44,144 research outputs found
Incremental Sparse GP Regression for Continuous-time Trajectory Estimation & Mapping
Recent work on simultaneous trajectory estimation and mapping (STEAM) for
mobile robots has found success by representing the trajectory as a Gaussian
process. Gaussian processes can represent a continuous-time trajectory,
elegantly handle asynchronous and sparse measurements, and allow the robot to
query the trajectory to recover its estimated position at any time of interest.
A major drawback of this approach is that STEAM is formulated as a batch
estimation problem. In this paper we provide the critical extensions necessary
to transform the existing batch algorithm into an extremely efficient
incremental algorithm. In particular, we are able to vastly speed up the
solution time through efficient variable reordering and incremental sparse
updates, which we believe will greatly increase the practicality of Gaussian
process methods for robot mapping and localization. Finally, we demonstrate the
approach and its advantages on both synthetic and real datasets.Comment: 10 pages, 10 figure
Stochastic 2-microlocal analysis
A lot is known about the H\"older regularity of stochastic processes, in
particular in the case of Gaussian processes. Recently, a finer analysis of the
local regularity of functions, termed 2-microlocal analysis, has been
introduced in a deterministic frame: through the computation of the so-called
2-microlocal frontier, it allows in particular to predict the evolution of
regularity under the action of (pseudo-) differential operators. In this work,
we develop a 2-microlocal analysis for the study of certain stochastic
processes. We show that moments of the increments allow, under fairly general
conditions, to obtain almost sure lower bounds for the 2-microlocal frontier.
In the case of Gaussian processes, more precise results may be obtained: the
incremental covariance yields the almost sure value of the 2-microlocal
frontier. As an application, we obtain new and refined regularity properties of
fractional Brownian motion, multifractional Brownian motion, stochastic
generalized Weierstrass functions, Wiener and stable integrals.Comment: 35 page
Learning in the Wild with Incremental Skeptical Gaussian Processes
The ability to learn from human supervision is fundamental for personal
assistants and other interactive applications of AI. Two central challenges for
deploying interactive learners in the wild are the unreliable nature of the
supervision and the varying complexity of the prediction task. We address a
simple but representative setting, incremental classification in the wild,
where the supervision is noisy and the number of classes grows over time. In
order to tackle this task, we propose a redesign of skeptical learning centered
around Gaussian Processes (GPs). Skeptical learning is a recent interactive
strategy in which, if the machine is sufficiently confident that an example is
mislabeled, it asks the annotator to reconsider her feedback. In many cases,
this is often enough to obtain clean supervision. Our redesign, dubbed ISGP,
leverages the uncertainty estimates supplied by GPs to better allocate labeling
and contradiction queries, especially in the presence of noise. Our experiments
on synthetic and real-world data show that, as a result, while the original
formulation of skeptical learning produces over-confident models that can fail
completely in the wild, ISGP works well at varying levels of noise and as new
classes are observed.Comment: 7 pages, 3 figures, code:
https://gitlab.com/abonte/incremental-skeptical-g
Extremes of Independent Gaussian Processes
For every , let be independent copies of a
zero-mean Gaussian process . We describe all processes
which can be obtained as limits, as , of the process
, where and are
normalizing constants. We also provide an analogous characterization for the
limits of the process , where .Comment: 19 page
Mutual Information and Minimum Mean-square Error in Gaussian Channels
This paper deals with arbitrarily distributed finite-power input signals
observed through an additive Gaussian noise channel. It shows a new formula
that connects the input-output mutual information and the minimum mean-square
error (MMSE) achievable by optimal estimation of the input given the output.
That is, the derivative of the mutual information (nats) with respect to the
signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input
statistics. This relationship holds for both scalar and vector signals, as well
as for discrete-time and continuous-time noncausal MMSE estimation. This
fundamental information-theoretic result has an unexpected consequence in
continuous-time nonlinear estimation: For any input signal with finite power,
the causal filtering MMSE achieved at SNR is equal to the average value of the
noncausal smoothing MMSE achieved with a channel whose signal-to-noise ratio is
chosen uniformly distributed between 0 and SNR
A general approach to small deviation via concentration of measures
We provide a general approach to obtain upper bounds for small deviations in different norms, namely the supremum
and - H\"older norms. The large class of processes under
consideration takes the form , where and
are two possibly dependent stochastic processes. Our approach provides an upper
bound for small deviations whenever upper bounds for the \textit{concentration
of measures} of - norm of random vectors built from increments of the
process and \textit{large deviation} estimates for the process are
available. Using our method, among others, we obtain the optimal rates of small
deviations in supremum and - H\"older norms for fractional Brownian
motion with Hurst parameter . As an application, we discuss
the usefulness of our upper bounds for small deviations in pathwise stochastic
integral representation of random variables motivated by the hedging problem in
mathematical finance
- …