Machine Learning for Signal Processing Neural Networks Continue Instructor: Bhiksha Raj Slides by Najim Dehak 1 Dec 2016 1 So what are neural networks?? Voice N.Net Image N.Net Text caption signal Transcription Game N.Net State Next move • What are these boxes? 18797/11755 2 So what are neural networks?? • It began with this.. • Humans are very good at the tasks we just saw • Can we model the human brain/ human intelligence? – An old question – dating18797/11755 back to Plato and Aristotle.. 3 MLP - Recap • MLPs are Boolean machines – They represent Boolean functions over linear boundaries – They can represent arbitrary boundaries • Perceptrons are correlation filters – They detect patterns in the input • MLPs are Boolean formulae over patterns detected by perceptron – Higher-level perceptrons may also be viewed as feature detectors • MLPs are universal approximators – Can model any function to arbitrary precision • Extra: MLP in classification – The network will fire if the combination of the detected basic features matches an “acceptable” pattern for a desired class of signal • E.g. Appropriate combinations of (Nose, Eyes, Eyebrows, Cheek, Chin) Face 4 MLP - Recap • MLPs are Boolean machines – They represent arbitrary Boolean functions over arbitrary linear boundaries • Perceptrons are pattern detectors – MLPs are Boolean formulae over these patterns • MLPs are universal approximators – Can model any function to arbitrary precision • MLPs are very hard to train – Training data are generally many orders of magnitude too few – Even with optimal architectures, we could get rubbish – Depth helps greatly! – Can learn functions that regular classifiers cannot 5 What is a deep network? Deep Structures • In any directed network of computational elements with input source nodes and output sink nodes, “depth” is the length of the longest path from a source to a sink • Left: Depth = 2. Right: Depth = 3 Deep Structures • Layered deep structure • “Deep” Depth > 2 MLP as a continuous-valued regression T1 T1 x 1 1 f(x) x T1 T2 x 1 -1 T2 + T2 • MLPs can actually compose arbitrary functions to arbitrary precision – Not just classification/Boolean functions • 1D example – Left: A net with a pair of units can create a pulse of any width at any location – Right: A network of N such pairs approximates the function with N scaled pulses 9 MLP features DIGIT OR NOT? • The lowest layers of a network detect significant features in the signal • The signal could be reconstructed using these features – Will retain all the significant components of the signal 10 Making it explicit: an autoencoder 푿 푻 푾 풀 푾 푿 • A neural network can be trained to predict the input itself • This is an autoencoder • An encoder learns to detect all the most significant patterns in the signals • A decoder recomposes the signal from the patterns 11 Deep Autoencoder DECODER ENCODER What does the AE learn 푿 푻 푾 풀 푾 푿 푇 2 퐘 = 퐖퐗 퐗 = 퐖푇퐘 퐸 = 퐗 − 퐖 퐖퐗 Find W to minimize Avg[E] • In the absence of an intermediate non-linearity • This is just PCA 13 The AE DECODER ENCODER • With non-linearity – “Non linear” PCA – Deeper networks can capture more complicated manifolds 14 The Decoder: DECODER • The decoder represents a source-specific generative dictionary • Exciting it will produce typical signals from the source! 15 The AE DECODER Cut the AE ENCODER 16 The Decoder: Sax dictionary DECODER • The decoder represents a source-specific generative dictionary • Exciting it will produce typical signals from the source! 17 The Decoder: Clarinet dictionary DECODER • The decoder represents a source-specific generative dictionary • Exciting it will produce typical signals from the source! 18 NN for speech enhancement 19 Story so far • MLPs are universal classifiers – They can model any decision boundary • Neural networks are universal approximators – They can model any regression • The decoder of MLP autoencoders represent a non-linear constructive dictionary! 