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Lecture 7: Convolutional Networks Justin Johnson Lecture 7 - 1 September 24, 2019 Reminder: A2 Due Monday, September 30, 11:59pm (Even if you enrolled late!) Your submission must pass the validation script Justin Johnson Lecture 7 - 2

  1. Lecture 7: Convolutional Networks Justin Johnson Lecture 7 - 1 September 24, 2019

  2. Reminder: A2 Due Monday, September 30, 11:59pm (Even if you enrolled late!) Your submission must pass the validation script Justin Johnson Lecture 7 - 2 September 24, 2019

  3. Slight schedule change Content originally planned for today got split into two lectures Pushes the schedule back a bit: A4 Due Date: Friday 11/1 -> Friday 11/8 A5 Due Date: Friday 11/15 -> Friday 11/22 A6 Due Date: Still Friday 12/6 Justin Johnson Lecture 7 - 3 September 24, 2019

  4. Last Time: Backpropagation During the backward pass, each node in the graph receives upstream gradients Represent complex expressions and multiplies them by local gradients to as computational graphs compute downstream gradients x s (scores) * hinge L + loss W R f Downstream gradients Local gradients Forward pass computes outputs Upstream gradient Backward pass computes gradients Justin Johnson Lecture 7 - 4 September 24, 2019

  5. Stretch pixels into column f(x,W) = Wx 56 Problem : So far our 56 231 classifiers don’t 231 respect the spatial 24 2 24 structure of images! Input image 2 (2, 2) (4,) Input: x h s W 1 W 2 3072 Output: 10 Hidden layer: 100 Justin Johnson Lecture 7 - 5 September 24, 2019

  6. Stretch pixels into column f(x,W) = Wx 56 Problem : So far our 56 231 classifiers don’t 231 respect the spatial 24 2 24 structure of images! Input image 2 (2, 2) Solution : Define new computational nodes (4,) Input: x that operate on images! h s W 1 W 2 3072 Output: 10 Hidden layer: 100 Justin Johnson Lecture 7 - 6 September 24, 2019

  7. Components of a Full-Connected Network Fully-Connected Layers Activation Function x h s Justin Johnson Lecture 7 - 7 September 24, 2019

  8. Components of a Convolutional Network Fully-Connected Layers Activation Function x h s Pooling Layers Normalization Convolution Layers Justin Johnson Lecture 7 - 8 September 24, 2019

  9. Components of a Convolutional Network Fully-Connected Layers Activation Function x h s Pooling Layers Normalization Convolution Layers Justin Johnson Lecture 7 - 9 September 24, 2019

  10. Fully-Connected Layer 32x32x3 image -> stretch to 3072 x 1 Input Output 1 1 10 x 3072 3072 10 weights Justin Johnson Lecture 7 - 10 September 24, 2019

  11. Fully-Connected Layer 32x32x3 image -> stretch to 3072 x 1 Input Output 1 1 10 x 3072 3072 10 weights 1 number: the result of taking a dot product between a row of W and the input (a 3072- dimensional dot product) Justin Johnson Lecture 7 - 11 September 24, 2019

  12. Convolution Layer 3x32x32 image: preserve spatial structure 32 height width 32 depth / 3 channels Justin Johnson Lecture 7 - 12 September 24, 2019

  13. Convolution Layer 3x32x32 image 3x5x5 filter Convolve the filter with the image i.e. “slide over the image spatially, 32 height computing dot products” width 32 depth / 3 channels Justin Johnson Lecture 7 - 13 September 24, 2019

  14. Convolution Layer Filters always extend the full depth of the input volume 3x32x32 image 3x5x5 filter Convolve the filter with the image i.e. “slide over the image spatially, 32 height computing dot products” width 32 depth / 3 channels Justin Johnson Lecture 7 - 14 September 24, 2019

  15. Convolution Layer 3x32x32 image 3x5x5 filter 1 number: 32 the result of taking a dot product between the filter and a small 3x5x5 chunk of the image (i.e. 3*5*5 = 75-dimensional dot product + bias) 32 3 Justin Johnson Lecture 7 - 15 September 24, 2019

  16. Convolution Layer 1x28x28 activation map 3x32x32 image 3x5x5 filter 28 convolve (slide) over 32 all spatial locations 28 32 1 3 Justin Johnson Lecture 7 - 16 September 24, 2019

  17. Convolution Layer two 1x28x28 activation map Consider repeating with 3x32x32 image a second (green) filter: 3x5x5 filter 28 28 convolve (slide) over 32 all spatial locations 28 32 1 1 3 Justin Johnson Lecture 7 - 17 September 24, 2019

  18. Convolution Layer 6 activation maps, each 1x28x28 3x32x32 image Consider 6 filters, each 3x5x5 Convolution Layer 32 6x3x5x5 32 Stack activations to get a filters 3 6x28x28 output image! Justin Johnson Lecture 7 - 18 September 24, 2019

  19. Convolution Layer 6 activation maps, each 1x28x28 3x32x32 image Also 6-dim bias vector: Convolution Layer 32 6x3x5x5 32 Stack activations to get a filters 3 6x28x28 output image! Justin Johnson Lecture 7 - 19 September 24, 2019

  20. 28x28 grid, at each Convolution Layer point a 6-dim vector 3x32x32 image Also 6-dim bias vector: Convolution Layer 32 6x3x5x5 32 Stack activations to get a filters 3 6x28x28 output image! Justin Johnson Lecture 7 - 20 September 24, 2019

  21. Convolution Layer 2x6x28x28 Batch of outputs 2x3x32x32 Also 6-dim bias vector: Batch of images Convolution Layer 32 6x3x5x5 32 filters 3 Justin Johnson Lecture 7 - 21 September 24, 2019

