particle flow for nonlinear filters
TRANSCRIPT
particle flow for
nonlinear filters
Fred Daum
19 June 2012
Copyright © 2012 Raytheon Company. All rights reserved.
Customer Success Is Our Mission is a trademark of Raytheon Company.
1
discrete time nonlinear filter problem
k
k
k
kkkk
z
w
tx
wttxFtx
tat timet vector measuremen
tat time vector noise process
tat time vectorstate)(
),),(()(
k
k
k
1
=
=
=
=+estimate x
given noisy
measurements
k
k
kk
k
kkkk
k
zzz
Z
Ztxp
v
vttxHz
,...,,Z
tsmeasuremen all ofset
given Z tat time x ofdensity y probabilit),(
tat time vector noiset measuremen
),),((
21k
kk
k
k
=
=
=
=
=
2
many applications of filtering & prediction
roboticsnavigation communicationscontrol of chemical,
mechanical, electrical & nuclear plants
guidance
tracking missiles, satellites, aircraft, ground vehicles,
artillery shells, comets, people, cows, salmon
weather & climate prediction
predicting ionosphere, thermosphere, troposphere
fancy signal processing
science imagingmedicine (e.g., MRI, surgery, pace makers,
drug design, diagnosis)
fancy signal processing
(e.g., MIMO radar, MIMO sonar, MIMO comm, MIMO nav)
oil & mineral exploration
financial engineering adaptive antennasaudio & video signal
processing
Bayesian decisions multi-sensor data fusion compressive sensing other….
3
type of
nonlinear
filter
statistics computed computational
complexity
estimation
accuracy
representation
of probability
density
extended
Kalman filters
mean vector &
covariance matrix d³
sometimes good
but often highly
suboptimal
mean vector &
covariance matrix
unscented
Kalman filters
mean vector &
covariance matrix d³
sometimes better
than EKF but
sometimes worse
mean vector &
covariance matrix
batch least
squares
mean vector &
covariance matrix d³
sometimes better
than EKF but
sometimes worse
mean vector &
covariance matrixd³sometimes worse
numerical
solution of
Fokker-Planck
PDE
full conditional
probability density of
statecurse of
dimensionality
optimal* points in state space
and/or smooth
functions
particle filters
full conditional
probability density of
state
curse of
dimensionality
optimal* particles
exact recursive
filters (Kalman,
Beneš, Daum,
Wonham, Yau)
full conditional
probability density of
state
polynomial in d
(for special
problems)
optimal
(for special
problems)
sufficient
statistics
4
dimension of
the state vector
extended
Kalman
filters
particle flow
filters
10
100
5
nonlinearity or non-Gaussianity
standard particle filters
filters
1
10
exact flow filter is many orders of magnitude faster per
particle than standard particle filters
- - - - -
bootstrap
EKF proposal
incomp flow
exact flow
3
104
105
106
107
Med
ian
Co
mp
uta
tio
n T
ime
fo
r 30 U
pd
ate
s (
sec)
d = 30
d = 20
d = 10
d = 5 bootstrap
particle filter
EKF proposal
* Intel Corel 2 CPU, 1.86GHz, 0.98GB of RAM, PC-MATLAB version 7.7
25 Monte Carlo trials10
210
310
410
510
-1
100
101
102
103
Number of Particles
Med
ian
Co
mp
uta
tio
n T
ime
fo
r 30 U
pd
ate
s (
sec)
6
exact flow
EKF proposal
incompressible
flow
particle flow filter is many orders of magnitude faster
real time computation (for the same or better
estimation accuracy)
3 or 4 orders of
3 or 4 orders of magnitude faster
per particle
avoids bottleneck in
many orders of
magnitude faster
3 or 4 orders of magnitude
fewer particles
bottleneck in parallel
processing due to resampling
7
10-1
100
101
102
EKF
PF Incompressible
new filter improves accuracy by
two orders of magnitude
median error
N = 500 particles
extended Kalman filter
standard particle filter
0 2 4 6 8 1010
-3
10-2
10
Time (sec)
PF Incompressible
PF Ax+BN = 500 particles
new filter
key idea: small curvature flow
(inspired by fluid dynamics) to make
solution of PDE div(pf) = η much faster
Euler’s equations:
3121233
2313122
1232311
)(
)(
)(
MIII
MIII
MIII
=−+
=−+
=−+
ωωω
ωωω
ωωω
&
&
&
8
Why engineers like particle filters:
• Very easy to code
• Extremely general dynamics & measurements: nonlinear & non-Gaussian
• Optimal estimation accuracy (if you use enough • Optimal estimation accuracy (if you use enough particles….)
• You don’t need to know anything about stochastic differential equations or any fancy numerical methods for solving PDEs
• Some people (erroneously) think that PFs beat the curse of dimensionality
9
curse of dimensionality for
classic particle filter
optimal
accuracy:
r = 1.0
10
prediction of
conditional
probability
density from
tk-1 to tk
nonlinear filter
measurements
:rule Bayes'
solution of
Fokker-Planck
equation
),(),(),(
:rule Bayes'
1 kkkkkk txzpZtxpZtxp −=
11
particle degeneracy*
likelihood of
measurementprior
density
particles to represent the prior
12
*Daum & Huang, “Particle degeneracy: root cause & solution,” SPIE Proceedings 2011.
chicken & egg problem
How do you pick a
good way to represent
the product of two
functions before you functions before you
compute the product
itself?
13
induced flow of particles
for Bayes’ rule
pdf pdfflow of density
prior posterior
)(log)(log),(log xhxgxp λλ +=
particles particles
flow of particles
sample from
density
sample from
density
λ=0 λ=1
)(log)(log),(log xhxgxp λλ +=
),( λλ
xfd
dx=
14
derivation of PDE for exact particle flow:
∂
∂−=
∂
∂
∂
∂−=
∂
∂
=
λλ
λ
λ
λ
λλ
x
pfTrxp
xp
x
pfTr
xp
xfd
dx
)(),(
),(log
)(),(
),( Fokker-Planck
equation with zero
diffusion
definition
of p(x,λ)
−−=
=
−=
∂
∂−
−+=
∂∂
λ
λλη
η
λλ
λ
λλλ
λ
d
Kdxhxp
pfdiv
pfdivxpK
xh
Kxhxgxp
x
)(log)(log),(
)(
)(),()(log
)(log
)(log)(log)(log),(log
15
definition
of η
PDE for
f given p
linear first order highly underdetermined PDE:
d
d
x
q
x
q
x
q
d
Kdxhxpx
xxqdiv
∂
∂++
∂
∂+
∂
∂=
−−=
=
...
