Tensor methods applied in Artificial Intelligence and data analysis.

Tensor methods applied in Artificial Intelligenceand data analysis.

Kim Batselier

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Introduction

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Introduction

Data-driven technology

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Introduction

The nature of data

Data comes in many forms:

wordsnumbersimagessound....

Organized in multi-dimensional arrays = tensors

My research is about developing data-driven technology withtensors

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Introduction

The nature of data

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Introduction

The nature of data

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Introduction

321× 481 image, 95% missing.

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Introduction

What is an image?

An array I of numbers

Each pixel is uniquely determined by its location and its color.

The location of each pixel is determined by 2 numbers(height, width)

The color of each pixel is determined by 3 numbers (redintensity, green intensity, blue intensity)

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Introduction

What is an image?

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Introduction

What is an image?

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Introduction

An image as a tensor

All location and color information of a whole image can bestored into a 3-dimensional array

Image I of height H and width W ⇒ I ∈ RH×W×3

Small example

For an image I of size 321× 481, we have

I(:, :, 1) ∈ R321×481 = Red intensities of all pixels.

I(:, :, 2) ∈ R321×481 = Green intensities of all pixels.

I(:, :, 3) ∈ R321×481 = Blue intensities of all pixels.

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Introduction

Problem statement

Given an image I and a partially observed instance X = (I)Ω.Can we use the observed X to fill-in the missing parts?

Solution strategy

Formulate as a optimization problem. Find the tensor A thatminimizes ||(A− I)Ω||

Problem: We don’t know I! So instead we find a tensor Athat minimizes ||(A−X )Ω||This is an ill-posed problem: the number of possible solutionsis infinite

We want to add the additional constraint that neighbouringpixels of A should be interrelated

⇒ How to introduce thisconstraint? Answer: Tensor Networks

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Introduction

Problem statement

Solution strategy

Formulate as a optimization problem. Find the tensor A thatminimizes ||(A− I)Ω||

Problem: We don’t know I! So instead we find a tensor Athat minimizes ||(A−X )Ω||This is an ill-posed problem: the number of possible solutionsis infinite

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Introduction

Problem statement

Solution strategy

Formulate as a optimization problem. Find the tensor A thatminimizes ||(A− I)Ω||Problem: We don’t know I! So instead we find a tensor Athat minimizes ||(A−X )Ω||

This is an ill-posed problem: the number of possible solutionsis infinite

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Introduction

Problem statement

Solution strategy

Formulate as a optimization problem. Find the tensor A thatminimizes ||(A− I)Ω||Problem: We don’t know I! So instead we find a tensor Athat minimizes ||(A−X )Ω||This is an ill-posed problem: the number of possible solutionsis infinite

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Introduction

Problem statement

Solution strategy

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Introduction

Problem statement

Solution strategy

We want to add the additional constraint that neighbouringpixels of A should be interrelated ⇒ How to introduce thisconstraint?

Answer: Tensor Networks

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Introduction

Problem statement

Solution strategy

We want to add the additional constraint that neighbouringpixels of A should be interrelated ⇒ How to introduce thisconstraint? Answer: Tensor Networks

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Tensor networks: a gentle introduction

Tensors

Multi-dimensional arrays are also called tensorsA ∈ RI1×I2×I3×···×Id

d = the order of the tensor/ the number of modes.

I1, I2, . . . , Id = the dimensions of the tensor.

Some familiar examples

d = 1⇒

a vector, e.g. a ∈ R5

d = 2⇒

a matrix, e.g. A ∈ R10×3

d = 0⇒

a scalar/a number, e.g. a ∈ R1

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Tensors

d = 1⇒

d = 2⇒

d = 0⇒

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Tensors

d = 1⇒

d = 2⇒

d = 0⇒

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Tensors

d = 1⇒

d = 2⇒

d = 0⇒

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Tensors

d = 1⇒

d = 2⇒

d = 0⇒

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Tensors

d = 1⇒ a vector, e.g. a ∈ R5

d = 2⇒

d = 0⇒

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Tensors

d = 2⇒

d = 0⇒

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Tensors

d = 2⇒ a matrix, e.g. A ∈ R10×3

d = 0⇒

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Tensors

d = 2⇒ a matrix, e.g. A ∈ R10×3

d = 0⇒

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Tensors

d = 2⇒ a matrix, e.g. A ∈ R10×3

d = 0⇒ a scalar/a number, e.g. a ∈ R1

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Tensor network building blocks

Aa a A

Tensor networks

C(i, j) =∑

k A(i, k) B(k, j)

is visually represented by

A Bi k j

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Aa a A

Tensor networks

C(i, j) =∑

k A(i, k) B(k, j)

is visually represented by

A Bi k j

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Aa a A

Tensor networks

C(i, j) =∑

k A(i, k) B(k, j) is visually represented by

A Bi k j

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Tensor networks

c =∑

k a(k) b(k) = aT b

Tensor networks

D(i, j) =∑

k,l A(k, j, l) B(i, k) c(l)

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Tensor networks

c =∑

k a(k) b(k) = aT b

Tensor networks

D(i, j) =∑

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Tensor networks

c =∑

k a(k) b(k) = aT b

Tensor networks

D(i, j) =∑

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Tensor networks

c =∑

k a(k) b(k) = aT b

Tensor networks

D(i, j) =∑

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The total number of elements in a tensor of order d anddimensions I1 = I2 = · · · = Id = I is Id.

⇒ This is known as the curse of dimensionality

If the tensor entries are interrelated, then the number of degrees offreedom can be drastically reduced via tensor decompositions.

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Small 4× 3 example

How many degrees of freedom does this matrix have?54 −30 −18−45 25 1581 −45 −279 −5 −3

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(9 −5 −3)

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Small 4× 3 example

How many degrees of freedom does this matrix have?54 −30 −18−45 25 1581 −45 −279 −5 −3

6−591

(9 −5 −3)

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Small 4× 3 example

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(9 −5 −3)= a bT

C(i, j) =1∑

a(i, k) b(k, j)

Take-away message

Tensor networks with small interconnection dimensions haveinterrelated entries.

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Small 4× 3 example

6−591

(9 −5 −3)= a bT

C(i, j) =

1∑k=1

a(i, k) b(k, j)

Take-away message

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Small 4× 3 example

6−591

(9 −5 −3)= a bT

C(i, j) =

1∑k=1

a(i, k) b(k, j)

Take-away message

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Small 4× 3 example

6−591

(9 −5 −3)= a bT

C(i, j) =

1∑k=1

a(i, k) b(k, j)

Take-away message

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Conclusion

Problem statement

Solution strategy

We want to add the additional constraint that neighbouringpixels of A should be interrelated ⇒ How to introduce thisconstraint? Answer: Tensor Networks

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Conclusion

For our 321× 481 image:

321 481 3321

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Conclusion

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Conclusion

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Conclusion

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Conclusion

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Final words :)

My research

Compressing data Learning nonlinearclassifiers from data

Estimation ofmodels from data

Converting data intotensor networks

web: https://sites.google.com/view/kim-batselier/home

e-mail: k dφτ batselier ατ tudelft dφτ nl

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Tensor methods applied in Artificial Intelligence and data analysis. · 2018. 11. 23. · Tensor...

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