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Autoencoders, Unsupervised Learning, and Deep Architectures P. Baldi University of California, Irvine

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1.General Definition 2.Historical Motivation (50s,80s,2010s) 3.Linear Autoencoders over Infinite Fields 4.Non-Linear Autoencoders: the Boolean Case 5.Summary and Speculations

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General Definition x 1,,x M training vectors in E N (e.g. E=IR or {0,1}) Learn A and B to minimize : i Δ[ F AB (x i )-x i ] B A H N N Key scaling parameters: N, H, M

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Autoencoder Zoo

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Historical Motivation Three time periods: 1950s, 1980s, 2010s. Three motivations: –Fundamental Learning Problem (1950s) –Unsupervised Learning (1980s) –Deep Architectures (2010s)

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2010: Deep Architectures

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1950s

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Where do you store your telephone number?

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THE SYNAPTIC BASIS OF MEMORY CONSOLIDATION © 2007, Paul De Koninck © 2004, Graham Johnson

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Scales Size in Metersx10 6 Diameter of Atom Hair Diameter of DNA Diameter of Synapse Fist Diameter of Axon Diameter of Neuron Room Length of Axon Park-Nation Length of Brain State Length of Body110 6 Nation

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The Organization of Behavior: A Neuropsychological Theory (1949) Let us assume that the persistence or repetition of a reverberatory activity (or “trace”) tends to induce lasting cellular changes that add to its stability…….When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process of metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased. Δw ij ~ x i x j

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1980s Hopfield PDP group

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Back-Propagation (1985)

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First Autoencoder x 1,,x M training points (real-valued vectors) Learn A and B to minimize i ||F AB (x i )-x i || 2 B A sigmoidal neurons H N N

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Linear Autoencoder x 1,…,x M training vectors over IR N Find two matrices A and B that minimize: i || AB(x i )-x i || 2 B A N N H

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Linear Autoencoder Theorem (IR) A and B are defined only up to group multiplication by an invertible HxH matrix C: W = AB = (AC -1) CB. Although the cost function is quadratic and the transformation W=AB is linear, the problem is NOT convex. The problem becomes convex if A or B is fixed. Assuming Σ XX is invertible and the covariance matrix has full rank : B*=(A t A) -1 A t and A*= Σ XX B t (B Σ XX Bt) -1. Alternate minimization of A and B is an EM algorithm. B A

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Linear Autoencoder Theorem (IR) The overall landscape of E has no local minima. All the critical points where the gradient is 0 are associated with projections onto subspaces associated with H eigenvectors of the covariance matrix. At any critical point: A=U I C and B=C -1 U I where the columns of U I are the H eigenvectors of Σ XX associated with the index set I. In this case, W = AB = P U I correspond to a projection. Generalization is easy to measure and understand. Projections onto the top H eigenvectors correspond to a global minimum. All other critical points are saddle points. B A H N N

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Landscape of E B A

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Linear Autoencoder Theorem (IR) Thus any critical point performs a form of clustering by hyperplane. For any vector x, all the vectors of the form x+KerB are mapped onto the same vector y=AB(x)=AB(x+ KerB). At any critical point where C=Identity A=B t. The constraint A=B t can be imposed during learning by weight sharing, or symmetric connections, and is consistent with a Hebbian rule that is symmetric between pre-and post- synaptic units (folded autoencoder, or clamping input and output units). B A N N H

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Linear Autoencoder Theorem (IR) At any critical point, reverberation is stable for every x (AB) 2 x=ABx The global minimum remains the same if additional matrices or rank >=H are introduced anywhere in the architecture. There is no gain in expressivity by adding such matrices. However such matrices could be introduced for other reasons. Vertical Composition law: “NH1HH1N ~NH1N + H1HH1” Results can be extended to linear case with given output targets and to the complex field. B A H N N

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Vertical Composition NH1HH1N ~ NH1N + H1HH1 H1 N H N N N N H N H