)(log)(log),(),(
),()),((
2
2
1
1η
λ
λλλη
ληλ
like Gauss’ divergence
law in electromagneticsdxxx ∂∂∂ 21
function values are
only known at
random points in d-
dimensional space
16
q = pf
f = unknown function
p & η = known at random points
We want dx/dλ = f(x,λ)
to be a stable dynamical
system.
law in electromagnetics
likelihood of
measurementprior
densityoptimal
accuracyg h
root cause of
curse of dimensionality:
curse of dimensionality:
particles to represent the prior
pdf pdf
particles particles
flow of density
flow of particles
sample from
density
sample from
density
λ=0 λ=1
prior posterior
)(log)(log),(log xhxgxp λλ +=
),( λλ
xfd
dx=
f.for PDE above thesolving
by flow particle design the We
loglog)(
+−=
λd
Kdhppfdiv
17
method to solve PDE how to pick unique solution comments
1. generalized inverse of linear differential operator minimum norm* very difficult to design robust stable & fast algorithm
2. Poisson’s equation gradient of potential*(assume irrotational flow)
very difficult to design robust stable & fast algorithm
3. generalized inverse of gradient of log-homotopy assume incompressible flow & pick minimum L² norm solution
workhorse for multimodal densities
4. generalization of method #3 most robustly stable filter or random pick, etc.
workhorse for multimodal densities
5. separation of variables (Gaussian) pick solution of specific form (polynomial) extremely fast & hard to beat in accuracy for many problems
6. separation of variables
(exponential family)
pick solution of specific form (finite basis functions)
needs theoretical work & numerical experiments
7. variational formulation (Gauss & Hertz) convex function minimization needs work
8. optimal control formulation convex functional minimization (e.g., least action like Monge-Kantorovich)
very high computational complexityaction like Monge-Kantorovich)
9. direct integration (of first order linear PDE in divergence form)
choice of d-1 arbitrary functions should work with enforcement of neutral charge density & importance sampling
10. generalized method of characteristics more conditions (e.g., small curvature or specify curl, or use Lorentz invariance)
needs theoretical work & numerical experiments
11. another homotopy (inspired by Gromov’s
h-principle) like Feynman’s QED perturbation
initial condition of ODE &
uniqueness of sol. to ODE
needs theoretical work & numerical experiments
12. finite dimensional parametric flow
(e.g., f = Ax+b with A & b parameters)
non-singular matrix to invert needs numerical experiments
13. Fourier transform of PDE (divergence form of linear PDE has constant coefficients!)
minimum norm* or most stable flow very difficult to design robust stable & fast algorithm
14. small curvature flow & assumed prior density solve d x d system of linear equations new in 2012. Beats other methods for difficult nonlinear problems.
15. small curvature flow & homotopy for inverse of A + B (sum of two linear operators)
numerically integrate ODE new in 2012. extremely cool theory.
16. small curvature flow & homotopy for generalized inverse of A + B (sum of two linear operators)
numerically integrate ODE new in 2012
exact particle flow for Gaussian densities:
fx
pfdiv
d
Kdh
xfd
dx
∂
∂−−=−
=
:exactly ffor solvecan weGaussian,h & gfor
log)(
)(log)log(
),(
λ
λ
λλ
[ ]
( ) ( )[ ]xAzRPHAIAIb
HRHPHPHA
bAxf
T
TT
+++=
+−=
+=
−
−
1
1
2
2
1
:exactly ffor solvecan weGaussian,h & gfor
λλ
λ
19
automatically stable
under very mild
conditions &
extremely fast
incompressible particle flow
∂
∂
∂
−=
gradient zero-nonfor
),(log
))(log(dx
2
x
xp
xh
T
λ
λ
∂
∂−
=
otherwise 0
gradient zero-nonfor ),(log
))(log(
d
dx 2
x
xpxh
λλ
20
irrotational particle flow:
∫
∫ −
∂
=
≥−
−=
=
∂
∂
∂
∂==
d
T
dyc
dyyx
cyxV
xx
xVTr
xpx
xVxf
d
dx
)logK(
-logh(y))p(y,)V(x,
3 dfor ),(),(
),(),(
),(/),(
),(
2
2
2
λλλ
ληλ
ληλ
λλ
λλ Poisson’s
equation
∑
∫
∫
∈
−
−
−−
∂
∂−≈
∂
∂
−
−−
∂
∂−=
∂
∂
−
−−
∂
∂−=
∂
∂
− ∂
=
iSjd
ji
T
ji
ji
d
T
d
T
d
xx
xxdcKxh
Mx
xV
yx
yxdcKyhE
x
xV
dyyx
yxdcKyhyp
x
xV
dyyx
))(2()
)(log)((log
1),(
))(2()
)(log)((log
),(
))(2()(log)(log),(
),(
-logh(y))p(y,)V(x,2
λ
λλ
λ
λλ
λ
λλ
λ
λλλ
21
like
Coulomb’s
law
0 5 10 15 20 25 30 35 400
100
200
300N = 1000
An
gle
Err
or
(de
g)
EKF
PF
0 5 10 15 20 25 30 35 400
5
10
15
Time
An
gle
Ra
te E
rro
r (d
eg
/se
c)
EKF
PF
22
100
200
300
400Time = 1, Frame 1
An
gle
Ra
te (
de
g/s
ec)
initial probability distribution of particles:
λ = 0.0
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
23
100
200
300
400Time = 1, Frame 2
An
gle
Ra
te (
de
g/s
ec)
λ = 0.1
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
24
100
200
300
400Time = 1, Frame 2
An
gle
Ra
te (
de
g/s
ec)
λ = 0.2
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
25
100
200
300
400Time = 1, Frame 3
An
gle
Ra
te (
de
g/s
ec)
λ = 0.3
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
26
100
200
300
400Time = 1, Frame 4
An
gle
Ra
te (
de
g/s
ec)
λ = 0.4
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
27
100
200
300
400Time = 1, Frame 5
An
gle
Ra
te (
de
g/s
ec)
λ = 0.5
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
28
100
200
300
400Time = 1, Frame 6
An
gle
Ra
te (
de
g/s
ec)
λ = 0.6
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
29
100
200
300
400Time = 1, Frame 7
An
gle
Ra
te (
de
g/s
ec)
λ = 0.7
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
30
100
200
300
400Time = 1, Frame 8
An
gle
Ra
te (
de
g/s
ec)
λ = 0.8
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
31
100
200
300
400Time = 1, Frame 9
An
gle
Ra
te (
de
g/s
ec)
λ = 0.9
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
32
100
200
300
400Time = 1, Frame 10
An
gle
Ra
te (
de
g/s
ec)
final probability distribution of particles (resulting from
one noisy measurement of sin(θ) with Bayes’ rule):
λ = 1
-200 -150 -100 -50 0 50 100 150 200-300
-200
-100
0
Angle (deg)
An
gle
Ra
te (
de
g/s
ec)
33
induced flow of particles
for Bayes’ rule
