machine learning with signal processing [0em] part i: signal...
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Machine Learning with Signal ProcessingPart I: Signal Processing Tooling
Arno SolinAssistant Professor in Machine Learning
Department of Computer ScienceAalto University
ICML 2020 TUTORIAL
7@arnosolin B arno.solin.fi
Machine learning with signal processing: Part IArno Solin
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CC-NC: https://xkcd.com/1524/
Machine learning with signal processing: Part IArno Solin
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Scope of this tutorial
This tutorial is about:I (Mostly) temporal modelsI Various tools in signal
processingI Pointers to application
areasI Links to aspects in
statistical ML
This tutorial is not about:I Audio processingI Image processingI Encyclopedic overview
Machine learning with signal processing: Part IArno Solin
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Goals
I Teach basic principles of direct links between signalprocessing and machine learning
I Provide an intuitive hands-on understanding of whatstochastic differential equations are all about
I Show how these methods have real benefits in speedingup learning, improving inference, and model building
Machine learning with signal processing: Part IArno Solin
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Disclaimer
I I do not have time to discuss many important andrelevant works
I If you think I should have included some of those, pleasesend me email and I will try to include it the next time
I The content of the tutorial is based on my own biasedopinion (and expertise)
I Many examples are based on my own work
Machine learning with signal processing: Part IArno Solin
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Structure
Part I
Tools anddiscrete-time
models
Part II
SDEs(continuous-time
models)
Part III
Gaussianprocesses
�Part IV
Applicationexamples
Machine learning with signal processing: Part IArno Solin
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Outline
Temporal models
Tools for workingwith time-series
Spectral methods
State spacemodels
Filtering andsmoothing
Non-linearestimation
Sensor fusion
Machine learning with signal processing: Part IArno Solin
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Motivation: Temporal models
/ One-dimensional problems(the data has a natural ordering)
/ Spatio-temporal models(something developing over time)
/ Long / unbounded data(sensor data streams, daily observations, etc.)
Explaining changes in number of births in the US
12 24 36 48 60 72 84 96 108 1200
100
200
300
Time (months)
Pre
cip
ita
tio
n (
mm
)
True
Estimate
Spatio-temporal modelling of precipitationBrain data (or why not neural networks)Sensor data modelling
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Machine learning
MLML
Data
Models Algorithms
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Tools for dealing with time-series
I Moment representationConsidering the statistical properties ofthe input data jointly over time
I Spectral (Fourier) representationAnalyzing the frequency-spacerepresentation of the problem/data
I State space (path) representationDescription of sample behaviour as adynamic system over time
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Spectral (Fourier) representation
I Fourier transform F [·]:
f (ω) =
∫f (x) exp(−i ωTx)dx
I Analyzing properties of ‘systems’ (input–output mappings)by transfer functions:
H(s) =Y (s)X (s)
=L[y(t)](s)L[x(t)](s)
,
where L[·] is the Laplace transform
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Discrete-time state space models
x0 x1
y1
x2
y2
x3
y3
x4
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x5
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· · ·
I A canonical state space model:
Dynamics: xk = f(xk−1,qk ), qk ∼ N(0,Qk ),
Measurement: yk = h(xk , rk ), rk ∼ N(0,Rk )
I The key to efficiency is the directed graph:The Markov property.
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Kalman filtering and smoothing
x0 x1
y1
x2
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x3
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y4
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· · ·xk
I Closed-form solution to linear-Gaussian filtering problems
xk = A xk−1 + qk , qk ∼ N(0,Qk ),
yk = H xk + rk , rk ∼ N(0,Rk )
I Filtering solution: p(xk | y1:k ) = N(xk | mk |k ,Pk |k )
I Smoothing solution: p(xk | y1:T ) = N(xk | mk |T ,Pk |T )
Machine learning with signal processing: Part IArno Solin
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Non-linear filtering
x0 x1
y1
x2
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y3
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y4
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· · ·xk
State Gaussian even if dynamicsand measurements non-linear
I Typically xk is assumed Gaussian:
xk = f(xk−1,qk ), qk ∼ N(0,Qk ),
yk = h(xk , rk ), rk ∼ N(0,Rk )
I Filtering solution: p(xk | y1:k ) ' N(xk | mk |k ,Pk |k )
I Smoothing solution: p(xk | y1:T ) ' N(xk | mk |T ,Pk |T )
Machine learning with signal processing: Part IArno Solin
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Sensor fusion by non-linear Kalman filtering
I Dynamics f(·, ·) governedby high-frequencyaccelerometer andgyroscope sensor samples(Newton’s laws)
I Observations yk arecamera frames
I Online learning problem(sensor noises and biases)
Real-time visual-inertialmotion tracking
Probabilistic inertial-visual odometryfor occlusion-robust navigation (https://youtu.be/_ywmtVzxURk)
Machine learning with signal processing: Part IArno Solin
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Non-linear filtering by sequential Monte Carlo
x0 x1
y1
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· · ·xk
If a Gaussian approximation is toorestrictive, resort to particle filtering
I Sequential Monte Carlo characterizes the state distributionwith a swarm of ‘particles’ (particle filtering)
I Can deal with multi-modality, severe non-linearities, etc.I All the usual MC caveats apply
Machine learning with signal processing: Part IArno Solin
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Sensor fusion by sequential Monte Carlo
I Absolute position tracking of asmartphone by using thecompass sensor (magnetometer)
I Match observations to a map oflocal anomalies of the magneticfield inside a building
I The ‘anomaly track’ becomesunique when the phone hasmoved a long-enough distance
Terrain matching in the magnetic landscapeby sequential Monte Carlo (https://youtu.be/UuUo9Q0OT1Q)
Machine learning with signal processing: Part IArno Solin
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Up next
2I Going from discrete-time to
continuous-timeI A gentle introduction to stochastic
differential equations (SDEs)
Machine learning with signal processing: Part IArno Solin
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Bibliography
These references are sources for finding a more detailed overview on thetopics of this part :
� T. Glad and L. Ljung (2000). Control Theory: Multivariable andNonlinear Methods. Taylor & Francis, New York.
� G. Wahba (1990). Spline Models for Observational Data. Siam.
� S. Sarkka (2013). Bayesian Filtering and Smoothing. CambridgeUniversity Press. Cambridge, UK.
� T. B. Schon et al. (2011). The particle filter in practice. The OxfordHandbook of Nonlinear Filtering. Oxford University Press, UK.