20 The need for shift invariance = • In many problems the location of a pattern is not important – Only the presence of the pattern • Conventional MLPs are sensitive to the location of the pattern – Moving it by one component results in an entirely different input that the MLP wont recognize • Requirement: Network must be shift invariant Convolutional Neural Networks History Hubel and Wiesel: 1959 (biological model), Fukushima: 1980 (computational model), Altas: 1988, Lecunn: 1989 (Backprop in convnets) Yann LeCun Kunihiko Fukushima Convolutional Neural Networks • A special kind of multi-layer neural networks. • Implicitly extract relevant features. • A feed-forward network that can extract topological properties from an image. • CNNs are also trained with a version of back-propagation algorithm. Connectivity & weight sharing All different weights All different weights Shared weights Convolution layer has much smaller number of parameters by local connection and weight sharing Fully Connected Layer Example: 200x200 image 40K hidden units ~2B parameters!!! - Spatial correlation is local - Waste of resources + we have not enough training samples anyway.. Ranzato 25 Locally Connected Layer Example: 200x200 image 40K hidden units Filter size: 10x10 4M parameters Note: This parameterization is good when input image is registered (e.g., face recognition). Ranzato 26 Locally Connected Layer STATIONARITY? Statistics is similar at different locations Example: 200x200 image 40K hidden units Filter size: 10x10 4M parameters Ranzato 27 Convolutional Layer Share the same parameters across different locations (assuming input is stationary): Convolutions with learned kernels Ranzato 28 Convolution Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Ranzato Convolutional Layer Learn multiple filters. E.g.: 200x200 image 100 Filters Filter size: 10x10 10K parameters Ranzato 46 Convolutional Layers before: output layer input layer hidden layer now: Convolution Layer 32x32x3 image 32 height 32 width 3 depth Convolution Layer 32x32x3 image 5x5x3 filter 32 Convolve the filter with the image i.e. “slide over the image spatially, computing dot products” 32 3 Convolution Layer Filters always extend the full depth of the input volume 32x32x3 image 5x5x3 filter 32 Convolve the filter with the image i.e. “slide over the image spatially, computing dot products” 32 3 Convolution Layer 32x32x3 image 5x5x3 filter 32 1 number: the result of taking a dot product between the filter and a small 5x5x3 chunk of the image 32 (i.e. 5*5*3 = 75-dimensional dot product + bias) 3 Convolution Layer activation map 32x32x3 image 5x5x3 filter 32 28 convolve (slide) over all spatial locations 32 28 3 1 Convolution Layer consider a second, green filter 32x32x3 image activation maps 5x5x3 filter 32 28 convolve (slide) over all spatial locations 32 28 3 1 Convolution Layer For example, if we had 6 5x5 filters, we’ll get 6 separate activation maps: activation maps 32 28 Convolution Layer 32 28 3 6 We stack these up to get a “new image” of size 28x28x6! CNN Preview: ConvNet is a sequence of Convolution Layers, interspersed with activation functions 32 28 CONV, ReLU e.g. 6 5x5x3 32 filters 28 3 6 CNN Preview: ConvNet is a sequence of Convolutional Layers, interspersed with activation functions 32 28 24 …. CONV, CONV, CONV, ReLU ReLU ReLU e.g. 6 e.g. 10 5x5x3 5x5x6 32 filters 28 filters 24 3 6 10 Pooling Layer Let us assume filter is an “eye” detector. Q.: how can we make the detection robust to the exact location of the eye? Ranzato 57 Pooling Layer By “pooling” (e.g., taking max) filter responses at different locations we gain robustness to the exact spatial location of features. Ranzato 58 Pooling Layer - makes the representations smaller and more manageable - operates over each activation map independently: Max Pooling Single depth slice 1 1 2 4 x max pool with 2x2 filters 5 6 7 8 and stride 2 6 8 3 2 1 0 3 4 1 2 3 4 y ConvNets: Typical Stage One stage (zoom) Convol. Pooling courtesy of K. Kavukcuoglu Ranzato 61 Digit classification ImageNet • 1.2 million high-resolution images from ImageNet LSVRC-2010 contest • 1000 different classes (sofmax layer) • NN configuration • NN contains 60 million parameters and 650,000 neurons, • 5 convolutional layers, some of which are followed by max-pooling layers • 3 fully-connected layers Krizhevsky, A., Sutskever, I. and Hinton, G. E. “ImageNet Classification with Deep Convolutional Neural Networks” NIPS 2012: Neural Information Processing Systems, Lake Tahoe, Nevada ImageNet Figure 3: 96 convolutional kernels of size 11×11×3 learned by the first convolutional layer on the 224×224×3 input images. The top 48 kernels were learned on GPU 1 while the bottom 48 kernels were learned on GPU 2. See Section 6.1 for details. Krizhevsky, A., Sutskever, I. and Hinton, G. E. “ImageNet Classification with Deep Convolutional Neural Networks” NIPS 2012: Neural Information Processing Systems, Lake Tahoe, Nevada ImageNet Eight ILSVRC-2010 test images and the five Five ILSVRC-2010 test images in the first labels considered most probable by our model. column. The remaining columns show the six The correct label is written under each image, training images that produce feature vectors