424 research outputs found
Magnetic miniband and magnetotransport property of a graphene superlattice
The eigen energy and the conductivity of a graphene sheet subject to a
one-dimensional cosinusoidal potential and in the presence of a magnetic field
are calculated. Such a graphene superlattice presents three distinct magnetic
miniband structures as the magnetic field increases. They are, respectively,
the triply degenerate Landau level spectrum, the nondegenerate minibands with
finite dispersion and the same Landau level spectrum with the pristine
graphene. The ratio of the magnetic length to the period of the potential
function is the characteristic quantity to determine the electronic structure
of the superlattice. Corresponding to these distinct electronic structures, the
diagonal conductivity presents very strong anisotropy in the weak and moderate
magnetic field cases. But the predominant magnetotransport orientation changes
from the transverse to the longitudinal direction of the superlattice. More
interestingly, in the weak magnetic field case, the superlattice exhibits
half-integer quantum Hall effect, but with large jump between the Hall
plateaux. Thus it is different from the one of the pristine graphene.Comment: 7 pages, 5 figure
Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations
Low rank matrix approximation is an important tool in machine learning. Given
a data matrix, low rank approximation helps to find factors, patterns and
provides concise representations for the data. Research on low rank
approximation usually focus on real matrices. However, in many applications
data are binary (categorical) rather than continuous. This leads to the problem
of low rank approximation of binary matrix. Here we are given a
binary matrix and a small integer . The goal is to find two binary
matrices and of sizes and respectively, so
that the Frobenius norm of is minimized. There are two models of this
problem, depending on the definition of the dot product of binary vectors: The
model and the Boolean semiring model. Unlike low rank
approximation of real matrix which can be efficiently solved by Singular Value
Decomposition, approximation of binary matrix is -hard even for .
In this paper, we consider the problem of Column Subset Selection (CSS), in
which one low rank matrix must be formed by columns of the data matrix. We
characterize the approximation ratio of CSS for binary matrices. For
model, we show the approximation ratio of CSS is bounded by
and this bound is asymptotically tight. For
Boolean model, it turns out that CSS is no longer sufficient to obtain a bound.
We then develop a Generalized CSS (GCSS) procedure in which the columns of one
low rank matrix are generated from Boolean formulas operating bitwise on
columns of the data matrix. We show the approximation ratio of GCSS is bounded
by , and the exponential dependency on is inherent.Comment: 38 page
Algorithmic Regularization in Model-free Overparametrized Asymmetric Matrix Factorization
We study the asymmetric matrix factorization problem under a natural
nonconvex formulation with arbitrary overparametrization. The model-free
setting is considered, with minimal assumption on the rank or singular values
of the observed matrix, where the global optima provably overfit. We show that
vanilla gradient descent with small random initialization sequentially recovers
the principal components of the observed matrix. Consequently, when equipped
with proper early stopping, gradient descent produces the best low-rank
approximation of the observed matrix without explicit regularization. We
provide a sharp characterization of the relationship between the approximation
error, iteration complexity, initialization size and stepsize. Our complexity
bound is almost dimension-free and depends logarithmically on the approximation
error, with significantly more lenient requirements on the stepsize and
initialization compared to prior work. Our theoretical results provide accurate
prediction for the behavior gradient descent, showing good agreement with
numerical experiments.Comment: 30 pages, 7 figure
Rethinking Lipschitz Neural Networks and Certified Robustness: A Boolean Function Perspective
Designing neural networks with bounded Lipschitz constant is a promising way
to obtain certifiably robust classifiers against adversarial examples. However,
the relevant progress for the important perturbation setting is
rather limited, and a principled understanding of how to design expressive
Lipschitz networks is still lacking. In this paper, we bridge the
gap by studying certified robustness from a novel perspective of
representing Boolean functions. We derive two fundamental impossibility results
that hold for any standard Lipschitz network: one for robust classification on
finite datasets, and the other for Lipschitz function approximation. These
results identify that networks built upon norm-bounded affine layers and
Lipschitz activations intrinsically lose expressive power even in the
two-dimensional case, and shed light on how recently proposed Lipschitz
networks (e.g., GroupSort and -distance nets) bypass these
impossibilities by leveraging order statistic functions. Finally, based on
these insights, we develop a unified Lipschitz network that generalizes prior
works, and design a practical version that can be efficiently trained (making
certified robust training free). Extensive experiments show that our approach
is scalable, efficient, and consistently yields better certified robustness
across multiple datasets and perturbation radii than prior Lipschitz networks.
