26,694 research outputs found
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
Noise threshold for universality of 2-input gates
Evans and Pippenger showed in 1998 that noisy gates with 2 inputs are
universal for arbitrary computation (i.e. can compute any function with bounded
error), if all gates fail independently with probability epsilon and
epsilon<theta, where theta is roughly 8.856%.
We show that formulas built from gates with 2 inputs, in which each gate
fails with probability at least theta cannot be universal. Hence, there is a
threshold on the tolerable noise for formulas with 2-input gates and it is
theta. We conjecture that the same threshold also holds for circuits.Comment: International Symposium on Information Theory, 2007, minor
corrections in v
Prediction Scores as a Window into Classifier Behavior
Most multi-class classifiers make their prediction for a test sample by
scoring the classes and selecting the one with the highest score. Analyzing
these prediction scores is useful to understand the classifier behavior and to
assess its reliability. We present an interactive visualization that
facilitates per-class analysis of these scores. Our system, called Classilist,
enables relating these scores to the classification correctness and to the
underlying samples and their features. We illustrate how such analysis reveals
varying behavior of different classifiers. Classilist is available for use
online, along with source code, video tutorials, and plugins for R, RapidMiner,
and KNIME at https://katehara.github.io/classilist-site/.Comment: Presented at NIPS 2017 Symposium on Interpretable Machine Learnin
Second-order Quantile Methods for Experts and Combinatorial Games
We aim to design strategies for sequential decision making that adjust to the
difficulty of the learning problem. We study this question both in the setting
of prediction with expert advice, and for more general combinatorial decision
tasks. We are not satisfied with just guaranteeing minimax regret rates, but we
want our algorithms to perform significantly better on easy data. Two popular
ways to formalize such adaptivity are second-order regret bounds and quantile
bounds. The underlying notions of 'easy data', which may be paraphrased as "the
learning problem has small variance" and "multiple decisions are useful", are
synergetic. But even though there are sophisticated algorithms that exploit one
of the two, no existing algorithm is able to adapt to both.
In this paper we outline a new method for obtaining such adaptive algorithms,
based on a potential function that aggregates a range of learning rates (which
are essential tuning parameters). By choosing the right prior we construct
efficient algorithms and show that they reap both benefits by proving the first
bounds that are both second-order and incorporate quantiles
pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems
pde2path is a free and easy to use Matlab continuation/bifurcation package
for elliptic systems of PDEs with arbitrary many components, on general two
dimensional domains, and with rather general boundary conditions. The package
is based on the FEM of the Matlab pdetoolbox, and is explained by a number of
examples, including Bratu's problem, the Schnakenberg model, Rayleigh-Benard
convection, and von Karman plate equations. These serve as templates to study
new problems, for which the user has to provide, via Matlab function files, a
description of the geometry, the boundary conditions, the coefficients of the
PDE, and a rough initial guess of a solution. The basic algorithm is a one
parameter arclength continuation with optional bifurcation detection and
branch-switching. Stability calculations, error control and mesh-handling, and
some elementary time-integration for the associated parabolic problem are also
supported. The continuation, branch-switching, plotting etc are performed via
Matlab command-line function calls guided by the AUTO style. The software can
be downloaded from www.staff.uni-oldenburg.de/hannes.uecker/pde2path, where
also an online documentation of the software is provided such that in this
paper we focus more on the mathematics and the example systems
A Formal, Resource Consumption-Preserving Translation of Actors to Haskell
We present a formal translation of an actor-based language with cooperative
scheduling to the functional language Haskell. The translation is proven
correct with respect to a formal semantics of the source language and a
high-level operational semantics of the target, i.e. a subset of Haskell. The
main correctness theorem is expressed in terms of a simulation relation between
the operational semantics of actor programs and their translation. This allows
us to then prove that the resource consumption is preserved over this
translation, as we establish an equivalence of the cost of the original and
Haskell-translated execution traces.Comment: Pre-proceedings paper presented at the 26th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh,
Scotland UK, 6-8 September 2016 (arXiv:1608.02534
Minimum Description Length Induction, Bayesianism, and Kolmogorov Complexity
The relationship between the Bayesian approach and the minimum description
length approach is established. We sharpen and clarify the general modeling
principles MDL and MML, abstracted as the ideal MDL principle and defined from
Bayes's rule by means of Kolmogorov complexity. The basic condition under which
the ideal principle should be applied is encapsulated as the Fundamental
Inequality, which in broad terms states that the principle is valid when the
data are random, relative to every contemplated hypothesis and also these
hypotheses are random relative to the (universal) prior. Basically, the ideal
principle states that the prior probability associated with the hypothesis
should be given by the algorithmic universal probability, and the sum of the
log universal probability of the model plus the log of the probability of the
data given the model should be minimized. If we restrict the model class to the
finite sets then application of the ideal principle turns into Kolmogorov's
minimal sufficient statistic. In general we show that data compression is
almost always the best strategy, both in hypothesis identification and
prediction.Comment: 35 pages, Latex. Submitted IEEE Trans. Inform. Theor
Extrapolation-Based Implicit-Explicit Peer Methods with Optimised Stability Regions
In this paper we investigate a new class of implicit-explicit (IMEX) two-step
methods of Peer type for systems of ordinary differential equations with both
non-stiff and stiff parts included in the source term. An extrapolation
approach based on already computed stage values is applied to construct IMEX
methods with favourable stability properties. Optimised IMEX-Peer methods of
order p = 2, 3, 4, are given as result of a search algorithm carefully designed
to balance the size of the stability regions and the extrapolation errors.
Numerical experiments and a comparison to other implicit-explicit methods are
included.Comment: 21 pages, 6 figure
High accuracy semidefinite programming bounds for kissing numbers
The kissing number in n-dimensional Euclidean space is the maximal number of
non-overlapping unit spheres which simultaneously can touch a central unit
sphere. Bachoc and Vallentin developed a method to find upper bounds for the
kissing number based on semidefinite programming. This paper is a report on
high accuracy calculations of these upper bounds for n <= 24. The bound for n =
16 implies a conjecture of Conway and Sloane: There is no 16-dimensional
periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5
+ 61440q^6 + ...Comment: 7 pages (v3) new numerical result in Section 4, to appear in
Experiment. Mat
The Role of Monotonicity in the Epistemic Analysis of Strategic Games
It is well-known that in finite strategic games true common belief (or common
knowledge) of rationality implies that the players will choose only strategies
that survive the iterated elimination of strictly dominated strategies. We
establish a general theorem that deals with monotonic rationality notions and
arbitrary strategic games and allows to strengthen the above result to
arbitrary games, other rationality notions, and transfinite iterations of the
elimination process. We also clarify what conclusions one can draw for the
customary dominance notions that are not monotonic. The main tool is Tarski's
Fixpoint Theorem.Comment: 20 page
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