pdf pdfflow of density
prior posterior
)(log)(log),(log xhxgxp λλ +=
particles particles
flow of particles
sample from
density
sample from
density
λ=0 λ=1
)(log)(log),(log xhxgxp λλ +=
),( λλ
xfd
dx=
34
103
104
d = 12, ny = 3, y = x2, SNR = 20dB
Dim
en
sio
nle
ss E
rro
r
extended Kalman filter (EKF)
particle filter beats the
EKF by two orders of
magnitude in accuracy
102
103
104
105
102
Dim
en
sio
nle
ss E
rro
r
Number of Particles
EKF
PF
quadratic measurement
nonlinearity 35
particle filter
magnitude in accuracy
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
36
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
37
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
38
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
39
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
40
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
41
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
42
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
43
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
44
0
0.2
0.4
0.6
0.8
1x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5
x 105
-1
-0.8
-0.6
-0.4
-0.2
45
105
106
107
d = 12, ny = 3, y = x
3, SNR = 20dB
Dim
en
sio
nle
ss
Err
or
particle filter beats the
EKF by two orders of
magnitude in accuracy
extended Kalman filter (EKF)
102
103
104
105
103
104
Dim
en
sio
nle
ss
Err
or
Number of Particles
EKF
PF
46
magnitude in accuracy
particle filter
cubic measurement
nonlinearity
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
47
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
48
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
49
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
50
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
51
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
52
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
53
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
54
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
55
0
0.5
1
1.5x 10
5 Time = 1, Magenta: truth, Green: PF estimate, Black: KF
-1 -0.5 0 0.5 1 1.5 2
x 105
-1.5
-1
-0.5
56
direct integration of PDE
∂
∂++
∂
∂+
∂
∂=
=
d
d
x
q
x
q
x
q
xxqdiv
...
),()),((
:PDE theof form divergence theuse
2
2
1
1η
ληλ
∫ −+=
=
=
==
∂∂∂
jx
j
d
dxxx
q
xxx
),(~),(q~q
q~ q ofcomponent onebut allpick
~-)q~-div(q
~)q~div( problem related ofsolution exact ~
jj
21
ληλη
ηη
η
57
most general solution for exact flow:
)(log-Cf
:issolution generalmost the
log)()log(
),(
## −+=
∂
∂−−=
=
yCCIh
fx
pfdivh
xfd
dxλ
λ
fundamental
PDE for exact
particle flow
vectorldimensiona-darbitrary y
C of inverse dgeneralize
x
logpC
:operator aldifferentilinear a is Cin which
)(log-Cf
#
##
=
=
+∂
∂=
−+=
C
div
yCCIh
could pick y to
robustly stabilize the
filter or random or
other
58
computing the generalized inverse of A + B:
#
#
#
0)(
BA)C( :homotopy a define
case) in this isit but truegenerally (not 1CC
C of inverse pseudoC
B A C define
d
CCd=
+=
=
=
+=
µ
µµpick A that is easy
to invert & pick
any B (i.e., could
( ) #####
###
#
##
0
0
BCCCCd
dC
Cd
dCC
d
dCCC
Cd
dC
d
dCC
d
−=
−=
=+
=
µ
µµ
µµ
µ
59
solve by numerical
integration from
µ = 0 to 1
any B (i.e., could
be hard to invert)
nonlinear filter performance (accuracy wrt
optimal & computational complexity)
DIMENSIONprocess noise
initial uncertainty
of state vector
measurementexploit
smoothness
sparseness
measurement
noise
stability
& mixing
of dynamics
multi-modal
nonlinearityill-conditioning
quality of
proposal density
concentration
of measure
exploit
structure (e.g.
exact filters)
60
variation in initial uncertainty of x
1010
1015
1020
Dim
en
sio
nle
ss E
rro
r
N = 1000, Stable, d = 10, Quadratic
Huge Initial Uncertainty
Large Initial Uncertainty
Medium Initial Uncertainty
Small Initial Uncertainty
, λλλλ = 0.6
25 Monte Carlo Trials
0 5 10 15 20 25 3010
-5
100
105
Time
Dim
en
sio
nle
ss E
rro
r
61
variation in eigenvalues of the plant (λ)
106
108
1010
1012
Dim
en
sio
nle
ss E
rro
r
N = 1000, d = 10, Cubic
λλλλ = 0.1
0.5
1.0
1.1
1.2
25 Monte Carlo Trials
0 5 10 15 20 25 3010
-2
100
102
104
Time
Dim
en
sio
nle
ss E
rro
r
62
variation in dimension of x
106
108
1010
1012
Dim
en
sio
nle
ss E
rro
r
N = 1000, λλλλ = 1.0, Cubic
Dimension = 5
10
15
20
25 Monte Carlo Trials
0 5 10 15 20 25 3010
-2
100
102
104
Time
Dim
en
sio
nle
ss E
rro
r
63
numerical results so far
applications:
• long range wideband radar
tracking TBMs & ICBMs
• angles only tracking (multiple
radars & 1 radar)
key parameters:
• dimension of state vector of
dynamical model (plant)
• initial uncertainty in the state
vectorradars & 1 radar)
• discrimination for wideband
radar (BMD)
• linear systems
• quadratic nonlinearity
• cubic nonlinearity
• cosine nonlinearity
• Euler’s equations (6 DOF)
vector
• stability of the plant
• signal-to-noise ratio of
measurements
• trajectory of objects
• radar-object geometry
64
particle flow filter is many orders of magnitude faster
real time computation (for the same or better
estimation accuracy)
3 or 4 orders of
3 or 4 orders of magnitude faster
per particle
avoids bottleneck in
many orders of
magnitude faster
3 or 4 orders of magnitude
fewer particles
bottleneck in parallel
processing due to resampling
65
many new ideas
direct integration of PDE but enforce
neutral charge & exploit known exact solutions
separation of variables
(exponential family & arbitrary densities)
method of characteristics (take
gradient to get d eqs in d unknowns)
small curvature flow (take gradient to get d eqs. in d unknowns)
variance reduction for Poisson’s eq exploiting
f = Ax+b or other
log speed & unit vector for Coulomb’s law
inverse or generalized inverse of A+B like Feynman’s QED
perturbation theory
Gelman & Mengoptimal flow
Lagrange’s trick assumed densities Lagrange’s trick
Gromov’s h-principle
renormalization group flow
assumed densities (Gauss-arbitrary,
exponential family-arbitrary, Gauss-
exponential family, etc.)
