POEM: 1-Bit Point-Wise Operations Based on E-M for Point Cloud Processing

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Algorithm 12 POEM training. L is the loss function (summation of LS and LR) and N

is the number of layers. Binarize() binarizes the filters obtained using the binarization Eq.

6.36, and Update() updates the parameters according to our update scheme.

Input: a minibatch of inputs and their labels, unbinarized weights w, scale factor α,

learning rates η.

Output:

updated

unbinarized

weights

w^t+1,

updated

scale

factor

α^t+1.

1: {1. Computing gradients with aspect to the parameters:}

2: {1.1. Forward propagation:}

3: for i =1 to N do

4:

b^wi ←Binarize(wi) (using Eq. 6.36)

5:

Bi-FC features calculation using Eq. 6.87 – 6.72

6:

Loss calculation using Eq. 6.88 – 6.44

7: end for

8: {1.2. Backward propagation:}

9: for i =N to 1 do

10:

{Note that the gradients are not binary.}

11:

Computing δw using Eq. 6.89 – 6.59

12:

Computing δα using Eq. 6.60 – 6.62

13:

Computing δp using Eq. 6.63 – 6.64

14: end for

15: {Accumulating the parameters gradients:}

16: for i = 1 to N do

17:

w^t+1 ←Update(δw, η) (using Eq. 6.89)

18:

α^t+1 ←Update(δα, η) (using Eq. 6.61)

19:

p^t+1 ←Update(δw, η) (using Eq. 6.64)

20:

η^t+1 ←Update(η) according to learning rate schedule

21: end for

Then, we optimize w^j

i ^as

δwj

i ^{= ∂LS}

∂w^j

i

+ λ^∂LR

∂w^j

i

+ τEM(w^j

i ^),

(6.58)

where τ is the hyperparameter to control the proportion of the Expectation-Maximization

operator EM(w^j

i ^{). EM(wj}

i ^{) is defined as}

EM(w^j

i ^{) =}

 2

k=1 ^ˆξjk

i ^(ˆμk

i ^−wj

i ^),

ˆμ¹

i ^{< wj}

i ^{< ˆμ2}

i

0,

else

.

(6.59)

Updating αi: We further update the scale factor αi with wi fixed. δαi is defined as the

gradient of αi, and we have

δαi = ^∂LS

∂αi

+ λ^∂LR

∂αi

(6.60)

αi ←|αi −ηδαi|,

(6.61)

where η is the learning rate. The gradient derived from softmax loss can be easily calculated

on the basis of backpropagation. Based on Eq. 6.44, we have

∂LR

∂αi

= (wi −αi ◦b^wi) · b^wi.

(6.62)