SurReal: Fréchet Mean and Distance Transform for Complex ...stellayu/... · 1 New complex-valued...
1
SurReal: Fréchet Mean and Distance Transform for Complex-Valued Deep Learning Rudrasis Chakraborty, Jiayun Wang, Stella X. Yu UC Berkeley / ICSI Overview 1 New complex-valued deep learning theory that handles scaling ambiguity with equivariance and invariance properties on a manifold. 2 Sur-real experimental validation with significant performance gain (94% → 98%) at a fraction (8%) of the baseline model size. New Deep Learning Theory 1 New manifold representation R + × SO(2). a + ib F 7→ ( r , R(θ )) , r =( a + ib )= p a 2 + b 2 θ = arg( a + ib )= atan2( b , a ) R(θ )= cos(θ ) - sin(θ ) sin(θ ) cos(θ ) . 2 New group action G for complex scaling: the product of planar rotation and scaling. 3 New convolution that is equivariant to G : weighted Fréchet Mean filtering: wFM ({z i } , { w i })= arg min m∈C K X i =1 w i d 2 (z i , m) , where K X i =1 w i = 1, w i ∈ (0, 1], ∀ i d (z 1 , z 2 )= v u t log 2 r 2 r 1 + klogm R -1 1 R 2 k 2 . 4 New fully connected layer operator that is invariant to G : distance to the wFM. m = wFM({t i }, { v i }) u i = d (t i , m). 5 New nonlinear activation function: ReLU in the tangent space, log/exp maps back to manifold. ( r , R) 7→ (exp (ReLU (log( r ))) , expm (ReLU (logm (R))) Schematic of Our CNN wFM C t 1 t d t u y c y 1 Invariant layer FC layer Conv-layers Conv-layers Cascaded Conv-layers x 1;1 x 7;8 wFM wFM (fX g ; ) x 1;1 x 7;8 Equivariance of Fréchet Mean Filtering Invariance of Distance Transform Rotation and scaling wFM(z;w ) z 1 z 4 g:z 1 g:z 2 g:wFM(z;w ) z 2 z 3 z 5 g:z 3 g:z 5 g:z 4 d 1 d 2 d 3 d 4 d 5 d 1 d 2 d 3 d 4 d 5 Tangent ReLU for the Complex Plane Our Results on MSTAR 100 × 100 × 1 49 × 49 × 16 23 × 23 × 32 11 × 11 × 64 4 × 4 × 128 256 FC FC 512 CBRM CBRM CBRM CBRM CBRM Softmax layer FC 10 100 × 100 × 1 CCtR 20 × 20 × 50 CCtR CCtR 4 × 4 × 100 dFM 200 Softmax layer FC 10 0% 20% 40% 60% 80% 100% (a, b) r (a, b, r) (r, θ) z (a, b) z raw data unit-norm data 98.2% 97.0% z z 93.5% 96.9% 94.5% 89.8% 46.0% Our Results on RadioML 128 × 2 CBRM CBRM 130 × 256 FC FC 11 Softmax layer 256 128 × 1 CCtR 25 × 64 CCtR CCtR 5 × 128 dFM 640 Softmax layer FC 11
Transcript of SurReal: Fréchet Mean and Distance Transform for Complex ...stellayu/... · 1 New complex-valued...
SurReal: Fréchet Mean and Distance Transform for Complex-Valued Deep LearningRudrasis Chakraborty, Jiayun Wang, Stella X. Yu
UC Berkeley / ICSI
Overview
1 New complex-valued deep learning theory thathandles scaling ambiguity with equivarianceand invariance properties on a manifold.
2 Sur-real experimental validation withsignificant performance gain (94%→ 98%) ata fraction (8%) of the baseline model size.
New Deep Learning Theory
1 New manifold representation R+× SO(2).
a+ i bF7→ (r, R(θ )) ,
r = (a+ i b) =p
a2+ b2
θ = arg(a+ i b) = atan2(b, a)
R(θ ) =
�
cos(θ ) − sin(θ )sin(θ ) cos(θ )
�
.
2 New group action G for complex scaling: theproduct of planar rotation and scaling.
3 New convolution that is equivariant to G:weighted Fréchet Mean filtering:
wFM ({zi} , {wi}) = argminm∈C
K∑
i=1
wid2 (zi,m) ,
whereK∑
i=1
wi = 1, wi ∈ (0,1],∀i
d(z1,z2) =
√
√
√
log2
�
r2
r1
�
+ ‖logm�
R−11 R2
�
‖2.
4 New fully connected layer operator that isinvariant to G: distance to the wFM.
m= wFM({ti}, {vi})ui = d(ti,m).
5 New nonlinear activation function: ReLU in thetangent space, log/exp maps back to manifold.
(r, R) 7→ (exp (ReLU (log(r))) , expm (ReLU (logm (R))) 2005/06/28ver : 1.3sub f i gpackage
Schematic of Our CNN
wFM
C
t1
td
tu
yc
y1
Invariant layer
FC
layer
Conv-layers
Conv-layers
Cascaded Conv-layers
x1;1
x7;8
wFM
wFM (fXg ; )
x1;1
x7;8
Equivariance of Fréchet Mean FilteringInvariance of Distance Transform
Rotation and scaling
wFM(z; w)
z1
z4
g:z1
g:z2
g:wFM(z; w)
z2z3
z5
g:z3
g:z5
g:z4
d1
d2d3
d4 d5
d1
d2
d3
d4
d5
Tangent ReLU for the Complex Plane
Our Results on MSTAR
100× 100× 1 49× 49× 1623× 23× 32
11× 11× 64
4× 4× 128
256
FCFC
512
CBRMCBRM
CBRMCBRM
CBRM
Softmax layer
FC
10
100× 100× 1
CCtR
20× 20× 50
CCtRCCtR
4× 4× 100
dFM
200
Softmax layer
FC
10
0%
20%
40%
60%
80%
100%
(a, b) r (a, b, r) (r, θ) z (a, b) z
raw data unit-norm data
98.2% 97.0%
z z
93.5%96.9%94.5%89.8% 46.0%
Our Results on RadioML
128× 2
CBRM
CBRM
130× 256
FC
FC
11
Softmax layer
256
128× 1
CCtR
25× 64
CCtRCCtR
5× 128
dFM
640
Softmax layer
FC
11
![On Translation Invariance in CNNs: Convolutional Layers ...€¦ · and subsampling in CNN layers translation equivariance is lost [5, 109]. In our paper, we also investigate the](https://static.fdocuments.us/doc/165x107/60868128fa11d83f925c9ce6/on-translation-invariance-in-cnns-convolutional-layers-and-subsampling-in-cnn.jpg)
![arXiv:1907.08175v3 [cs.CV] 24 Dec 2019propose the Fréchet Joint Distance (FJD), which is defined as the Fréchet distance between joint distributions of images and conditioning,](https://static.fdocuments.us/doc/165x107/5e6f310792ebeb584c4cc14b/arxiv190708175v3-cscv-24-dec-2019-propose-the-frchet-joint-distance-fjd.jpg)
