8,554 research outputs found
OPML: A One-Pass Closed-Form Solution for Online Metric Learning
To achieve a low computational cost when performing online metric learning
for large-scale data, we present a one-pass closed-form solution namely OPML in
this paper. Typically, the proposed OPML first adopts a one-pass triplet
construction strategy, which aims to use only a very small number of triplets
to approximate the representation ability of whole original triplets obtained
by batch-manner methods. Then, OPML employs a closed-form solution to update
the metric for new coming samples, which leads to a low space (i.e., )
and time (i.e., ) complexity, where is the feature dimensionality.
In addition, an extension of OPML (namely COPML) is further proposed to enhance
the robustness when in real case the first several samples come from the same
class (i.e., cold start problem). In the experiments, we have systematically
evaluated our methods (OPML and COPML) on three typical tasks, including UCI
data classification, face verification, and abnormal event detection in videos,
which aims to fully evaluate the proposed methods on different sample number,
different feature dimensionalities and different feature extraction ways (i.e.,
hand-crafted and deeply-learned). The results show that OPML and COPML can
obtain the promising performance with a very low computational cost. Also, the
effectiveness of COPML under the cold start setting is experimentally verified.Comment: 12 page
Rigorous a-posteriori analysis using numerical eigenvalue bounds in a surface growth model
In order to prove numerically the global existence and uniqueness of smooth
solutions of a fourth order, nonlinear PDE, we derive rigorous a-posteriori
upper bounds on the supremum of the numerical range of the linearized operator.
These bounds also have to be easily computable in order to be applicable to our
rigorous a-posteriori methods, as we use them in each time-step of the
numerical discretization. The final goal is to establish global bounds on
smooth local solutions, which then establish global uniqueness.Comment: 19 pages, 9 figure
Robust Low-Rank Subspace Segmentation with Semidefinite Guarantees
Recently there is a line of research work proposing to employ Spectral
Clustering (SC) to segment (group){Throughout the paper, we use segmentation,
clustering, and grouping, and their verb forms, interchangeably.}
high-dimensional structural data such as those (approximately) lying on
subspaces {We follow {liu2010robust} and use the term "subspace" to denote both
linear subspaces and affine subspaces. There is a trivial conversion between
linear subspaces and affine subspaces as mentioned therein.} or low-dimensional
manifolds. By learning the affinity matrix in the form of sparse
reconstruction, techniques proposed in this vein often considerably boost the
performance in subspace settings where traditional SC can fail. Despite the
success, there are fundamental problems that have been left unsolved: the
spectrum property of the learned affinity matrix cannot be gauged in advance,
and there is often one ugly symmetrization step that post-processes the
affinity for SC input. Hence we advocate to enforce the symmetric positive
semidefinite constraint explicitly during learning (Low-Rank Representation
with Positive SemiDefinite constraint, or LRR-PSD), and show that factually it
can be solved in an exquisite scheme efficiently instead of general-purpose SDP
solvers that usually scale up poorly. We provide rigorous mathematical
derivations to show that, in its canonical form, LRR-PSD is equivalent to the
recently proposed Low-Rank Representation (LRR) scheme {liu2010robust}, and
hence offer theoretic and practical insights to both LRR-PSD and LRR, inviting
future research. As per the computational cost, our proposal is at most
comparable to that of LRR, if not less. We validate our theoretic analysis and
optimization scheme by experiments on both synthetic and real data sets.Comment: 10 pages, 4 figures. Accepted by ICDM Workshop on Optimization Based
Methods for Emerging Data Mining Problems (OEDM), 2010. Main proof simplified
and typos corrected. Experimental data slightly adde
Network Density of States
Spectral analysis connects graph structure to the eigenvalues and
eigenvectors of associated matrices. Much of spectral graph theory descends
directly from spectral geometry, the study of differentiable manifolds through
the spectra of associated differential operators. But the translation from
spectral geometry to spectral graph theory has largely focused on results
involving only a few extreme eigenvalues and their associated eigenvalues.
Unlike in geometry, the study of graphs through the overall distribution of
eigenvalues - the spectral density - is largely limited to simple random graph
models. The interior of the spectrum of real-world graphs remains largely
unexplored, difficult to compute and to interpret.
In this paper, we delve into the heart of spectral densities of real-world
graphs. We borrow tools developed in condensed matter physics, and add novel
adaptations to handle the spectral signatures of common graph motifs. The
resulting methods are highly efficient, as we illustrate by computing spectral
densities for graphs with over a billion edges on a single compute node. Beyond
providing visually compelling fingerprints of graphs, we show how the
estimation of spectral densities facilitates the computation of many common
centrality measures, and use spectral densities to estimate meaningful
information about graph structure that cannot be inferred from the extremal
eigenpairs alone.Comment: 10 pages, 7 figure
A stability criterion for high-frequency oscillations
We show that a simple Levi compatibility condition determines stability of
WKB solutions to semilinear hyperbolic initial-value problems issued from
highly-oscillating initial data with large amplitudes. The compatibility
condition involves the hyperbolic operator, the fundamental phase associated
with the initial oscillation, and the semilinear source term; it states roughly
that hyperbolicity is preserved around resonances.
If the compatibility condition is satisfied, the solutions are defined over
time intervals independent of the wavelength, and the associated WKB solutions
are stable under a large class of initial perturbations. If the compatibility
condition is not satisfied, resonances are exponentially amplified, and
arbitrarily small initial perturbations can destabilize the WKB solutions in
small time.
The amplification mechanism is based on the observation that in frequency
space, resonances correspond to points of weak hyperbolicity. At such points,
the behavior of the system depends on the lower order terms through the
compatibility condition.
The analysis relies, in the unstable case, on a short-time Duhamel
representation formula for solutions of zeroth-order pseudo-differential
equations.
Our examples include coupled Klein-Gordon systems, and systems describing
Raman and Brillouin instabilities.Comment: Final version, to appear in M\'em. Soc. Math. F
An inverse Sturm-Liouville problem with a fractional derivative
In this paper, we numerically investigate an inverse problem of recovering
the potential term in a fractional Sturm-Liouville problem from one spectrum.
The qualitative behaviors of the eigenvalues and eigenfunctions are discussed,
and numerical reconstructions of the potential with a Newton method from finite
spectral data are presented. Surprisingly, it allows very satisfactory
reconstructions for both smooth and discontinuous potentials, provided that the
order of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Computational Physic
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