machine learning with signal processing [0em] part i: signal...

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

https://twitter.com/arnosolin

http://arno.solin.fi

Machine learning with signal processing: Part IArno Solin

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CC-NC: https://xkcd.com/1524/

https://xkcd.com/1524/


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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


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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


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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


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Structure

Part I

Tools anddiscrete-time

models

Part II

SDEs(continuous-time

models)

Part III

Gaussianprocesses

�Part IV

Applicationexamples


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Outline

Temporal models

Tools for workingwith time-series

Spectral methods

State spacemodels

Filtering andsmoothing

Non-linearestimation

Sensor fusion


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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

y4

x5

y5

· · ·

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

y2

x3

y3

x4

y4

x5

y5

· · ·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 )


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Non-linear filtering

x0 x1

y1

x2

y2

x3

y3

x4

y4

x5

y5

· · ·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 )


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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)

https://youtu.be/_ywmtVzxURk


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Non-linear filtering by sequential Monte Carlo

x0 x1

y1

x2

y2

x3

y3

x4

y4

x5

y5

· · ·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


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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)

https://youtu.be/UuUo9Q0OT1Q


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Up next

2I Going from discrete-time to

continuous-timeI A gentle introduction to stochastic

differential equations (SDEs)


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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.

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