Le Gland more general homotopy than log-
homotopy (e,g, log-log)
harvest ideas from Monge-Kantorovich
optimal trasnport
big/small parameter approximations (far field, narrow band, steady state, large
number of particles, 1/137, etc.)
parametric flow to enforce stability
non-unique solutions of PDE (mixed in x, λ &
components of x)
Galerkin or
homotopy-Galerkin or differential quadrature
Q ≥ 0 with Dirac approximation to solve
Fokker-Planck PDE
Q ≥ 0 with Daum exact solutions of Fokker-
Planck (1986)
HYBRIDS
of above
obtain unique solution to Gauss div law by enforcing Lorentz invariance (like Maxwell’s eqs) 66
small curvature particle flow:
x
f
x
pf
x
p
x
fdiv
fx
pfdiv
d
Kdh
+=
∂
∂
∂
∂−
∂
∂−
∂
∂−=
∂
∂
∂
∂−−=−
:flow) ibleincompress and bAx f (and flow curvature smallfor
loglog)(
x
logh
:PDE above ofgradient thecompute
log)(
)(log)log(
2
2
λ
λ
T
x
f
x
p
x
h
x
pf
∂
∂
∂
∂+
∂
∂
∂
∂−=
=∂
∂
+=
−
logloglog
:hence
0x
div(f)
:flow) ibleincompress and bAx f (and flow curvature smallfor
1
2
2
67
• extremely fast to compute
• Hessian of logp is
always non-singular
• similar to Fisher matrix
• generalizes our two
favorite flows!
10-1
100
101
102
EKF
PF Incompressible
new filter improves accuracy by
two orders of magnitude
median error
N = 500 particles
extended Kalman filter
standard particle filter
0 2 4 6 8 1010
-3
10-2
10
Time (sec)
PF Incompressible
PF Ax+BN = 500 particles
new filter
key idea: small curvature flow
(inspired by fluid dynamics) to make
solution of PDE div(pf) = η much faster
Euler’s equations:
3121233
2313122
1232311
)(
)(
)(
MIII
MIII
MIII
=−+
=−+
=−+
ωωω
ωωω
ωωω
&
&
&
68
0div(f)
:flow ibleincompress
=)b()xA(f
:flowGaussian
+= λλ
small curvature flow:
69
0div(f) =Tr(A)div(f)
)b()xA(f
=
+= λλ
0)(
=∂
∂
x
fdiv
small curvature flow (even better):
T
x
f
x
p
x
pf
x
fdiv
fx
pfdiv
d
Kdh
+=
∂
∂
∂
∂−
∂
∂−
∂
∂−=
∂
∂
∂
∂−−=−
:flow) ibleincompress and bAx f (and flow curvature smallfor
loglog)(
x
logh
:PDE above ofgradient thecompute
log)(
)(log)log(
2
2
λ
λ
T
x
h
xx
p
x
pf
∂
∂
∂
∂
∂
∂+
∂
∂−=
=∂
∂
+=
−
logloglog
:hence
0x
div(f)
:flow) ibleincompress and bAx f (and flow curvature smallfor
1
2
2
70
computing the inverse of A + B:
1
1-
1-
0)(
BA)G( :homotopy a define
GG
G of inverse G
B A G define
−
=
+=
=
=
+=
d
GGd
I
µ
µµ
pick A that is easy
to invert & pick
A & B are linear operators (e.g., matrices or
differential operators)
111
111
1
11
0
0
−−−
−−−
−
−−
−=
−=
=+
=
BGGd
dG
Gd
dGG
d
dGGG
Gd
dG
d
dGG
d
µ
µµ
µµ
µ
solve by numerical integration from
µ = 0 to 1with the obvious initial condition
to invert & pick any B (i.e., could
be hard to invert)
71
new nonlinear filter: particle flow
new particle flow filter standard particle filters
many orders of magnitude faster than
standard particle filters
suffers from curse of dimensionality
3 to 4 orders of magnitude faster per
particle
suffers from “particle degeneracy”
3 to 4 orders of magnitude fewer particles
required to achieve optimal accuracy
requires millions or billions of particles
for high dimensional problems
Bayes’ rule is computed using particle
flow (like physics)
Bayes’ rule is computed using a pointwise
multiplication of two functions
no proposal density depends on proposal density (e.g.,
Gaussian from EKF or UKF)
no resampling of particles resampling is needed to repair the damage
done by Bayes’ rule
embarrassingly parallelizable suffers from bottleneck due to resampling
computes log of unnormalized density suffers from severe numerical problems
due to computation of normalized density72
History of Mathematics
1. Creation of the integers
2. Invention of counting
3. Invention of addition as a fast
method of countingmethod of counting
4. Invention of multiplication as a
fast method of addition
5. Invention of particle flow as a
fast method of multiplication*
73
Fred Daum, Jim Huang & Arjang Noushin, “exact particle flow for nonlinear filters,” Proceedings of SPIE Conference, Orlando Florida,
April 2010.
Fred Daum & Jim Huang, “particle degeneracy: root cause & solution,” Proceedings of SPIE Conference, Orlando Florida, April 2011.
