2,846 research outputs found
Equivalent Properties of CD Inequality on Graph
We study some equivalent properties of the curvature-dimension conditions
inequality on infinite, but locally finite graph. These equivalences
are gradient estimate, Poincar\'e type inequalities and reverse Poincar\'e
inequalities. And we also obtain one equivalent property of gradient estimate
for a new notion of curvature-dimension conditions at the
same assumption of graphs.Comment: 13 page
The first laws of thermodynamics of the (2+1)-dimensional BTZ black holes and Kerr-de Sitter spacetimes
We investigate the first law of thermodynamics in the case of the
(2+1)-dimensional BTZ black holes and Kerr-de Sitter spacetimes, in particular,
we focus on the integral mass formulas. It is found that by assuming the
cosmological constant as a variable state parameter, both the differential and
integral mass formulas of the first law of black hole thermodynamics in the
asymptotic flat spacetimes can be directly extended to those of rotating black
holes in anti-de Sitter and de Sitter backgrounds. It should be pointed that
these formulas come into existence in any dimensions also.Comment: 3 pages, no figure, revtex4, references added, to appear in CP
Ultracontractivity and functional inequalities on infinite graphs
In this paper, we prove the equivalent of ultracontractive bound of heat
semigroup or the uniform upper bound of the heat kernel with the Nash
inequality, Log-Sobolev inequalities on graphs. We also show that under the
assumption of volume growth and nonnegative curvature the Sobolev
inequality, Nash inequality, Faber-Krahn inequality, Log-Sobolev inequalities,
discrete and continuous-time uniform upper estimate of heat kernel are all true
on graph.Comment: 13 page
A gradient estimate for positive functions on graphs
We derive a gradient estimate for positive functions, in particular for
positive solutions to the heat equation, on finite or locally finite graphs.
Unlike the well known Li-Yau estimate, which is based on the maximum principle,
our estimate follows from the graph structure of the gradient form and the
Laplacian operator. Though our assumption on graphs is slightly stronger than
that of Bauer, Horn, Lin, Lippner, Mangoubi, and Yau (J. Differential Geom. 99
(2015) 359-405), our estimate can be easily applied to nonlinear differential
equations, as well as differential inequalities.
As applications, we estimate the greatest lower bound of Cheng's eigenvalue
and an upper bound of the minimal heat kernel, which is recently studied by
Bauer, Hua and Yau (Preprint, 2015) by the Li-Yau estimate. Moreover,
generalizing an earlier result of Lin and Yau (Math. Res. Lett. 17 (2010)
343-356), we derive a lower bound of nonzero eigenvalues by our gradient
estimate.Comment: 11 page
Global gradient estimate on graph and its applications
Continuing our previous work (arXiv:1509.07981v1), we derive another global
gradient estimate for positive functions, particularly for positive solutions
to the heat equation on finite or locally finite graphs. In general, the
gradient estimate in the present paper is independent of our previous one. As
applications, it can be used to get an upper bound and a lower bound of the
heat kernel on locally finite graphs. These global gradient estimates can be
compared with the Li-Yau inequality on graphs contributed by Bauer, Horn, Lin,
Lipper, Mangoubi and Yau (J. Differential Geom. 99 (2015) 359-409). In many
topics, such as eigenvalue estimate and heat kernel estimate (not including the
Liouville type theorems), replacing the Li-Yau inequality by the global
gradient estimate, we can get similar results.Comment: 7 page
The Second Order Linear Model
We study a fundamental class of regression models called the second order
linear model (SLM). The SLM extends the linear model to high order functional
space and has attracted considerable research interest recently. Yet how to
efficiently learn the SLM under full generality using nonconvex solver still
remains an open question due to several fundamental limitations of the
conventional gradient descent learning framework. In this study, we try to
attack this problem from a gradient-free approach which we call the
moment-estimation-sequence (MES) method. We show that the conventional gradient
descent heuristic is biased by the skewness of the distribution therefore is no
longer the best practice of learning the SLM. Based on the MES framework, we
design a nonconvex alternating iteration process to train a -dimension
rank- SLM within memory and one-pass of the dataset. The proposed
method converges globally and linearly, achieves recovery error
after retrieving samples.
