1,130 research outputs found
On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes
In this paper, we present an improved union bound on the Linear Programming
(LP) decoding performance of the binary linear codes transmitted over an
additive white Gaussian noise channels. The bounding technique is based on the
second-order of Bonferroni-type inequality in probability theory, and it is
minimized by Prim's minimum spanning tree algorithm. The bound calculation
needs the fundamental cone generators of a given parity-check matrix rather
than only their weight spectrum, but involves relatively low computational
complexity. It is targeted to high-density parity-check codes, where the number
of their generators is extremely large and these generators are spread densely
in the Euclidean space. We explore the generator density and make a comparison
between different parity-check matrix representations. That density effects on
the improvement of the proposed bound over the conventional LP union bound. The
paper also presents a complete pseudo-weight distribution of the fundamental
cone generators for the BCH[31,21,5] code
Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes
We introduce the notion of the stopping redundancy hierarchy of a linear
block code as a measure of the trade-off between performance and complexity of
iterative decoding for the binary erasure channel. We derive lower and upper
bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and
Bonferroni-type inequalities, and specialize them for codes with cyclic
parity-check matrices. Based on the observed properties of parity-check
matrices with good stopping redundancy characteristics, we develop a novel
decoding technique, termed automorphism group decoding, that combines iterative
message passing and permutation decoding. We also present bounds on the
smallest number of permutations of an automorphism group decoder needed to
correct any set of erasures up to a prescribed size. Simulation results
demonstrate that for a large number of algebraic codes, the performance of the
new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on
Information Theor
Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control
We introduce a framework for calibrating machine learning models so that
their predictions satisfy explicit, finite-sample statistical guarantees. Our
calibration algorithm works with any underlying model and (unknown)
data-generating distribution and does not require model refitting. The
framework addresses, among other examples, false discovery rate control in
multi-label classification, intersection-over-union control in instance
segmentation, and the simultaneous control of the type-1 error of outlier
detection and confidence set coverage in classification or regression. Our main
insight is to reframe the risk-control problem as multiple hypothesis testing,
enabling techniques and mathematical arguments different from those in the
previous literature. We use our framework to provide new calibration methods
for several core machine learning tasks with detailed worked examples in
computer vision and tabular medical data.Comment: Code available at https://github.com/aangelopoulos/lt
The correlation space of Gaussian latent tree models and model selection without fitting
We provide a complete description of possible covariance matrices consistent
with a Gaussian latent tree model for any tree. We then present techniques for
utilising these constraints to assess whether observed data is compatible with
that Gaussian latent tree model. Our method does not require us first to fit
such a tree. We demonstrate the usefulness of the inverse-Wishart distribution
for performing preliminary assessments of tree-compatibility using
semialgebraic constraints. Using results from Drton et al. (2008) we then
provide the appropriate moments required for test statistics for assessing
adherence to these equality constraints. These are shown to be effective even
for small sample sizes and can be easily adjusted to test either the entire
model or only certain macrostructures hypothesized within the tree. We
illustrate our exploratory tetrad analysis using a linguistic application and
our confirmatory tetrad analysis using a biological application.Comment: 15 page
Target Enumeration via Euler Characteristic Integrals
We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, landmarks) in a region using local counts performed by a network of sensors, each of which measures the number of targets nearby but neither their identities nor any positional information. We formulate and solve several such problems based on the types of sensors and mobility of the targets. The main contribution of this paper is the adaptation of a topological sheaf integration theory — integration with respect to Euler characteristic — to yield complete solutions to these problems
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