Fred Daum & Jim Huang “numerical experiments for nonlinear filters with exact particle flow induced by log-homotopy,” Proceedings of SPIE Conference, Orlando Florida, April 2010.Conference, Orlando Florida, April 2010.
Fred Daum & Jim Huang, “exact particle flow for nonlinear filters: seventeen dubious solutions to a linear first order underdetermined PDE,” Proceedings of IEEE Conference, Asilomar California, November 2010.
Fred Daum, Jim Huang & Arjang Noushin, “Coulomb’s law particle flow for nonlinear filters,” Proceedings of SPIE Conference, San Diego California, August 2011.
74
BACKUPBACKUP
75
fluid dynamics
electro-magnetics
plasma physics
quantum field
theorybig bang
76
1. derive PDE
2. solve PDE
77
3. test solution
physics & nonlinear filters
Metropolis algorithm &
particle filters
PDEs
(Fokker-Planck, Gauss div law, Maxwell’s eqs,
continuity equation, Einstein field eqs., etc.)
Beneš advice:
“think like a fluid mechanic”
Coulomb’s law &
particle flow
path integrals for NLF (Pierre Del
Moral & Bhashyam Balaji & Mike Schilder, et al.)
Why is the maximum speed of particles c ?
incompressible flow
irrotational flow
small curvature flow
Dan Zwillinger’s
advice: “think like a physicist”
Lie algebras & gauge theory for NLF
(Brockett & Mitter & Marcus & Ocone et al.)
1/N
Dudley Herschbach
Mark Kac & Ed Witten
BIG BANG
inflation
cosmic acceleration
quantization of physical particles & particles in
filters also!
(via resampling)
renormalization group flow & particle flow
(Bhashyam Balaji)
Hamiltonian Monte Carlo
“the physicists are always 20 years
ahead of us”
Prof. Jun Liu
strings vs. particles
(Francois Le Gland 2001)
78
numerical results so far
applications:
• long range wideband radar
tracking TBMs & ICBMs
• angles only tracking (multiple
radars & 1 radar)
key parameters:
• dimension of state vector of
dynamical model (plant)
• initial uncertainty in the state
vectorradars & 1 radar)
• discrimination for wideband
radar (BMD)
• linear systems
• quadratic nonlinearity
• cubic nonlinearity
• cosine nonlinearity
• Euler’s equations (6 DOF)
vector
• stability of the plant
• signal-to-noise ratio of
measurements
• trajectory of objects
• radar-object geometry
79
104
d = 12, ny = 3, y = x2, SNR = 20dB
Dim
en
sio
nle
ss E
rro
r
quadratic measurement nonlinearity
extended Kalman filter (EKF)
particle filter beats the
102
103
104
105
102
103
Dim
en
sio
nle
ss E
rro
r
Number of Particles
EKF
PF
particle filter
particle filter beats the
EKF by two orders of
magnitude in accuracy
80
106
107
d = 12, ny = 3, y = x
3, SNR = 20dB
Dim
en
sio
nle
ss
Err
or
cubic measurement nonlinearity
particle filter beats the
EKF by two orders of
magnitude in accuracy
extended Kalman filter (EKF)
102
103
104
105
103
104
105
Dim
en
sio
nle
ss
Err
or
Number of Particles
EKF
PF
magnitude in accuracy
particle filter
81
0 5 10 15 20 25 30 35 400
100
200
300N = 1000
An
gle
Err
or
(de
g)
EKF
PF
nonlinear measurement (sine function of θ)
0 5 10 15 20 25 30 35 40
0 5 10 15 20 25 30 35 400
5
10
15
Time
An
gle
Ra
te E
rro
r (d
eg
/se
c)
EKF
PF
82
100
101
102
Euler’s equations of rotational motion
with wideband range measurements
median
error in
state
vector
extended Kalman filter
standard particle filter
new filter
0 2 4 6 8 1010
-3
10-2
10-1
Time (sec)
EKF
PF Incompressible
PF Ax+B
N = 500 particles
50 Monte Carlo runs
new particle filter
new filter
improves
accuracy by two
orders of
magnitude
83
103
104
105
Ve
loc
ity
Err
or
(m/s
ec
)
N = 100, σσσσr = 100m
EKF
Inexact Flow
Exact Flow
Exact Flow with Redraw
radar tracking ballistic missile
(d =6 & N = 100 particles)
incompressible
flow
0 20 40 60 80 10010
0
101
102
Time (sec)
Ve
loc
ity
Err
or
(m/s
ec
)
exact
flow
EKF
84
103
104
km
median range estimation error (angles only data)
0 50 100 150 200 250 300 350 40010
1
102
Time (sec)
km
EKF: solid
Log Homotopy PF: dashed
angle = 90 deg
angle = 75 deg
angle = 25 deg
angle = 10 deg
both EKF & PF
use modified spherical
coordinates
85
103
104
angle = 10 deg
median range estimation error (angles only data)
extended Kalman
filter
0 50 100 150 200 250 300 350 40010
1
102
Time (sec)
km
both EKF & PF
use modified spherical
coordinates
exact flow filter
86
100
101
102
3D position error, angle 1 sigma = 0.1 mrad
Radar separation = 0 m
10 m
100 m
1000 m
two radars air targets angles only
0 20 40 60 80 100 12010
-4
10-3
10-2
10-1
Time (sec)
km
excellent
accuracy
from two
radars
87
100
101
102
3D position error, angle 1 sigma = 0.1 mrad
Radar separation = 1000 m
2000 m
5000 m
10000 mexcellent
accuracy
from two
radars
two radars air targets angles only
0 20 40 60 80 100 12010
-4
10-3
10-2
10-1
Time (sec)
km
88
100
101
102
3D position error, angle 1 sigma = 1.0 mrad
Radar separation = 0 m
10 m
100 m
1000 m
two radars air targets angles only
0 20 40 60 80 100 12010
-3
10-2
10-1
Time (sec)
km
excellent
accuracy
from two
radars
89
100
101
102
Euler’s equations of rotational motion
with wideband range measurements
median
error in
state
vector
extended Kalman filter
standard particle filter
new filter
0 2 4 6 8 1010
-3
10-2
10-1
Time (sec)
EKF
PF Incompressible
PF Ax+B
N = 500 particles
50 Monte Carlo runs
new particle filter
new filter
improves
accuracy by two
orders of
magnitude
90
104
106
108
Dim
en
sio
nle
ss E
rro
r
Stable, d = 30, Linear
EKF
Inexact Flow
Exact Flow
Exact Flow with Redraw
only need N = 100
particles for optimal
accuracy for d = 30
102
103
104
100
102
Number of Particles
Dim
en
sio
nle
ss E
rro
r
accuracy for d = 30
dimensional problem
91
exact flow: performance vs. number of particles
101
102
Dim
en
sio
nle
ss E
rro
r A
fter
30 U
pd
ate
s
λλλλ = 1.2, Linear, Large Initial Uncertainty
Dimension = 5
10
15
20
30extremely
unstable
plant
25 Monte Carlo Trials
102
103
104
10-1
100
Number of Particles
Dim
en
sio
nle
ss E
rro
r A
fter
30 U
pd
ate
s
92
93
item exact particle flow Monge-Kantorovich
optimal transport
1. purpose fix particle degeneracy due
to Bayes’ rule
move physical objects with
minimal effort from p1 to p2
2. conservation of
probability mass along flow
yes yes
3. deterministic yes* yes
4. homotopy of density no yes
5. log-homotopy of
density
yes no
6. optimality criteria none minimum action, etc.6. optimality criteria none minimum action, etc.
