22,556 research outputs found
The Average Sensitivity of an Intersection of Half Spaces
We prove new bounds on the average sensitivity of the indicator function of
an intersection of halfspaces. In particular, we prove the optimal bound of
. This generalizes a result of Nazarov, who proved the
analogous result in the Gaussian case, and improves upon a result of Harsha,
Klivans and Meka. Furthermore, our result has implications for the runtime
required to learn intersections of halfspaces
The Correct Exponent for the Gotsman-Linial Conjecture
We prove a new bound on the average sensitivity of polynomial threshold
functions. In particular we show that a polynomial threshold function of degree
in at most variables has average sensitivity at most
. For fixed the exponent
in terms of in this bound is known to be optimal. This bound makes
significant progress towards the Gotsman-Linial Conjecture which would put the
correct bound at
Learning Geometric Concepts with Nasty Noise
We study the efficient learnability of geometric concept classes -
specifically, low-degree polynomial threshold functions (PTFs) and
intersections of halfspaces - when a fraction of the data is adversarially
corrupted. We give the first polynomial-time PAC learning algorithms for these
concept classes with dimension-independent error guarantees in the presence of
nasty noise under the Gaussian distribution. In the nasty noise model, an
omniscient adversary can arbitrarily corrupt a small fraction of both the
unlabeled data points and their labels. This model generalizes well-studied
noise models, including the malicious noise model and the agnostic (adversarial
label noise) model. Prior to our work, the only concept class for which
efficient malicious learning algorithms were known was the class of
origin-centered halfspaces.
Specifically, our robust learning algorithm for low-degree PTFs succeeds
under a number of tame distributions -- including the Gaussian distribution
and, more generally, any log-concave distribution with (approximately) known
low-degree moments. For LTFs under the Gaussian distribution, we give a
polynomial-time algorithm that achieves error , where
is the noise rate. At the core of our PAC learning results is an efficient
algorithm to approximate the low-degree Chow-parameters of any bounded function
in the presence of nasty noise. To achieve this, we employ an iterative
spectral method for outlier detection and removal, inspired by recent work in
robust unsupervised learning. Our aforementioned algorithm succeeds for a range
of distributions satisfying mild concentration bounds and moment assumptions.
The correctness of our robust learning algorithm for intersections of
halfspaces makes essential use of a novel robust inverse independence lemma
that may be of broader interest
A new Edge Detector Based on Parametric Surface Model: Regression Surface Descriptor
In this paper we present a new methodology for edge detection in digital
images. The first originality of the proposed method is to consider image
content as a parametric surface. Then, an original parametric local model of
this surface representing image content is proposed. The few parameters
involved in the proposed model are shown to be very sensitive to
discontinuities in surface which correspond to edges in image content. This
naturally leads to the design of an efficient edge detector. Moreover, a
thorough analysis of the proposed model also allows us to explain how these
parameters can be used to obtain edge descriptors such as orientations and
curvatures.
In practice, the proposed methodology offers two main advantages. First, it
has high customization possibilities in order to be adjusted to a wide range of
different problems, from coarse to fine scale edge detection. Second, it is
very robust to blurring process and additive noise. Numerical results are
presented to emphasis these properties and to confirm efficiency of the
proposed method through a comparative study with other edge detectors.Comment: 21 pages, 13 figures and 2 table
Moment-Matching Polynomials
We give a new framework for proving the existence of low-degree, polynomial
approximators for Boolean functions with respect to broad classes of
non-product distributions. Our proofs use techniques related to the classical
moment problem and deviate significantly from known Fourier-based methods,
which require the underlying distribution to have some product structure.
Our main application is the first polynomial-time algorithm for agnostically
learning any function of a constant number of halfspaces with respect to any
log-concave distribution (for any constant accuracy parameter). This result was
not known even for the case of learning the intersection of two halfspaces
without noise. Additionally, we show that in the "smoothed-analysis" setting,
the above results hold with respect to distributions that have sub-exponential
tails, a property satisfied by many natural and well-studied distributions in
machine learning.
Given that our algorithms can be implemented using Support Vector Machines
(SVMs) with a polynomial kernel, these results give a rigorous theoretical
explanation as to why many kernel methods work so well in practice
On the use of simulated experiments in designing tests for material characterization from full-field measurements
The present paper deals with the use of simulated experiments to improve the design of an actual mechanical test. The analysis focused on the identification of the orthotropic properties of composites using the unnotched Iosipescu test and a full-field optical technique, the grid method. The experimental test was reproduced numerically by finite element analysis and the recording of deformed grey level images by a CCD camera was simulated trying to take into account the most significant parameters that can play a role during an actual test, e.g. the noise, the failure of the specimen, the size of the grid printed on the surface, etc. The grid method then was applied to the generated synthetic images in order to extract the displacement and strain fields and the Virtual Fields Method was finally used to identify the material properties and a cost function was devised to evaluate the error in the identification. The developed procedure was used to study different features of the test such as the aspect ratio and the fibre orientation of the specimen, the use of smoothing functions in the strain reconstruction from noisy data, the influence of missing data on the identification. Four different composite materials were considered and, for each of them, a set of optimized design variables was found by minimization of the cost function
The Phoenix Deep Survey: The 1.4 GHz microJansky catalogue
The initial Phoenix Deep Survey (PDS) observations with the Australia
Telescope Compact Array have been supplemented by additional 1.4 GHz
observations over the past few years. Here we present details of the
construction of a new mosaic image covering an area of 4.56 square degrees, an
investigation of the reliability of the source measurements, and the 1.4 GHz
source counts for the compiled radio catalogue. The mosaic achieves a 1-sigma
rms noise of 12 microJy at its most sensitive, and a homogeneous radio-selected
catalogue of over 2000 sources reaching flux densities as faint as 60 microJy
has been compiled. The source parameter measurements are found to be consistent
with the expected uncertainties from the image noise levels and the Gaussian
source fitting procedure. A radio-selected sample avoids the complications of
obscuration associated with optically-selected samples, and by utilising
complementary PDS observations including multicolour optical, near-infrared and
spectroscopic data, this radio catalogue will be used in a detailed
investigation of the evolution in star-formation spanning the redshift range 0
< z < 1. The homogeneity of the catalogue ensures a consistent picture of
galaxy evolution can be developed over the full cosmologically significant
redshift range of interest. The 1.4 GHz mosaic image and the source catalogue
are available on the web at http://www.atnf.csiro.au/~ahopkins/phoenix/ or from
the authors by request.Comment: 16 pages, 11 figures, 4 tables. Accepted for publication by A
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