Cheng Soon Ong & Christfried Webers. Canberra February June 2016

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1 c Cheng Soon Ong & Christfried Webers Research Group and College of Engineering and Computer Science Canberra February June (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 31

2 c Part I Neural Network 3 2of 31

3 Number of layers c expression of a function is compact when it has few computational elements, i.e. few degrees of freedom that need to be tuned by learning for a fixed number of training examples, expect that compact representations of the target function would yield better generalization Depends on the base operations allowed affine operations, sigmoid linear regression and logistic regression have depth 1 fixed kernel, affine operations kernel machines have two levels affine transformation followed by a non-linearity multi-layer perceptron usually has two layers 3of 31

4 Easier to represent with more layers c functions that can be compactly represented by a depth k architecture might require an exponential number of computational elements to be represented by a depth k 1 architecture Example, the d bit parity function { d parity : (b 1,..., b d ) {0, 1} d 1 if i=1 b i 0 otherwise is even Theorem: d-bit parity circuits of depth 2 have exponential size 4of 31

5 Recall: Multi-layer Neural Network Architecture M y k (x, w) = g w (2) kj h j=0 ( D i=0 ) w (1) ji x i where w now contains all weight and bias parameters. x D hidden units z M w (1) MD w (2) KM c y K inputs outputs y 1 x 1 x 0 z 1 w (2) 10 z 0 We could add more hidden layers 5of 31

6 Empirical observations - pre 2006 c Deep architectures get stuck in local minima or plateaus As architecture gets deeper, more difficult to obtain good generalisation Hard to initialise random weights well 1 or 2 hidden layers seem to perform better 2006: Unsupervised pre-training, find distributed representation 6of 31

7 Deep representation - intuition c Bengio, "Learning Deep Architectures for AI", of 31

8 Deep representation - practice c AlexNet / VGG-F network visualized by mneuron. 8of 31

9 Recall: PCA Idea: Linearly project the data points onto a lower dimensional subspace such that the variance of the projected data is maximised, or the distortion error from the projection is minimised. Both formulation lead to the same result. Need to find the lower dimensional subspace, called the principal subspace. c u 1 x 2 x n x n x 1 9of 31

10 Multiple PCA layers? - linear transforms Principle Component Analysis is a linear transformation (because it is a projection) The composite of two linear transformations is linear Linear transformations M : R m R n are matrices Let S and T be matrices of appropriate dimension such that ST is defined We show the property of additivity c ST(X + X ) = S(T(X + X )) = S(T(X) + T(X )) = S(T(X)) + S(T(X )) = ST(X) + ST(X ) Similarly for multiplication with a scalar 10of 31

11 Multiple PCA layers? - projection Let X T X = UΛU T be the eigenvalue decomposition of the covariance matrix. Define U k to be the matrix formed by the first k columns of U, corresponding to the k largest eigenvalues. Define Λ k similarly Consider the features formed by projecting onto the principal components c Z = XU k We perform PCA a second time, Z T Z = VΛ Z V T. By the definition of Z and X T X, we have Z T Z = (XU k ) T (XU k ) = U T k X T XU k = U T k UΛU T U k Since the rows and columns respectively of U T k U are orthonormal, the right expression is Λ k. Hence Λ Z = Λ k and V is the identity, therefore the second PCA has no effect 11of 31

12 An autoencoder is trained to encode the input x into some representation c(x) so that the input can be reconstructed from that representation the target output of the autoencoder is the autoencoder input itself With one linear hidden layer and the mean squared error criterion, the k hidden units learn to project the input in the span of the first k principal components of the data If the hidden layer is nonlinear, the autoencoder behaves differently from PCA, with the ability to capture multimodal aspects of the input distribution Let f be the decoder. We want to minimise the reconstruction error c N l (x n, f (c(x n ))) n=1 12of 31

13 Cost function Recall: f (c(x)) is the reconstruction produced by the network Minimisation of the negative log likelihood of the reconstruction, given the encoding c(x) RE = log P(x c(x)) c If x c(x) is Gaussian, we recover the familiar squared error If the inputs x i are either binary or considered to be binomial probabilities, then the loss function would be the cross entropy log P(x c(x)) = x i log f i (c(x)) + (1 x i ) log(1 f i (c(x))) where f i ( ) is the i th component of the decoder 13of 31

14 Undercomplete c Consider a small number of hidden units. c(x) is viewed as a lossy compression of x Cannot have small loss for all x, so focus on training examples Hope code c(x) is a distributed representation that captures the main factors of variation in the data 14of 31

15 Stacking autoencoders Let c j and f j be the encoder and corresponding decoder of the j th layer Let z j be the representation after the encoder c j We can define multiple layers of autoencoders recursively. For example, let z 1 = c 1 (x), and z 2 = c 2 (z 1 ), the corresponding decoding is given by f 1 (f 2 (z 2 )) Because of non-linear activation functions, the latent feature z 2 can capture more complex patterns than z 1. c 15of 31