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Linear Autoencoder Theorem (IR) At any critical point, reverberation is stable (AB) 2 x=ABx The global minimum remains the same if additional matrices or rank >=H are introduced anywhere in the architecture. There is no gain in expressivity by adding such matrices. However such matrices could be introduced for other reasons. VerticalcComposition law: “NH1HH1N ~NH1N + H1HH1” Results can be extended to linear case with given output targets and to the complex field. Provides some intuition for the non-linear case. B A H N N

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Boolean Autoencoder

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x 1,…,x M training vectors over IH N (binary) Find Boolean functions A and B that minimize: i H[ AB(x i ),x i ] H= Hamming Distance Variation 1: Enforce AB(x i ) {x 1,…,x M } Variation 2: Restrict A and B (connectivity, threshold gates, etc)

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Boolean Autoencoder Fix A

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Boolean Autoencoder Fix A h=10010

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Boolean Autoencoder Fix A h=10010 y=A(h)=

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Boolean Autoencoder Fix A h=10010 y=A(h)= A(h1) A(h2) A(h3)

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Autoencoder Fix A h=10010 y=A(h)= B({Voronoi A(h)}) =h A(h1) A(h2) A(h3)

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Autoencoder Fix A h=10010 y=A(h)= B({Voronoi A(h)}) =h A(h1) A(h2) A(h3)

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Boolean Autoencoder Fix B

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Boolean Autoencoder Fix B h=10100 A

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Boolean Autoencoder Fix B h=10100 A A(h)=?

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Boolean Autoencoder Fix B h=10100 A A(h)=?

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Boolean Autoencoder Fix B h=10100 A A(h)=

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Boolean Autoencoder Fix B h=10100 A A(h)= A(h)=Majority[B -1 (h)]

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Boolean Autoencoder Theorem A and B are defined only up to the group of permutations of the 2H points in the H-dimensional hypercube of the hidden layer. The overal optimization problem is non trivial. Polynomial time solutions exist when H is held constant (centroids in the training set). When H~εLogN the problem becomes NP-complete. The problem has a simple solution when A is fixed or B is fixed: A*(h)=Majority {B -1 (h)} B*{Voronoi A(h)}=h [B*(x)=h such that A(h) is closest to x among {A(h)}]. Every “critical point” (A* and B*) correspond to a clustering into K=2 H clusters. The optimum correspond to the best clustering. (Maximum?) Plenty of approximate algorithms (k means, hierarchical clustering, belief propagation (centroids in training set). Generalization is easy to measure and understand.

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Boolean Autoencoder Theorem At any critical point, reverberation is stable. The global minimum remains the same if additional Boolean functions with layers >=H are introduced anywhere in the architecture. There is no gain in expressivity by adding such functions. However such functions could be introduced for other reasons. Composition law: “NH1HH1N ~NH1N + H1HH1”. Can achieve hierarchical clustering in input space. Results can be extended to the case with given output targets.

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Learning Complexity Linear autoencoder over infinite fields can be solved analytically Boolean autoencoder is NP complete as soon as the number of clusters (K=2 H ) scales like M ε (for ε>0). It is solvable in polynomial time when K is fixed. Linear autoencoder over finite fields is NP complete in the general case. RBM learning is NP complete in the general case.

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Embedding of Square Lattice in Hypercube 4x3 square lattice with embedding in H

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Vertical Composition

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Horizontal Composition

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Autoencoders with H>N Identity provides trivial solution Regularization//Horizontal Composition//Noise

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Information and Coding (Transmission and Storage) message parity bits noisy channel decoded message

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Summary and Speculations

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Unsupervised Learning Autoencoders Clustering Hebbian Learning

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Information and Coding Theory Autoencoders Compression Communication

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Deep Architectures Autoencoders Vertical Composition Horizontal Composition

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Summary and Speculations Unsupervised Learning: Hebb, Autoencoders, RBMs, Clustering Conceptually clustering is the fundamental operation Clustering can be combined with targets Clustering is composable: horizontally, vertically, recursively, etc. Autoencoders implement clustering and labeling simultaneously Deep architecture conjecture

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