7. how to pick a solution a dozen distinct methods minimum action, etc.
8. stability of flow explicitly
considered
yes rarely
9. high dimensional
applications solved
successfully
yes (d ≤ 30) no (d = 1, 2 or 3)
10. computational
complexity
numerical integration of
ODE for each particle
solution of Monge-Ampere
nonlinear PDE or other PDE
11. solution of PDE for nice
special cases
many (incompressible,
irrotational, Gaussian, etc.)
only irrotational94
fast Ewald’s method vs. Coulomb’s law
item fast Ewald method
in physics &
chemistry*
Coulomb’s law
with fast k-NN
comments
1. dimension of x d = 3 d = 3 to 30 rapid decay of
Coulomb kernel in
higher d helps!
2. relative error
desired
0.0001 or better 1% to 10% all Ewald methods the
same for 1% accuracy
3. cut-off in x space fixed distance random per k-NN automatic space-taper
to weight convolution
4. desired force on mesh at particles big difference!
5. neutral charge locally enforced locally enforced crucial
6. smoothing charge Gaussian no explicit smoothing
7. k-space or real-
space
both real space (x) no FFT needed for
Coulomb
95
*Shan, Klepeis, Eastwood, Dror & Shaw, “Gaussian split Ewald: a fast Ewald mesh
method for molecular simulation,” Journal of Chemical Physics, 2005.
new exact particle flow: operator-valued homotopy
Improved version of incompressible flow (does not assume that div(f) = 0 and makes no other assumptions about the problem)
Avoids computing p(x,λ) or dividing by p (unlike Coulomb’s law), but rather it uses the gradient of the log of p(x,λ)
Does not depend on EKF or UKF
Completely general (highly non-Gaussian multimodal densities and compressible flow)
Compute the flow dx/dλ using the generalized inverse of the sum of two linear differential operators (A + B) using an ODE derived with a homotopy (similar to matrix inversion using ODE derived by homotopy); exploits easy computation of generalized inverse of A; A = gradient of log p(x,λ) as in incompressible flow & B = divergence
Inspired by: (1) Feynman’s perturbation approximation for QED but we do not approximate by exploiting small parameter 1/137 (see Kaku‘s book on QFT page 137) and (2) homotopy for matrix inverse and (3) generalized inverse of sum of two matrices in certain special cases (see page 50 in Campbell & Meyer book), as well as (4) Gromov’s h-principle; but our algorithm is new & distinct from these methods.
96
105
106
107
d = 12, ny = 3, y = x
3, SNR = 20dB
Dim
en
sio
nle
ss
Err
or cubic measurement
nonlinearity
102
103
104
105
103
104
Dim
en
sio
nle
ss
Err
or
Number of Particles
EKF
PF
97
finite dimensional parametric approximation:
2
2
)()(x
logplog(h)J
)b()xA()f(x,let
)(log
)log(
rATrbAx
rfdivfx
phJ
+++∂
∂+=
+≈
++∂
∂+=
λλλ
98
3
6
j
T
donly toreduced becan thisA sparse
forbut ,d is complexity nalcomputatio however, )5(
)(Tr(A) div(f) that note (4)
stablerobustly flow make penalty to add also could (3)
flow ofstability force to-BBAlet could (2)
xparticleeach at J minimize tob & Afor solve )1(
x
A∑==
=
∂
λ
:flow)exact our for usual (as EKFan usingGaussian
h& gith solution wexact our from computed are b &A in which
)c(x,)b()xA()f(x,let ,generality of lossWithout
)(log)(x
)logp(x,
:PDE following thesatisfies that )f(x, flow a find want toWe
xhfdivf
∂
++=
−=+∂
∂
λλλλ
λ
λ
hybrid method:
2#
#
#
log/
loglog
:(.) of inverse-pseudo thedenotes (.)in which
)log()()(log
x
logpc
:)c(x, norm minimum thecompute and 0,div(c) that Assume
)log()()()(x
logp
x
p
x
p
x
p
hATrbAxx
p
hcdivATrcbAx
T
∂
∂
∂
∂=
∂
∂
+++
∂
∂
∂
∂−≈
≈
−=++++∂
∂
λ
99
recall derivation of incompressible flow:
∂
∂−−=−
=
λ
λ
λλ
fx
pfdiv
d
Kdh
xfd
dx
log)(
)(log)log(
),(
−
∂
∂−=
=
∂
λ
λ
λ
d
Kdh
x
pf
xd
)(loglog
log
:inverse-pseudo theusing ffor solve and
0,div(f) ible,incompress is flow that theassume
#
100
recall incompressible particle flow:
2log
log)(log)(log
d
dx
∂
∂
∂
−−
= λ
λ
λ px
p
d
Kdxh
T
gradient zerofor 0
2logd
=
∂
∂=
λ
λ
d
dx
x
p
101
new derivation of compressible flow:
∂
∂−−=−
=
λ
λ
λλ
fx
pfdiv
d
Kdh
xfd
dx
log)(
)(log)log(
),(
−
+
∂
∂−=
∂
λ
λ
λ
d
Kdhdiv
x
pf
xd
)(loglog
log
:inverse-pseudo theusing ffor solve and
ible,incompress is flow that theassumenot willwe
#
102
particle flow filter
• orders of magnitude faster than standard particle filters
• orders of magnitude more accurate than the extended
Kalman filter for difficult nonlinear problems
• solves particle degeneracy problem using particle flow
induced by log-homotopy for Bayes’ rule
• no resampling of particles• no resampling of particles
• no proposal density
• no importance sampling & no MCMC methods
• unnormalized log probability density
• embarrassingly parallelizable w/o resampling bottleneck
(unlike other particle filters)
• exploits smoothness & regularity of densities
103
effect of divergence of f:
log)()log(
),(
#
fx
pfdivh
xfd
dx
∂
∂
∂−−=
= λλ
suppose
that div(f) is
given
[ ])(loglog
#
fdivhx
pf +
∂
∂−=
effect of div(f)
is to change
the speed of
particle flow
if div(f) = 0
we get
incompressible
flow104
incompressible particle flow
2