Furthermore, our theoretical analysis reveals that not all SLMs can be learned
on every sub-gaussian distribution. When the instances are sampled from a
so-called -MIP distribution, the SLM can be learned by
samples where and are positive constants depending on the skewness
and kurtosis of the distribution. For non-MIP distribution, an addition
diagonal-free oracle is necessary and sufficient to guarantee the learnability
of the SLM. Numerical simulations verify the sharpness of our bounds on the
sampling complexity and the linear convergence rate of our algorithm
Nonconvex One-bit Single-label Multi-label Learning
We study an extreme scenario in multi-label learning where each training
instance is endowed with a single one-bit label out of multiple labels. We
formulate this problem as a non-trivial special case of one-bit rank-one matrix
sensing and develop an efficient non-convex algorithm based on alternating
power iteration. The proposed algorithm is able to recover the underlying
low-rank matrix model with linear convergence. For a rank- model with
features and classes, the proposed algorithm achieves
recovery error after retrieving one-bit labels
within memory. Our bound is nearly optimal in the order of
. This significantly improves the state-of-the-art sampling
complexity of one-bit multi-label learning. We perform experiments to verify
our theory and evaluate the performance of the proposed algorithm
Li-Yau inequality for unbounded Laplacian on graphs
In this paper, we derive Li-Yau inequality for unbounded Laplacian on
complete weighted graphs with the assumption of the curvature-dimension
inequality , which can be regarded as a notion of curvature on
graphs. Furthermore, we obtain some applications of Li-Yau inequality,
including Harnack inequality, heat kernel bounds and Cheng's eigenvalue
estimate. These are first kind of results on this direction for unbounded
Laplacian on graphs.Comment: 19 page
Volume doubling, Poincar\'e inequality and Guassian heat kernel estimate for nonnegative curvature graphs
By studying the heat semigroup, we prove Li-Yau type estimates for bounded
and positive solutions of the heat equation on graphs, under the assumption of
the curvature-dimension inequality , which can be consider as a
notion of curvature for graphs. Furthermore, we derive that if a graph has
non-negative curvature then it has the volume doubling property, from this we
can prove the Gaussian estimate for heat kernel, and then Poincar\'e inequality
and Harnack inequality. As a consequence, we obtain that the dimension of space
of harmonic functions on graphs with polynomial growth is finite, which
original is a conjecture of Yau on Riemannian manifold proved by Colding and
Minicozzi. Under the assumption of positive curvature on graphs, we derive the
Bonnet-Myers type theorem that the diameter of graphs is finite and bounded
above in terms of the positive curvature by proving some Log Sobolev
inequalities.Comment: 45 pages. arXiv admin note: text overlap with arXiv:0801.0812,
arXiv:0911.1819 by other author
Transkernel: Bridging Monolithic Kernels to Peripheral Cores
Smart devices see a large number of ephemeral tasks driven by background
activities. In order to execute such a task, the OS kernel wakes up the
platform beforehand and puts it back to sleep afterwards. In doing so, the
kernel operates various IO devices and orchestrates their power state
transitions. Such kernel executions are inefficient as they mismatch typical
CPU hardware. They are better off running on a low-power, microcontroller-like
core, i.e., peripheral core, relieving CPU from the inefficiency.
We therefore present a new OS structure, in which a lightweight virtual
executor called transkernel offloads specific phases from a monolithic kernel.
The transkernel translates stateful kernel execution through cross-ISA, dynamic
binary translation (DBT); it emulates a small set of stateless kernel services
behind a narrow, stable binary interface; it specializes for hot paths; it
exploits ISA similarities for lowering DBT cost.
Through an ARM-based prototype, we demonstrate transkernel's feasibility and
benefit. We show that while cross-ISA DBT is typically used under the
assumption of efficiency loss, it can enable efficiency gain, even on
off-the-shelf hardware.Comment: The camera-ready version of this paper, will appear at USENIX ATC'1
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