16 Higher level image features - faces c codingplayground.blogspot.com 16of 31

17 Pre-training supervised neural networks c Latent features z j in layer j can capture high level patterns z j = c j (c j 1 ( c 2 (c 1 (x)) )) These features may be useful for supervised learning tasks. In contrast to the feed forward network, the features z j are constructed in an unsupervised fashion. Discard the decoding layers, and directly use z j with a supervised training method, such as logistic regression. 17of 31

18 Higher dimensional hidden layer c if there is no other constraint, then an autoencoder with d-dimensional input and an encoding of dimension at least d could potentially just learn the identity function Perfect reconstruction: small input weights, large output weights Avoid by: Regularisation Early stopping of stochastic gradient descent Add noise in the encoding Sparsity constraint on code c(x). 18of 31

19 Denoising autoencoder c Add noise to input, keeping perfect example as output tries to: 1 preserve information of input 2 undo stochastic corruption process Reconstruction log likelihood log P(x c(ˆx)) where x noise free, ˆx corrupted 19of 31

20 Image denoising Images with Gaussian noise added. c I SML Denoised using Stacked Sparse Denoising Images from Xie et. al. NIPS of 31

21 Inpainting - 1 Free a bird c I SML Image from http: //cimg.eu/greycstoration/demonstration.shtml 21of 31

22 Inpainting - 2 Undo text over image c I SML Image from Bach et. al. ICCV tutorial of 31

23 Recall: Basis functions c For fixed basis functions φ(x), we use domain knowledge for encoding features Neural networks use data to learn a set of transformations. φ i (x) is the i th component of the feature vector, and is learned by the network. The transformations φ i ( ) for a particular dataset may no longer be orthogonal, and furthermore may be minor variations of each other. We collect all the transformed features into a matrix Φ. 23of 31

24 Sparse representations Idea: Have many hidden nodes, but only a few active for a particular code c(x). Student t prior on codes l 1 penalty on coefficients α Given bases in matrix Φ, look for codes by choosing α such that input signal x is reconstructed with low l 2 reconstruction error, while w is sparse c min α N n=1 1 2 xn Φαn λ α 1 Φ is overcomplete, no longer orthogonal Sparse small number of non-zero α i. Exact recovery under certain conditions (coherence): l 1 l 0. 24of 31

25 Sparsity - probabilistic view Approximate maximum likelihood training of generative model with latent variables α Joint distribution with observed variables x Recall sum rule: p(x, α) = p(α)p(x α) c log p(x) = log α p(x, α) approximates the sum with a single (highly likely) value of α. Given α, maximise log p(x, α) with the Laplace prior p(α i ) = λ 2 exp( λ α i ) 25of 31

26 The image denoising problem c y = x }{{} orig + }{{} }{{} w measurements original image noise 26of 31

27 Sparsity assumption Only have noisy measurements y = x }{{} orig + }{{} }{{} w measurements original image noise Given Φ R m p, find α such that c α 0 is small for x Φα where 0 is the number of non-zero elements of α. Φ is not necessarily features constructed from training data. Minimise reconstruction error min α N n=1 1 2 x n Φα n λ α 0 27of 31

28 Convex relaxation Want to minimise number of components min α N n=1 but 0 is hard to optimise Relax to a convex norm 1 2 x n Φα n λ α 0 c min α N n=1 where α 1 = n α n. 1 2 x n Φα n λ α 1 When does minimisation with l 1 regularisation give the same solution as minimisation with l 0 regularisation (exact recovery)? 28of 31

29 Mutual coherence c Expect to be ok when columns of Φ not too parallel Assume columns of Φ are normalised to unit norm Let K = ΦΦ T be the Gram matrix, then K(i, j) is the value of the inner product between φ i and φ j. Define the mutual coherence M = M(Φ) = max K(i, j) i j If we have an orthogonal basis, then Φ is an orthogonal matrix, hence K(i, j) = 0 when i j. However, if we have very similar columns, then M 1. 29of 31

30 Exact recovery conditions Consider the convex relaxation 1 min α 2 x i Φα i λ α 1 i If a solution of the true l 0 problem, α satisfies c α 0 < 1 M then it is the unique sparsest solution. If α satisfies the stronger condition α 0 < 1 ( ) 2 M the unique sparsest solution is also the minimum of the above optimisation. 30of 31

31 References c Yoshua Bengio, "Learning Deep Architectures for AI", Foundations and Trends in, autoencoders.html Fuchs, "On Sparse Representations in Arbitrary Redundant Bases", IEEE Trans. Info. Theory, of 31

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