracted from deep convolutional networks
have been shown to be effective for texture generation, fine-grained
recognition, material classification, and scene understanding. In this paper,
we study a class of orderless aggregation functions designed to minimize
interference or equalize contributions in the context of second-order features
and we show that they can be computed just as efficiently as their first-order
counterparts and they have favorable properties over aggregation by summation.
Another line of work has shown that matrix power normalization after
aggregation can significantly improve the generalization of second-order
representations. We show that matrix power normalization implicitly equalizes
contributions during aggregation thus establishing a connection between matrix
normalization techniques and prior work on minimizing interference. Based on
the analysis we present {\gamma}-democratic aggregators that interpolate
between sum ({\gamma}=1) and democratic pooling ({\gamma}=0) outperforming both
on several classification tasks. Moreover, unlike power normalization, the
{\gamma}-democratic aggregations can be computed in a low dimensional space by
sketching that allows the use of very high-dimensional second-order features.
This results in a state-of-the-art performance on several datasets
Differentiable Programming Tensor Networks
Differentiable programming is a fresh programming paradigm which composes
parameterized algorithmic components and trains them using automatic
differentiation (AD). The concept emerges from deep learning but is not only
limited to training neural networks. We present theory and practice of
programming tensor network algorithms in a fully differentiable way. By
formulating the tensor network algorithm as a computation graph, one can
compute higher order derivatives of the program accurately and efficiently
using AD. We present essential techniques to differentiate through the tensor
networks contractions, including stable AD for tensor decomposition and
efficient backpropagation through fixed point iterations. As a demonstration,
we compute the specific heat of the Ising model directly by taking the second
order derivative of the free energy obtained in the tensor renormalization
group calculation. Next, we perform gradient based variational optimization of
infinite projected entangled pair states for quantum antiferromagnetic
Heisenberg model and obtain start-of-the-art variational energy and
magnetization with moderate efforts. Differentiable programming removes
laborious human efforts in deriving and implementing analytical gradients for
tensor network programs, which opens the door to more innovations in tensor
network algorithms and applications.Comment: Typos corrected, discussion and refs added; revised version accepted
for publication in PRX. Source code available at
https://github.com/wangleiphy/tensorgra
A pseudospectral matrix method for time-dependent tensor fields on a spherical shell
We construct a pseudospectral method for the solution of time-dependent,
non-linear partial differential equations on a three-dimensional spherical
shell. The problem we address is the treatment of tensor fields on the sphere.
As a test case we consider the evolution of a single black hole in numerical
general relativity. A natural strategy would be the expansion in tensor
spherical harmonics in spherical coordinates. Instead, we consider the simpler
and potentially more efficient possibility of a double Fourier expansion on the
sphere for tensors in Cartesian coordinates. As usual for the double Fourier
method, we employ a filter to address time-step limitations and certain
stability issues. We find that a tensor filter based on spin-weighted spherical
harmonics is successful, while two simplified, non-spin-weighted filters do not
lead to stable evolutions. The derivatives and the filter are implemented by
matrix multiplication for efficiency. A key technical point is the construction
of a matrix multiplication method for the spin-weighted spherical harmonic
filter. As example for the efficient parallelization of the double Fourier,
spin-weighted filter method we discuss an implementation on a GPU, which
achieves a speed-up of up to a factor of 20 compared to a single core CPU
implementation.Comment: 33 pages, 9 figure
Status and Future Perspectives for Lattice Gauge Theory Calculations to the Exascale and Beyond
In this and a set of companion whitepapers, the USQCD Collaboration lays out
a program of science and computing for lattice gauge theory. These whitepapers
describe how calculation using lattice QCD (and other gauge theories) can aid
the interpretation of ongoing and upcoming experiments in particle and nuclear
physics, as well as inspire new ones.Comment: 44 pages. 1 of USQCD whitepapers
A Distributed and Incremental SVD Algorithm for Agglomerative Data Analysis on Large Networks
In this paper, we show that the SVD of a matrix can be constructed
efficiently in a hierarchical approach. Our algorithm is proven to recover the
singular values and left singular vectors if the rank of the input matrix
is known. Further, the hierarchical algorithm can be used to recover the
largest singular values and left singular vectors with bounded error. We also
show that the proposed method is stable with respect to roundoff errors or
corruption of the original matrix entries. Numerical experiments validate the
proposed algorithms and parallel cost analysis
- …