Our code is available at https://github.com/zbh2047/SortNet.Comment: 37 pages; to appear in NeurIPS 2022 (Oral
Content-based Controls For Music Large Language Modeling
Recent years have witnessed a rapid growth of large-scale language models in
the domain of music audio. Such models enable end-to-end generation of
higher-quality music, and some allow conditioned generation using text
descriptions. However, the control power of text controls on music is
intrinsically limited, as they can only describe music indirectly through
meta-data (such as singers and instruments) or high-level representations (such
as genre and emotion). We aim to further equip the models with direct and
content-based controls on innate music languages such as pitch, chords and drum
track. To this end, we contribute Coco-Mulla, a content-based control method
for music large language modeling. It uses a parameter-efficient fine-tuning
(PEFT) method tailored for Transformer-based audio models. Experiments show
that our approach achieved high-quality music generation with low-resource
semi-supervised learning, tuning with less than 4% parameters compared to the
original model and training on a small dataset with fewer than 300 songs.
Moreover, our approach enables effective content-based controls, and we
illustrate the control power via chords and rhythms, two of the most salient
features of music audio. Furthermore, we show that by combining content-based
controls and text descriptions, our system achieves flexible music variation
generation and style transfer. Our source codes and demos are available online
Nonlinear dynamics of full-range CNNs with time-varying delays and variable coefficients
In the article, the dynamical behaviours of the full-range cellular neural networks (FRCNNs) with variable coefficients and time-varying delays are considered. Firstly, the improved model of the FRCNNs is proposed, and the existence and uniqueness of the solution are studied by means of differential inclusions and set-valued analysis. Secondly, by using the Hardy inequality, the matrix analysis, and the Lyapunov functional method, we get some criteria for achieving the globally exponential stability (GES). Finally, some examples are provided to verify the correctness of the theoretical results
A Validation Approach to Over-parameterized Matrix and Image Recovery
In this paper, we study the problem of recovering a low-rank matrix from a
number of noisy random linear measurements. We consider the setting where the
rank of the ground-truth matrix is unknown a prior and use an overspecified
factored representation of the matrix variable, where the global optimal
solutions overfit and do not correspond to the underlying ground-truth. We then
solve the associated nonconvex problem using gradient descent with small random
initialization. We show that as long as the measurement operators satisfy the
restricted isometry property (RIP) with its rank parameter scaling with the
rank of ground-truth matrix rather than scaling with the overspecified matrix
variable, gradient descent iterations are on a particular trajectory towards
the ground-truth matrix and achieve nearly information-theoretically optimal
recovery when stop appropriately. We then propose an efficient early stopping
strategy based on the common hold-out method and show that it detects nearly
optimal estimator provably. Moreover, experiments show that the proposed
validation approach can also be efficiently used for image restoration with
deep image prior which over-parameterizes an image with a deep network.Comment: 29 pages and 9 figure
A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equations
We propose a model-data asymptotic-preserving neural network(MD-APNN) method
to solve the nonlinear gray radiative transfer equations(GRTEs). The system is
challenging to be simulated with both the traditional numerical schemes and the
vanilla physics-informed neural networks(PINNs) due to the multiscale
characteristics. Under the framework of PINNs, we employ a micro-macro
decomposition technique to construct a new asymptotic-preserving(AP) loss
function, which includes the residual of the governing equations in the
micro-macro coupled form, the initial and boundary conditions with additional
diffusion limit information, the conservation laws, and a few labeled data. A
convergence analysis is performed for the proposed method, and a number of
numerical examples are presented to illustrate the efficiency of MD-APNNs, and
particularly, the importance of the AP property in the neural networks for the
diffusion dominating problems. The numerical results indicate that MD-APNNs
lead to a better performance than APNNs or pure data-driven networks in the
simulation of the nonlinear non-stationary GRTEs
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