Autoencoders, Unsupervised Learning, and Deep

Architectures

P. Baldi

University of California, Irvine

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

General Definition • x1, ,xM training vectors in EN (e.g. E=IR or {0,1})

• Learn A and B to minimize: i Δ[ FAB(xi)-xi]

B

A

H

N

N

Key scaling parameters: N, H, M

Autoencoder Zoo

Autoencoders

Linear Non-Linear

Complex Real Finite Fields (GF2) BooleanNeural Network

(sigmoidal)Boltzmann Machines

RBMsThreshold Gates

Boolean/Linear

Historical Motivation

• Three time periods: 1950s, 1980s, 2010s.

• Three motivations: – Fundamental Learning Problem (1950s)– Unsupervised Learning (1980s)– Deep Architectures (2010s)

2010: Deep Architectures

1950s

Where do you store your telephone number?

THE SYNAPTIC BASIS OF MEMORY CONSOLIDATION

© 2007, Paul De Koninck© 2004, Graham Johnson

Scales

Size in Meters x106

Diameter of Atom 10-10 10-4 Hair

Diameter of DNA 10-9 10-3

Diameter of Synapse

10-7 10-1 Fist

Diameter of Axon 10-6 1

Diameter of Neuron

10-5 10 Room

Length of Axon 10-3-100 103-106 Park-Nation

Length of Brain 10-1 105 State

Length of Body 1 106 Nation

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.

Δwij ~ xixj

1980s

• Hopfield

• PDP group

Back-Propagation (1985)BACK-PROPAGATION

ERROR E=F(w)

i

W ij

j

OUTPUT LAYER

INPUT LAYER

GRADIENT DESCENT: Δ w ij = µ outj єi

µ = learning rate

First Autoencoder • x1, ,xM training points (real-valued vectors)

• Learn A and B to minimize i ||FAB(xi)-xi||2

B

A

sigmoidal neurons

sigmoidal neurons

H

N

N

Linear Autoencoder

• x1,…,xM training vectors over IRN

• Find two matrices A and B that minimize:

i ||AB(xi)-xi||2

B

A

N

N

H

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 ΣXX is invertible and the covariance matrix has full rank : B*=(AtA)-1At and A*= ΣXX Bt(B ΣXX Bt)-1.

• Alternate minimization of A and B is an EM algorithm.

B

A

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=UI C and B=C-1UI where the columns of UI are the H eigenvectors of ΣXX associated with the index set I. In this case, W = AB = PUI

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

Landscape of E

B

A

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=Bt. The constraint A=Bt 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

Linear Autoencoder Theorem (IR)• At any critical point, reverberation is stable for every x

(AB)2x=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

Vertical Composition

• NH1HH1N ~ NH1N + H1HH1

H1

N

H1

H

H1

NN

H1

H1

N

H

N

H1

H

Linear Autoencoder Theorem (IR)• At any critical point, reverberation is stable (AB)2x=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

Boolean Autoencoder

Boolean Autoencoder

• x1,…,xM training vectors over IHN (binary)

• Find Boolean functions A and B that minimize:

i H[AB(xi),xi]

H= Hamming Distance

• Variation 1: Enforce AB(xi) {x1,…,xM}

• Variation 2: Restrict A and B (connectivity, threshold gates, etc)

Boolean Autoencoder

Fix A

Boolean Autoencoder

Fix A

h=10010

Boolean Autoencoder

Fix A

h=10010

y=A(h)=11010110010

Boolean Autoencoder

Fix A

h=10010

y=A(h)=11010110010

A(h1)

A(h2)

A(h3)

Autoencoder

Fix A

h=10010

y=A(h)=11010110010

B({Voronoi A(h)}) =h

A(h1)

A(h2)

A(h3)

Autoencoder

Fix A

h=10010

y=A(h)=11010110010

B({Voronoi A(h)}) =h

A(h1)

A(h2)

A(h3)

Boolean Autoencoder

Fix B

Boolean Autoencoder

Fix B

h=10100

A

Boolean Autoencoder

Fix B

h=10100

A

A(h)=?

Boolean Autoencoder

Fix B

00110101001

11010100101

10101010101

h=10100

A

A(h)=?

Boolean Autoencoder

Fix B

00110101001

11010100101

10101010101

h=10100

A

A(h)=10110100101

Boolean Autoencoder

Fix B

00110101001

11010100101

10101010101

h=10100

A

A(h)=10110100101 A(h)=Majority[B-1(h)]

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=2H 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.

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.

Learning Complexity• Linear autoencoder over infinite fields can be

solved analytically• Boolean autoencoder is NP complete as soon

as the number of clusters (K=2H) scales like Mε (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.

Embedding of Square Lattice in Hypercube

• 4x3 square lattice with embedding in H7

0000000

1111111

1111000

0000111

Vertical Composition

Horizontal Composition

Autoencoders with H>N

• Identity provides trivial solution• Regularization//Horizontal Composition//Noise

Information and Coding (Transmission and Storage)

message

parity bits

noisy channel

decoded message

Summary and Speculations

Autoencoders

Linear Non-Linear

Complex Real Finite Fields (GF(2)) BooleanNeural Network

(sigmoidal)Boltzmann Machines

RBMsBoolean/Linear

over GF(2)

Boolean/Linear over R or C

Threshold Gates

Unsupervised Learning

Autoencoders

Clustering

Hebbian Learning

Information and Coding Theory

Autoencoders

Compression

Communication

Deep Architectures

Autoencoders

Vertical Composition

HorizontalComposition

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