log)log(
dx
∂
∂
∂−
= x
ph
T
gradient zerofor 0
2logd
=
∂
∂ ∂=
λ
λ
d
dx
x
px
105
103
104
km
angles only tracking
d = 6 and N = 500 particles
exact flow filter
0 50 100 150 200 250 300 350 40010
1
102
Time (sec)
EKF: Solid
Log Homotopy PF: Dashed
Tincknell Angle = 90 deg
Tincknell Angle = 75 deg
Tincknell Angle = 25 deg
Tincknell Angle = 10 deg
median range error
(100 Monte Carlo runs)
106
103
104
km
Tincknell angle = 90 deg
angles only tracking
d = 6 and N = 500 particles
exact flow filter
0 50 100 150 200 250 300 350 40010
1
102
Time (sec)
km
EKF
Log Homotopy PF
median range error
(100 Monte Carlo runs)
107
103
104
km
Tincknell angle = 75 deg
angles only tracking
d = 6 and N = 500 particles
exact flow filter
0 50 100 150 200 250 300 350 40010
1
102
Time (sec)
km
EKF
Log Homotopy PF
median range error
(100 Monte Carlo runs)
108
103
104
km
Tincknell angle = 25 deg
angles only tracking
d = 6 and N = 500 particles
exact flow filter
0 50 100 150 200 250 300 350 40010
1
102
Time (sec)
km
EKF
Log Homotopy PF
median range error
(100 Monte Carlo runs)
109
103
104
km
Tincknell angle = 10 deg
angles only tracking
d = 6 and N = 500 particles
exact flow filter
0 50 100 150 200 250 300 350 40010
1
102
Time (sec)
km
EKF
Log Homotopy PF
median range error
(100 Monte Carlo runs)
110
exact particle flow for Gaussian densities:
fx
pfdivh
xfd
dx
∂
∂−−=
=
:exactly ffor solvecan weGaussian,h & gfor
log)()log(
),( λλ
[ ]
( ) ( )[ ]xAzRPHAIAIb
HRHPHPHA
bAxf
T
TT
+++=
+−=
+=
−
−
1
1
2
2
1
:exactly ffor solvecan weGaussian,h & gfor
λλ
λ
111
automatically stable
under very mild
conditions &
extremely fast
0
0.2
0.4
0.6
0.8Inside = 5 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
112
0
0.2
0.4
0.6
0.8Inside = 10.6 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
113
0
0.2
0.4
0.6
0.8Inside = 13.8 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
114
0
0.2
0.4
0.6
0.8Inside = 16.6 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
115
0
0.2
0.4
0.6
0.8Inside = 17.6 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
116
0
0.2
0.4
0.6
0.8Inside = 21 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
117
0
0.2
0.4
0.6
0.8Inside = 21 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
118
0
0.2
0.4
0.6
0.8Inside = 23.8 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
119
0
0.2
0.4
0.6
0.8Inside = 26.4 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
120
0
0.2
0.4
0.6
0.8Inside = 27.6 percent, Magenta: truth, Green: PF estimate, Black: KF
Ax+
b
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Ax+
b
121
incompressible particle flow
2
log)log(
dx
∂
∂
∂−
= x
ph
T
gradient zerofor 0
2logd
=
∂
∂ ∂=
λ
λ
d
dx
x
px
122
0
0.2
0.4
0.6
0.8Inside = 6 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
123
0
0.2
0.4
0.6
0.8Inside = 5.8 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
124
0
0.2
0.4
0.6
0.8Inside = 8.2 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
125
0
0.2
0.4
0.6
0.8Inside = 9.2 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
126
0
0.2
0.4
0.6
0.8Inside = 11.2 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
127
0
0.2
0.4
0.6
0.8Inside = 11.8 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
128
0
0.2
0.4
0.6
0.8Inside = 12.8 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
129
0
0.2
0.4
0.6
0.8Inside = 12 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
130
0
0.2
0.4
0.6
0.8Inside = 11.6 percent, Magenta: truth, Green: PF estimate, Black: KF
Hessia
n
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.8
-0.6
-0.4
-0.2
Hessia
n
131
variation in SNR
101
102
Dim
en
sio
nle
ss E
rro
r
N = 1000, Stable, d = 10, Linear
SNR = 10 dB
15 dB
20 dB
30 dB
, λλλλ = 0.6
25 Monte Carlo Trials
0 5 10 15 20 25 3010
-1
100
Time
Dim
en
sio
nle
ss E
rro
r
132
variation in process noise
103
104
105
106
107
Dim
en
sio
nle
ss E
rro
r
N = 1000, Stable, d = 10, Quadratic
σσσσq = 0.01X
0.1X
1X
10X
100X
, λλλλ = 0.6
25 Monte Carlo Trials
0 5 10 15 20 25 3010
-1
100
101
102
Time
Dim
en
sio
nle
ss E
rro
r
133
particle flow filter
• orders of magnitude faster than standard particle filters
• orders of magnitude more accurate than the extended
Kalman filter for difficult nonlinear problems
• solves particle degeneracy problem using particle flow
induced by log-homotopy for Bayes’ rule
• no resampling of particles• no resampling of particles
• no proposal density
• no importance sampling & no MCMC methods
• unnormalized log probability density
• embarrassingly parallelizable w/o resampling bottleneck
(unlike other particle filters)
• exploits smoothness & regularity of densities
134
fundamental PDE for exact particle flow:
x
pfTrxp
xp
x
pfTr
xp
xfd
dx
∂
∂−=
∂
∂
∂
∂−=
∂
∂
=
)(),(
),(log
)(),(
),(
λλ
λ
λ
λ
λλ
Fokker-Planck
equation with Q = 0
assume
log-homotopy
fx
pfdivh
fx
p
x
fTrxpxpxh
xhxgxp
xTrxp
∂
∂−−=
∂
∂−
∂
∂−=
+=
∂−=
∂
log)()log(
),(),()(log
)(log)(log),(log
),(
λλ
λλ
λλ
log-homotopy
first order linear
underdetermined
PDE in f(x,λ)135
direct integration of fundamental PDE:
...
