8. Convolutional NN

Good for graphs. Images = very structured graph

Goals:

• Want to predict the same label despite image translated/shifted by a few pixels
  • Invariance to translation
• Want segmentation (labeling pixels) to be translated along input images
  • Equivariance with translation

How?

• Augment with translated images
• Special regularization
• Invariance built into pre-processing
• Invariance in structure of NN ($⟹$ convolutional NN)

Convolution Layer

• Each unit only depends on all “closeby” inputs (eg. pixels
• Weights are reused (sparse)
• Convolution: weighted sum of nearby pixels
• Key idea: learn the filters

$$z_i=\sum _{j=\max (1,i-d+1)}^{\min (i,k)}{w}_j{x}_{i-j+1}$$

c: with padding, d: without padding

Ex.: $w=[-1,1]$ highlights edges. $[0,1,1]⇝[0,1,0]$ double check

$$\left(\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\end{array}\right)=\left[\begin{array}{ccc}{w}_2& 0& 0\\ w_1& w_2& 0\\ 0& w_1& w_2\\ 0& 0& w_1\end{array}\right]\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)$$

Stride: Skipping pixels in a convolutionary layers in regular steps

Computing output dimensions:

• Applying $m$ different $f\times f$ filters (convoluting by $f\times f\times m$ tensor)
• To an $n\times n$ image
• Padding $p$
• Stride $s$ $⇝$ $l\times l\times m$ tensor with $l=\frac{n+2p-f}{s}+1$

The # of inputs affecting each unit grows with every hidden unit.

Pooling Layers (Subsampling)

• Aggregate neighboring entries
• Consider either the average or maximum value of a group of neighboring pixels
  • Average is just another convolution

Common setup