Urbashi Mitra Ming Hsieh Department of Electrical Engineering - - PowerPoint PPT Presentation

Urbashi Mitra

Ming Hsieh Department of Electrical Engineering University of Southern California, Los Angeles, USA

Matrix factoriza@on techniques for data clustering and radio map reconstruc@on with applica@ons to UAV placement

Acknowledgements This research has been funded in part by one or more of the following grants: ONR N00014-15-1-2550, NSF CNS-1213128, NSF CCF-1718560, NSF CCF-1410009 , NSF CPS-1446901, and AFOSR FA9550-12-1-0215.

with Jun8ng Chen (USC), Omid Esrafilian (Eurecom) & David Gesbert (EURECOM)

Explora8on-Exploita8on

Autonomous Underwater Vehicle (AUV) Ummaned Aerial Vehicle (UAV)

AUVs and Communica8ons

−1 1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 X (km) Y (km) Z (km)

AUVs as data mules

¨ Autonomous

Underwater Vehicle (AUV)

¨ Sensors in a field ¨ AUV needs to collect

data from sensors

¨ Underwater

communica@on hinders data collec@on

¨ How to traverse

field?

§ Path planning problem

Vehicle start Sensor 1 Sensor 2 Sensor 3 Dynamic source Trajectory

UAV Relay Placement

5

Op@mize the posi@on of a drone for a dynamic network over complex terrain

Key & Challenge: Know the shadowing & avoid it

(More generally, learn the fine-grained environment & adapt to it)

1 mile 400 feet

intelligent transporta@on Device-to-device connec@on, mmWave applica@on, etc. Search and Surveillance

6

400 600 800 1000

Longitude [m]

400 600 800 1000

Latitude [m]

5 15 25 35 45 Building height [m]

City model, top view

user BS

How to Efficiently Learn Radio Map?

Simula@on of a theore@cally simplified throughput map

(source: Nokia)

Simulated SNR map in a more realis@c sedng

When a global radio map is available, op@mal UAV placement straigheorward Challenges: accurate modeling of buildings (materials) and incomplete data sets

Is there an underlying model?

7

Probabilis@c model

(Hourani et. al’14 TCOML, Mozaffari et. al’16 ICC)

“Fingerprint” model

(Romero et. al’17 TSP)

Over simplified Least structured High complexity 3D terrain-map based

(Monserrat et. al.’15 IJAP)

Not robust to missing data large bias

Segmented Propaga8on Model

8

q Classical model (large-scale fading): , with

xi , where depend on various “scenarios”

q Segmented model (proposed): q Par88on the space of x into K disjoint segments Dk, each with an

individual propaga@on law [dB]

User xu

x

Drone

D1

D2

D3

du(x, xu)

S T D

• 1

d B

(Simulated data over a 3D terrain model by Feng et. al’06VTC)

α, β, σ

g(1)

u (x) = β1 + α1du(x)

g(2)

u (x) = β2 + α2du(x)

y(x) =

K

X

k=1

⇣ g(k)

u (x) = βk + αkdu(x) + ξk

⌘ I n x ∈ Dk

• y = β + αdu(x) + ξ

ξ ∼ N(0, σ2)

ˆ gu(x) =

K

X

k=1

• βk + αkdu(x)
• I{x ∈ Dk}

The Learning Problem

9

q Data: SNR measurement data set {UAV-user posi@on , channel

quality , , } (unlabeled!)

q Model to learn: q Special case: If are independent Gaussian,

maximum-likelihood es@ma@on yields (Chen et. al’17 ICC, Globecom):

where

1.6 1.8 2 2.2 2.4 2.6 2.8 Drone-user distance in log scale

• 130
• 120
• 110
• 100
• 90
• 80
• 70
• 60

Channel gain in dB

pk(x, y) =

1 √ 2πσk exp

⇢ − (y−α(k)

1

du(x)−α(k)

2

)2 2σ2

k

• maximize

{α(k)

1

,α(k)

2

,σk,πk,¯ z(l)

k }

N

X

l=1 K

X

k=1

¯ z(l)

k

h log πk + log pk(x(l), y(l)) i x(l)

l = 1, 2, . . . , N y(l)

ξk

regression problem clustering problem

+

K

X

k=1

ξkI{x ∈ Dk}

noise

Example When MLE Fails

q What if you don’t have perfect informa@on of the sample

loca8on?

q Strong errors even if is Gaussian q Other issues: huge complexity for large data sets (due to EM

itera@on…), strong assump@ons (Gaussian, etc.)

10

1.6 1.8 2 2.2 2.4 2.6 2.8

Drone-user distance in log scale

• 130
• 120
• 110
• 100
• 90
• 80
• 70
• 60

Channel gain in dB

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Drone-user distance in log scale

• 130
• 120
• 110
• 100
• 90
• 80
• 70
• 60

Channel gain in dB

perfect xu noisy ˆ

xu

mis-clustering desired clustering

ξk

Can We Break the Dependence?

11

Clustering Gauss Cond.

(α1, β1) (α2, β2) D1 D2

… … Regression Data set {posi@on , measurement } x(l) y(l) Ques@on: Can we obtain the regression parameters without relying

• n the temporary clustering result and Gaussian assump@on?