),(),(
)),((
:PDE theof form divergence theuse
2
2
1
1
∂
∂−=
∂
∂
∂
∂++
∂
∂+
∂
∂=
=
∂
∂=
∑≠
η
η
ληλ
λ
dkj
d
d
x
q
x
q
x
q
x
q
x
q
xx
xqTrxqdiv
pick for best
stability of
particle flow
[ ]
.0(x)dx iff
exists )q(x,for solution a , and on conditions regularity Assuming
)conditionsity compatibilfor (except function arbitrary )(
)()()(
=
Ω
=∂
∂=
−=
∂∂
∫
∑
∫
∑
Ω
≠
≠
η
λη
θ
θη
d
jk k
kj
j
x
jj
jk kj
x
qx
dxxxxq
xx
j
particle flow
136
more details of direct integration:
∫ ∫Ω
=
≥
−=
=
such thatfunction arbitrary )(x
in which
2kfor )()()(
)()(
kρ
ηρη
η
k
x
kkk dxdxxxxq
xqdiv
k
k
∫
∫
Ω
Ω
=
Ω
=
0(x)dx that is q(x)solution a of
existence for thecondition sufficient &necessary a
set,smooth connected open, bounded, is and
support,compact with functionssmooth assuming
1)(xk
k
η
ρ kdx
k
137
Oh’s Formula for Monte Carlo errors
Nkk
kd
/21
exp21
1 22
+
+
+≈
εσ
Assumptions:
(1) Gaussian density (zero mean & unit covariance matrix)
(2) d-dimensional random variable
(3) Proposal density is also Gaussian with mean ε and covariance matrix kI, but it is not exact for k ≠ 1 or ε ≠ 0
(4) N = number of Monte Carlo trials
138
nonlinear filtering problem
x = d-dimensional state vector
t = time
w(t) = process noise vector
dx = F(x, t)dt + G(x, t) dwcontinuous
time
dynamicsdiscrete
time
measurements
z(tk) = m-dimensional measurement vector
tk = time of kth measurement
vk = measurement noise vector
p(x, t | Zk) = probability density of x at time t given Zk
Zk = set of all measurements up to & including time tk
z(tk) = H(x(tk), tk, vk)
measurements
139
difficulties for exact finite dimensional filters
vs. particle filters
Bayes’ update of
conditional density
of x
prediction of
conditional density
of x with time
1. exact filters
(e.g., Daum 1986)
easy hard
(e.g., Daum 1986)
2. particle filters hard easy
3. hybrid of exact
& particle filters
? ?
140
What is a particle filter?
141
Prediction of
conditional
probability
density from
tk-1 to tk
Update conditional
probability density
particles particles
Particle Filter
Probability
density is
represented
by particles
measurements
Importance sampling
from proposal density
(Monte Carlo or QMC
sampling)
probability density
using current
measurement
& Bayes’ rule
particles
Monte
Carlo
or QMC
simulation
of dynamics
142
103
104
105
Velo
cit
y E
rro
r (m
/sec)N = 100, σσσσ
r = 100m
EKF
Inexact Flow
Exact Flow
Exact Flow with Redraw
radar tracking ballistic missile
(d =6 & N = 100 particles)
incompressible
flow
0 20 40 60 80 10010
0
101
102
10
Time (sec)
Velo
cit
y E
rro
r (m
/sec)
143
exact
flow
EKF
method to solve PDE how to pick unique solution computation
1. generalized inverse of linear differential
operator
minimum norm* Coulomb’s law or fast Poisson solver
2. Poisson’s equation gradient of potential*
(assume irrotational flow)
Coulomb’s law or fast Poisson solver
3. generalized inverse of gradient of log-
homotopy
assume incompressible flow (i.e.,
divergence free flow)
fast (but need to compute the gradient
of logp(x, λ) from random points)
4. most general solution most robustly stable filter or random
pick, etc.
fast (but need to compute the gradient
of logp(x, λ) from random points)
5. separation of variables (Gaussian) pick solution of specific form
(polynomial)
extremely fast (formula for flow)
6. separation of variables
(exponential family)
pick solution of specific form (finite basis
functions)
very fast (formula for flow)
7. variational formulation (Gauss & Hertz) convex function minimization ODEs
8. optimal control formulation convex functional minimization (e.g.,
least action like Monge-Kantorovich)
HJB PDE or Euler-Lagrange PDEs
(or maybe ODES for nice problem?)
9. direct integration (of first order linear PDE in
divergence form)
choice of d-1 arbitrary functions one-dimensional integral
10. generalized method of characteristics more conditions (e.g., curl = given &
chain rule)
ODEs from chain rule
11. another homotopy (inspired by Gromov’s
h-principle) like Feynman’s QED perturbation
initial condition of ODE &
uniqueness of sol. to ODE
ODEs from homotopy
12. finite dimensional parametric flow
(e.g., f = Ax+b with A & b parameters)
non-singular matrix to invert d³ or d^6 (least squares for d or d² parameters, i.e., A & b)
13. Fourier transform of PDE (divergence form
of linear PDE has constant coefficients!)
minimum norm* or most stable flow Coulomb’s law or fast Poisson solver144
(1) Flavia Lanzara , Vladimir Maz’ya & Gunther Schmidt
“On the fast computation of high dimensional volume
potentials” arXiv:0911.0443v1 [math.NA] 2 Nov 2009
note: linear computational complexity in d for uniform grid,
and it can be extended to scattered data!
145
and it can be extended to scattered data!
(2) huge literature on fast Poisson solvers (e.g., FMM
Rokhlin, Beylkin, Coifman, Hackbush, et al.)