  22. Convolution Layer N x C out x H’ x W’ Batch of outputs N x C in x H x W Also C out -dim bias vector: Batch of images Convolution Layer H W C out x C in x K w x K h C out filters C in Justin Johnson Lecture 7 - 22 September 24, 2019

  23. Stacking Convolutions 32 28 26 …. Conv Conv Conv W 1 : 6x3x5x5 W 2 : 10x6x3x3 W 3 : 12x10x3x3 b 1 : 5 b 2 : 10 b 3 : 12 32 28 26 3 6 10 Input: First hidden layer: Second hidden layer: N x 3 x 32 x 32 N x 6 x 28 x 28 N x 10 x 26 x 26 Justin Johnson Lecture 7 - 23 September 24, 2019

  24. Q : What happens if we stack Stacking Convolutions two convolution layers? 32 28 26 …. Conv Conv Conv W 1 : 6x3x5x5 W 2 : 10x6x3x3 W 3 : 12x10x3x3 b 1 : 5 b 2 : 10 b 3 : 12 32 28 26 3 6 10 Input: First hidden layer: Second hidden layer: N x 3 x 32 x 32 N x 6 x 28 x 28 N x 10 x 26 x 26 Justin Johnson Lecture 7 - 24 September 24, 2019

  25. (Recall y=W 2 W 1 x is Q : What happens if we stack Stacking Convolutions a linear classifier) two convolution layers? A : We get another convolution! 32 28 26 …. Conv ReLU Conv ReLU Conv ReLU W 1 : 6x3x5x5 W 2 : 10x6x3x3 W 3 : 12x10x3x3 b 1 : 6 b 2 : 10 b 3 : 12 32 28 26 3 6 10 Input: First hidden layer: Second hidden layer: N x 3 x 32 x 32 N x 6 x 28 x 28 N x 10 x 26 x 26 Justin Johnson Lecture 7 - 25 September 24, 2019

  26. What do convolutional filters learn? 32 28 26 …. Conv ReLU Conv ReLU Conv ReLU W 1 : 6x3x5x5 W 2 : 10x6x3x3 W 3 : 12x10x3x3 b 1 : 6 b 2 : 10 b 3 : 12 32 28 26 3 6 10 Input: First hidden layer: Second hidden layer: N x 3 x 32 x 32 N x 6 x 28 x 28 N x 10 x 26 x 26 Justin Johnson Lecture 7 - 26 September 24, 2019

  27. What do convolutional filters learn? 32 28 Linear classifier: One template per class Conv ReLU W 1 : 6x3x5x5 b 1 : 6 32 28 3 6 Input: First hidden layer: N x 3 x 32 x 32 N x 6 x 28 x 28 Justin Johnson Lecture 7 - 27 September 24, 2019

  28. What do convolutional filters learn? MLP: Bank of whole-image templates 32 28 Conv ReLU W 1 : 6x3x5x5 b 1 : 6 32 28 3 6 Input: First hidden layer: N x 3 x 32 x 32 N x 6 x 28 x 28 Justin Johnson Lecture 7 - 28 September 24, 2019

  29. What do convolutional filters learn? First-layer conv filters: local image templates (Often learns oriented edges, opposing colors) 32 28 Conv ReLU W 1 : 6x3x5x5 b 1 : 6 32 28 3 6 Input: First hidden layer: AlexNet: 64 filters, each 3x11x11 N x 3 x 32 x 32 N x 6 x 28 x 28 Justin Johnson Lecture 7 - 29 September 24, 2019

  30. A closer look at spatial dimensions 32 28 Conv ReLU W 1 : 6x3x5x5 b 1 : 6 32 28 3 6 Input: First hidden layer: N x 3 x 32 x 32 N x 6 x 28 x 28 Justin Johnson Lecture 7 - 30 September 24, 2019

  31. A closer look at spatial dimensions Input: 7x7 Filter: 3x3 7 7 Justin Johnson Lecture 7 - 31 September 24, 2019

  32. A closer look at spatial dimensions Input: 7x7 Filter: 3x3 7 7 Justin Johnson Lecture 7 - 32 September 24, 2019

  33. A closer look at spatial dimensions Input: 7x7 Filter: 3x3 7 7 Justin Johnson Lecture 7 - 33 September 24, 2019

  34. A closer look at spatial dimensions Input: 7x7 Filter: 3x3 7 7 Justin Johnson Lecture 7 - 34 September 24, 2019

  35. A closer look at spatial dimensions Input: 7x7 Filter: 3x3 Output: 5x5 7 7 Justin Johnson Lecture 7 - 35 September 24, 2019

  36. A closer look at spatial dimensions Input: 7x7 Filter: 3x3 Output: 5x5 In general: Problem: Feature 7 maps “shrink” Input: W with each layer! Filter: K Output: W – K + 1 7 Justin Johnson Lecture 7 - 36 September 24, 2019

  37. A closer look at spatial dimensions 0 0 0 0 0 0 0 0 0 Input: 7x7 0 0 Filter: 3x3 0 0 Output: 5x5 0 0 In general: Problem: Feature 0 0 maps “shrink” Input: W with each layer! 0 0 Filter: K Output: W – K + 1 0 0 0 0 Solution: padding Add zeros around the input 0 0 0 0 0 0 0 0 0 Justin Johnson Lecture 7 - 37 September 24, 2019

  38. A closer look at spatial dimensions 0 0 0 0 0 0 0 0 0 Input: 7x7 0 0 Filter: 3x3 0 0 Output: 5x5 0 0 In general: Very common: 0 0 Set P = (K – 1) / 2 to Input: W make output have 0 0 Filter: K same size as input! 0 0 Padding: P Output: W – K + 1 + 2P 0 0 0 0 0 0 0 0 0 0 0 Justin Johnson Lecture 7 - 38 September 24, 2019

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