MLE Approach

Our Answer: Build A Feature Matrix

12

β0

1

β0

2

β0

M

β0 α

α2 α1 αM

V (αi, β0

j; λ) = N

X

l=1

exp

• − λ

⇥ y(l) − α ˆ du(x(l)) − β0⇤2

q Discre@ze the support of parameters q Determine how well the dataset agrees with αi β0

j

y = αdu(x) + β

S T D

• 1

d B

(Simulated data over a 3D terrain model by Feng et. al’06VTC)

Clustering as Peak Localiza8on from the Feature Matrix

13

1.6 1.8 2 2.2 2.4 2.6 2.8

Drone-user distance in log scale

• 130
• 120
• 110
• 100
• 90
• 80
• 70
• 60

Channel gain in dB

{(θ, c) : L(θ, c) → pi}

β0

1

β0

2

β0

M

β0

α2 α1 αM

(ˆ α(1), ˆ β(1)) (ˆ α(2), ˆ β(2))

feature matrix

• intui@on: subset à line model à local peak
• Joint regression clustering problem à mul@-peak localiza@on problem

log-distance channel quality y du(x) y = αdu(x) + β similarity func@on

• n model error

peaks at correct values scaqer plot of measurement data

Strategy

q Structured model: , where q Proposi8on (Chen&Mitra’18 SSP): Under small enough parameter

and some other regularity condi@ons, V(k) is a unimodal matrix

14

V = V(1) + V(2) + · · · + V(K) + N

Theorem (Chen&Mitra’18ArXiv): The dominant singular vectors of a unimodal matrix are unimodal

≈

K

X

k=1

vT

k

uk + · · · λ

V (k)

ij

=

N

X

l=1

exp

• − λ

⇥ y(l) − αi ˆ du(x(l)) − βj ⇤2 I{x(l) ∈ Dk}

component matrix for one data subset:

Induces a low-rank unimodal- structured model rank = number of clusters

V ≈

K

X

k=1

An Example

q Model to general training data:

where , for LOS condi@on, and

• therwise,

q Parameter:

15

y = α1 log10 d(x) + α2 + n

α(1) = (−22, −28), n ∼ N(0, 1)

α(2) = (−36, −22), n ∼ N(0, 82)

λ = 1/(2σ2), σ = 8 [dB]

”true” propaga@on map 20 * 20 feature matrix V

p1(x) = log10 d(x) − 2.2, p2(x) = 1

How to localize the peaks in a robust way?

Algorithm via Mul8-Peak Localiza8on

q Unimodal-constrained Matrix Factoriza8on (UMF) q Analy@cal form of the unimodal constraints q Projected gradient for UMF via fast unimodal

projec@on (Chen&Mitra’17 Asilomar)

16

minimize

{αk,uk,vk}

• PΩ( ˆ

H −

K

X

k=1

αkukvT

k )

• 2

F

subject to uk, vk are unimodal

0 ≤u1 ≤ u2 ≤ · · · ≤ us us+1 ≥ us+2 ≥ · · · ≥ uN ≥ 0

Unimodal cone intersect with a sphere in R3

sparse sample UMF

peak localzt.

{uk, vk}

(ˆ αk, ˆ βk) classifica@on

labels

ˆ Dk

KNN etc.

Algorithm: Projected Gradient

q Equivalent objec@ve func@on (S: “sample or not” indicator matrix) q Gradients q Projected gradient (to unimodal cone ) q Random ini@aliza@on to handle convergence issue q The projec@on onto the unimodal cone requires only O(M2K)

17

U

f(U, W) =

• S
• V UWT

2

F

∂ ∂Uf = 2(S V)W + 2(S (UWT))W ∂ ∂Wf = 2(ST VT)U + 2(ST (WUT))U U(t + 1) = PU n U(t) − µt ∂ ∂Uf(U(t), W(t))

• W(t + 1) = PU

n W(t) − νt ∂ ∂Wf(U(t + 1), W(t))

• PU{·}

Fast Unimodal Projec8on

q It is possible to do fast unimodal

projec@on with M2K complexity (Chen & Mitra ’17 Asilomar, Németh’10)

18

Unimodal cone intersec@ng with a sphere

Fast unimodal projec8on is hundred 8mes faster!

M M M M

minimize

y

kx yk2 subject to y 2 UM

PU

0 ≤u1 ≤ u2 ≤ · · · ≤ us us ≥ us+1 ≥ · · · ≥ uM ≥ 0

Applica8on to Radio Map Reconstruc8on w/o Knowing User Loca8on

19

Baseline: Naive matrix comple@on directly from

• riginal data

Target: Hidden radio map (noise corrupted) to be es@mated Proposed: Extract parameters from a transformed matrix à classifica@on on training data à KNN for LOS/NLOS classifica@on on global map

Sampling over 100 random loca@ons fails to recover the propaga@on paqern recover paqern + value

20

UAV placement vs Map building

peak finding trough finding

• p@mal placement of UAV relay is

where signal is weakest select the right similarity func@on

Summary

q A learning tool

§ Compressed data to a feature matrix to discover (clustered)

data structure

§ Proposed a unimodal-constrained matrix factoriza@on (UMF)

tool for component analysis

§ Provided a low complexity solu@on for non-parametric data

clustering

q Other applica8ons: nonparametric underwater source

localiza@on

q What next: go truly adap@ve/ac@ve, op@mize similarity

func@ons

21